UT Testing-Section 2 Physics of Ultrasound

Section 2: Physics of Ultrasound

Content: Section 2: Physics of Ultrasound 

2.0: Ultrasound Formula 

2.1: Wave Propagation 

2.2: Modes of Sound Wave Propagation 

2.3: Sound Propagation in Elastic Materials 

2.4: Properties of Acoustic Plane Wave 

2.5: Wavelength and Defect Detection 

2.6: Attenuation of Sound Waves 

2.7: Acoustic Impedance 

2.8: Reflection and Transmission Coefficients (Pressure) 

2.9: Refraction and Snell's Law 

2.10: Mode Conversion 

2.11: Signal-to-Noise Ratio 

2.12: The Sound Fields- Dead / Fresnel & Fraunhofer Zones 

2.13: Inverse Square Rule/ Inverse Rule 

2.14: Resonance 

2.15 Measurement of Sound 

2.16 Practice Makes Perfect

2.0: Ultrasound Formula 

http://www.ndt-ed.org/GeneralResources/Calculator/calculator.htm

Ultrasonic Formula

Ultrasonic Formula

Parameters of Ultrasonic Waves

2.1: Wave Propagation 

Ultrasonic testing is based on time-varying deformations or vibrations in 

materials, which is generally referred to as acoustics. All material substances 

are comprised of atoms, which may be forced into vibration motion about their 

equilibrium positions. Many different patterns of vibration motion exist at the 

atomic level, however, most are irrelevant to acoustics and ultrasonic testing. 

Acoustics is focused on particles that contain many atoms that move in 

unison to produce a mechanical wave. When a material is not stressed in 

tension or compression beyond its elastic limit, its individual particles perform 

elastic oscillations. When the particles of a medium are displaced from their 

equilibrium positions, internal (electrostatic) restoration forces arise. It is these 

elastic restoring forces between particles, combined with inertia of the 

particles, that leads to the oscillatory motions of the medium. 

Keywords: 

■ internal (electrostatic) restoration forces 

■ inertia of the particles

Acoustic Spectrum

Acoustic Spectrum

Acoustic Spectrum

Acoustic Wave – Node and Anti-Node 

The points where the two waves constantly cancel each other are called 

nodes, and the points of maximum amplitude between them, antinodes. 

http://www.physicsclassroom.com/Class/waves/u10l4c.cfm 

http://www.physicsclassroom.com/Class/waves/h4.gif

Acoustic Wave – Node and Anti-Node 

Formation of a standing wave by two waves from opposite directions

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html

Q151 A point, line or surface of a vibration body marked by absolute or 

relative freedom from vibratory motion (momentarily?) is referred to as: 

a) a node 

b) an antinode 

c) rarefaction 

d) compression

2.2: Modes of Sound Wave Propagation 

2.2.1 Modes of Ultrasound 

In solids, sound waves can propagate in four principle modes that are based 

on the way the particles oscillate. Sound can propagate as; 

• longitudinal waves, 

• shear waves, 

• surface waves, 

• and in thin materials as plate waves. 

Longitudinal and shear waves are the two modes of propagation most widely 

used in ultrasonic testing. The particle movement responsible for the 

propagation of longitudinal and shear waves is illustrated below.

2.2.2 Propagation & Polarization Vectors 

• Propagation Vector- The direction of wave propagation 

• Polarization Vector- The direction of particle motion

Longitudinal and shear waves

Longitudinal and shear waves- Defined the Vectors



2.2.3 Longitudinal Wave 

Also Knows as: 

• longitudinal waves, 

• pressure wave 

• compressional waves. 

• density waves 

can be generated in (1) liquids, as well as (2) solids 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Graphics/Flash/longitudinal.swf

In longitudinal waves, the oscillations occur in the longitudinal direction or the 

direction of wave propagation. Since compressional and dilational forces are 

active in these waves, they are also called pressure or compressional waves. 

They are also sometimes called density waves because their particle density 

fluctuates as they move. Compression waves can be generated in liquids, as 

well as solids because the energy travels through the atomic structure by a 

series of compressions and expansion (rarefaction) movements.

Longitudinal wave: Longitudinal waves (L-Waves) compress and decompress 

the material in the direction of motion, much like sound waves in air.

Longitudinal Wave

2.2.4 Shear waves (S-Waves) 

In air, sound travels by the compression and rarefaction of air molecules in 

the direction of travel. However, in solids, molecules can support vibrations in 

other directions, hence, a number of different types of sound waves are 

possible. Waves can be characterized in space by oscillatory patterns that 

are capable of maintaining their shape and propagating in a stable 

manner. The propagation of waves is often described in terms of what are 

called “wave modes.” 

As mentioned previously, longitudinal and transverse (shear) waves are most 

often used in ultrasonic inspection. However, at surfaces and interfaces, 

various types of elliptical or complex vibrations of the particles make other 

waves possible. Some of these wave modes such as (1) Rayleigh and (2) 

Lamb waves are also useful for ultrasonic inspection. 

Keywords: 

Compression 

Rarefaction

Shear waves vibrate particles at right angles compared to the motion of the 

ultrasonic wave. The velocity of shear waves through a material is 

approximately half that of the longitudinal waves. The angle in which the 

ultrasonic wave enters the material determines whether longitudinal, shear, or 

both waves are produced.

Shear waves

In the transverse or shear wave, the particles oscillate at a right angle or 

transverse to the direction of propagation. Shear waves require an 

acoustically solid material for effective propagation, and therefore, are not 

effectively propagated in materials such as liquids or gasses. Shear waves 

are relatively weak when compared to longitudinal waves. In fact, shear 

waves are usually generated in materials using some of the energy from 

longitudinal waves. 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Graphics/Flash/transverse.swf

Q10: For a shear wave travelling from steel to water incident on the boundary 

at 10 degrees will give a refracted shear wave in water with an angle of: 

A. 0 degrees 

B. 5 degrees 

C. 20 degrees 

D. none of the above

2.2.5 Rayleigh Characteristics 

Rayleigh waves are a type of surface wave that travel near the surface of 

solids. Rayleigh waves include both longitudinal and transverse motions that 

decrease exponentially in amplitude as distance from the surface increases. 

There is a phase difference between these component motions. In isotropic 

solids these waves cause the surface particles to move in ellipses in planes 

normal to the surface and parallel to the direction of propagation – the major 

axis of the ellipse is vertical. At the surface and at shallow depths this motion 

is retrograde 逆行 , that is the in-plane motion of a particle is counterclockwise 

when the wave travels from left to right. 

http://en.wikipedia.org/wiki/Rayleigh_wave

Rayleigh waves are a type of surface acoustic wave that travel on solids. 

They can be produced in materials in many ways, such as by a localized 

impact or by piezo-electric transduction, and are frequently used in nondestructive 

testing for detecting defects. They are part of the seismic waves 

that are produced on the Earth by earthquakes. When guided in layers they 

are referred to as Lamb waves, Rayleigh–Lamb waves, or generalized 

Rayleigh waves.

Rayleigh waves

Q29: The longitudinal wave incident angle which results in formation of a 

Rayleigh wave is called: 

A. Normal incidence 

B. The first critical angle 

C. The second critical angle 

D. Any angle above the first critical angle

Surface (or Rayleigh) waves travel the surface of a relatively thick solid 

material penetrating to a depth of one wavelength. 

Surface waves combine both (1) a longitudinal and (2) transverse motion to 

create an elliptic orbit motion as shown in the image and animation below. 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Graphics/Flash/rayleigh.swf

The major axis of the ellipse is perpendicular to the surface of the solid. As 

the depth of an individual atom from the surface increases the width of its 

elliptical motion decreases. Surface waves are generated when a 

longitudinal wave intersects a surface near the second critical angle and 

they travel at a velocity between .87 and .95 of a shear wave. Rayleigh 

waves are useful because they are very sensitive to surface defects (and 

other surface features) and they follow the surface around curves. 

Because of this, Rayleigh waves can be used to inspect areas that other 

waves might have difficulty reaching. 

Wave velocity: 

• Longitudinal wave velocity =1v, 

• The velocity of shear waves through a material is approximately half that 

of the longitudinal waves, (≈0.5v) 

• Surface waves are generated when a longitudinal wave intersects a 

surface near the second critical angle and they travel at a velocity 

between .87 and .95 of a shear wave. ≈(0.87~0.95)x0.5v

The major axis of the ellipse is perpendicular to the surface of the solid.

Surface wave

Surface wave or Rayleigh wave are formed when shear waves refract to 90. 

The whip-like particle vibration of the shear wave is converted into elliptical 

motion by the particle changing direction at the interface with the surface. The 

wave are not often used in industrial NDT although they do have some 

application in aerospace industry. Their mode of propagation is elliptical along 

the surface of material, penetrating to a depth of one wavelength. They will 

follow the contour of the surface and they travel at approximately 90% of the 

velocity of the shear waves. 

Depth of penetration of 

about one wavelength 

Direction of wave propagation

Surface wave has the ability to follow surface contour, until it meet a sharp 

change i.e. a surface crack/seam/lap. However the surface waves could be 

easily completely absorbed by excess couplant of simply touching the part 

ahead of the waves. 

Transducer 

Wedge 

Surface discontinuity 

Specimen

Surface wave - Following Contour 

Surface wave

Surface wave – One wavelength deep 

λ 

λ

Rayleigh Wave 

http://web.ics.purdue.edu/~braile/edumod/waves/Rwave_files/image001.gif

Love Wave 

http://web.ics.purdue.edu/~braile/edumod/waves/Lwave_files/image001.gif

Love Wave

Other Reading: Rayleigh Waves 

Surface waves (Rayleigh waves) are another type of ultrasonic wave used in 

the inspection of materials. These waves travel along the flat or curved 

surface of relatively thick solid parts. For the propagation of waves of this type, 

the waves must be traveling along an interface bounded on one side by the 

strong elastic forces of a solid and on the other side by the practically 

negligible elastic forces between gas molecules. Surface waves leak energy 

into liquid couplants and do not exist for any significant distance along the 

surface of a solid immersed in a liquid, unless the liquid covers the solid 

surface only as a very thin film. Surface waves are subject to attenuation in a 

given material, as are longitudinal or transverse waves. They have a velocity 

approximately 90% of the transverse wave velocity in the same material. The 

region within which these waves propagate with effective energy is not much 

thicker than about one wavelength beneath the surface of the metal.

At this depth, wave energy is about 4% of the wave energy at the surface, 

and the amplitude of oscillation decreases sharply to a negligible value at 

greater depths. Surface waves follow contoured surfaces. For example, 

surface waves traveling on the top surface of a metal block are reflected from 

a sharp edge, but if the edge is rounded off, the waves continue down the 

side face and are reflected at the lower edge, returning to the sending point. 

Surface waves will travel completely around a cube if all edges of the cube 

are rounded off. Surface waves can be used to inspect parts that have 

complex contours.

Q110: What kind of wave mode travel at a velocity slightly below the shear 

wave and their modes of propagation are both longitudinal and transverse 

with respect to the surface? 

a) Rayleigh wave 

b) Transverse wave 

c) L-wave 

d) Longitudinal wave

Q: Which of the following modes of vibration exhibits the shortest wavelength 

at a given frequency and in a given material? 

A. longitudinal wave 

B. compression wave 

C. shear wave 

D. surface wave

2.2.6 Lamb Wave: 

Lamb waves propagate in solid plates. They are elastic waves whose 

particle motion lies in the plane that contains the direction of wave 

propagation and the plate normal (the direction perpendicular to the plate). In 

1917, the english mathematician horace lamb published his classic analysis 

and description of acoustic waves of this type. Their properties turned out to 

be quite complex. An infinite medium supports just two wave modes traveling 

at unique velocities; but plates support two infinite sets of lamb wave modes, 

whose velocities depend on the relationship between wavelength and plate 

thickness.

Since the 1990s, the understanding and utilization of lamb waves has 

advanced greatly, thanks to the rapid increase in the availability of computing 

power. Lamb's theoretical formulations have found substantial practical 

application, especially in the field of nondestructive testing. 

The term rayleigh–lamb waves embraces the rayleigh wave, a type of wave 

that propagates along a single surface. Both rayleigh and lamb waves are 

constrained by the elastic properties of the surface(s) that guide them. 

http://en.wikipedia.org/wiki/Lamb_wave 

http://pediaview.com/openpedia/Lamb_waves

Types of Wave 

New! 

• Plate wave- Love 

• Stoneley wave 

• Sezawa

Plate or Lamb waves are the most commonly used plate waves in 

NDT. Lamb waves are complex vibrational waves that propagate parallel to 

the test surface throughout the thickness of the material. Propagation of Lamb 

waves depends on the density and the elastic material properties of a 

component. They are also influenced a great deal by the test frequency and 

material thickness. Lamb waves are generated at an incident angle in which 

the parallel component of the velocity of the wave in the source is equal to the 

velocity of the wave in the test material. Lamb waves will travel several 

meters in steel and so are useful to scan plate, wire, and tubes. 

Lamb wave influenced by: (Dispersive Wave) 

■ 

■ 

■ 

■ 

Density 

Elastic material properties 

Frequencies 

Material thickness

Plate or Lamb waves are similar to surface waves except they can only be 

generated in materials a few wavelengths thick. 

http://www.ndt.net/ndtaz/files/lamb_a.gif

Plate wave or Lamb wave are formed by the introduction of surface wave 

into a thin material. They are a combination of (1) compression and surface or 

(2) shear and surface waves causing the plate material to flex by totally 

saturating the material. The two types of plate waves:

With Lamb waves, a number of modes of particle vibration are possible, but 

the two most common are symmetrical and asymmetrical. The complex 

motion of the particles is similar to the elliptical orbits for surface 

waves. Symmetrical Lamb waves move in a symmetrical fashion about the 

median plane of the plate. This is sometimes called the extensional mode 

because the wave is “stretching and compressing” the plate in the wave 

motion direction. Wave motion in the symmetrical mode is most efficiently 

produced when the exciting force is parallel to the plate. The asymmetrical 

Lamb wave mode is often called the “flexural mode” because a large portion 

of the motion moves in a normal direction to the plate, and a little motion 

occurs in the direction parallel to the plate. In this mode, the body of the plate 

bends as the two surfaces move in the same direction. 

The generation of waves using both piezoelectric transducers and 

electromagnetic acoustic transducers (EMATs) are discussed in later sections. 

Keywords: 

Symmetrical = extensional mode 

Asymmetrical = flexural mode

When guided in layers they are referred to as Lamb waves, Rayleigh–Lamb 

waves, or generalized Rayleigh waves. 

Lamb waves – 2 modes





Symmetrical = extensional mode

Other Reading: Lamb Wave 

Lamb waves, also known as plate waves, are another type of ultrasonic wave 

used in the nondestructive inspection of materials. Lamb waves are 

propagated in plates (made of composites or metals) only a few wavelengths 

thick. A Lamb wave consists of a complex vibration that occurs throughout the 

thickness of the material. The propagation characteristics of Lamb waves 

depend on the density, elastic properties, and structure of the material as well 

as the thickness of the test piece and the frequency. Their behavior in general 

resembles that observed in the transmission of electromagnetic waves 

through waveguides. 

There are two basic forms of Lamb waves: 

• Symmetrical, or dilatational 

• Asymmetrical, or bending

The form is determined by whether the particle motion is symmetrical or 

asymmetrical with respect to the neutral axis of the test piece. Each form is 

further subdivided into several modes having different velocities, which can 

be controlled by the angle at which the waves enter the test piece. 

Theoretically, there are an infinite number of specific velocities at which Lamb 

waves can travel in a given material. Within a given plate, the specific 

velocities for Lamb waves are complex functions of plate thickness and 

frequency. 

In symmetrical (dilatational) Lamb waves, there is a compressional 

(longitudinal) particle displacement along the neutral axis of the plate and an 

elliptical particle displacement on each surface (Fig. 4a). In asymmetrical 

(bending) Lamb waves, there is a shear (transverse) particle displacement 

along the neutral axis of the plate and an elliptical particle displacement on 

each surface (Fig. 4b). The ratio of the major to minor axes of the ellipse is a 

function of the material in which the wave is being propagated.

Fig. 4 Diagram of the basic patterns of (a) symmetrical (dilatational) and (b) 

asymmetrical (bending) Lamb waves. The wavelength, , is the distance 

corresponding to one complete cycle.

Q1: The wave mode that has multiple or varying wave velocities is: 

A. Longitudinal waves 

B. Shear waves 

C. Transverse waves 

D. Lamb waves

2.2.7 Dispersive Wave: 

Wave modes such as those found in Lamb wave have a velocity of 

propagation dependent upon the operating frequency, sample thickness and 

elastic moduli. They are dispersive (velocity change with frequency) in that 

pulses transmitted in these mode tend to become stretched or dispersed.

Dispersion refers to the fact that in a real medium such as water, air, or glass, 

a wave traveling through that medium will have a velocity that depends upon 

its frequency. Dispersion occurs for any form of wave, acoustic, 

electromagnetic, electronic, even quantum mechanical. Dispersion is 

responsible for a prism being able to resolve light into colors and defines the 

maximum frequency of broadband pulses one can send down an optical fiber 

or through a copper wire. Dispersion affects wave and swell forecasts at 

sea and influences the design of sound equipment. Dispersion is a physical 

property of the medium and can combine with other properties to yield very 

strange results. For example, in the propagation of light in an optical fiber, the 

glass introduces dispersion and separates the wavelengths of light according 

to frequency, however if the light is intense enough, it can interact with the 

electrons in the material changing its refractive index. The combination of 

dispersion and index change can cancel each other leading to a wave that 

can propagate indefinitely maintaining a constant shape. Such a wave has 

been termed a soliton. 

http://www.rpi.edu/dept/chem-eng/WWW/faculty/plawsky/Comsol%20Modules/DispersiveWave/DispersiveWave.html

Plate or Lamb waves are generated at an incident angle in which the parallel 

component of the velocity of the wave in the source is equal to the velocity of 

the wave in the test material.

Thickness Limitation: 

One can not generate shear / surface (or Lamb?) wave on a plate that is 

thinner than ½ the wavelength.

2.3: Sound Propagation in Elastic Materials 

In the previous pages, it was pointed out that sound waves propagate due to 

the vibrations or oscillatory motions of particles within a material. An 

ultrasonic wave may be visualized as an infinite number of oscillating masses 

or particles connected by means of elastic springs. Each individual particle is 

influenced by the motion of its nearest neighbor and both (1) inertial and (2) 

elastic restoring forces act upon each particle. 

A mass on a spring has a single resonant frequency determined by its spring 

constant k and its mass m. The spring constant is the restoring force of a 

spring per unit of length. Within the elastic limit of any material, there is a 

linear relationship between the displacement of a particle and the force 

attempting to restore the particle to its equilibrium position. This linear 

dependency is described by Hooke's Law.

Spring model- A mass on a spring has a single resonant frequency 

determined by its spring constant k and its mass m.

Spring model- A mass on a spring has a single resonant frequency 

determined by its spring constant k and its mass m.

In terms of the spring model, Hooke's Law says that the restoring force due to 

a spring is proportional to the length that the spring is stretched, and acts in 

the opposite direction. Mathematically, Hooke's Law is written as F =-kx, 

where F is the force, k is the spring constant, and x is the amount of particle 

displacement. Hooke's law is represented graphically it the bottom. Please 

note that the spring is applying a force to the particle that is equal and 

opposite to the force pulling down on the particle.

Elastic Model

Elastic Model / Longitudinal Wave

Elastic Model / Longitudinal Wave

Elastic Model / Shear Wave

Elastic Model / Shear Wave

The Speed of Sound 

Hooke's Law, when used along with Newton's Second Law, can explain a few 

things about the speed of sound. The speed of sound within a material is a 

function of the properties of the material and is independent of the amplitude 

of the sound wave. Newton's Second Law says that the force applied to a 

particle will be balanced by the particle's mass and the acceleration of the 

particle. Mathematically, Newton's Second Law is written as F = ma. Hooke's 

Law then says that this force will be balanced by a force in the opposite 

direction that is dependent on the amount of displacement and the spring 

constant (F = -kx). Therefore, since the applied force and the restoring force 

are equal, ma = -kx can be written. The negative sign indicates that the force 

is in the opposite direction. 

F= ma = -kx

Since the mass m and the spring constant k are constants for any given 

material, it can be seen that the acceleration a and the displacement x are the 

only variables. It can also be seen that they are directly proportional. For 

instance, if the displacement of the particle increases, so does its acceleration. 

It turns out that the time that it takes a particle to move and return to its 

equilibrium position is independent of the force applied. So, within a given 

material, sound always travels at the same speed no matter how much force 

is applied when other variables, such as temperature, are held constant. 

a ∝ x

What properties of material affect its speed of sound? 

Of course, sound does travel at different speeds in different materials. This is 

because the (1) mass of the atomic particles and the (2) spring constants are 

different for different materials. The mass of the particles is related to the 

density of the material, and the spring constant is related to the elastic 

constants of a material. The general relationship between the speed of sound 

in a solid and its density and elastic constants is given by the following 

equation:

Elastic constant 

→ spring constants 

Density 

→ mass of the atomic particles

Where V is the speed of sound, C is the elastic constant, and p is the material 

density. This equation may take a number of different forms depending on the 

type of wave (longitudinal or shear) and which of the elastic constants that are 

used. The typical elastic constants of a materials include: 

• Young's Modulus, E: a proportionality constant between uniaxial stress 

and strain. 

• Poisson's Ratio, n: the ratio of radial strain to axial strain 

• Bulk modulus, K: a measure of the incompressibility of a body subjected to 

hydrostatic pressure. 

• Shear Modulus, G: also called rigidity, a measure of a substance's 

resistance to shear. 

• Lame's Constants, l and m: material constants that are derived from 

Young's Modulus and Poisson's Ratio.

Q163 Acoustic velocity of materials are primary due to the material's: 

a) density 

b) elasticity 

c) both a and b 

d) acoustic impedance

Q50: The principle attributes that determine the differences in ultrasonic 

velocities among materials are: 

A. Frequency and wavelength 

B. Thickness and travel time 

C. Elasticity and density 

D. Chemistry and permeability

When calculating the velocity of a longitudinal wave, Young's Modulus and 

Poisson's Ratio are commonly used. 

When calculating the velocity of a shear wave, the shear modulus is used. It 

is often most convenient to make the calculations using 

Lame's Constants, which are derived from Young's Modulus and Poisson's 

Ratio.

E/N/G

It must also be mentioned that the subscript ij attached to C (C ij ) in the above 

equation is used to indicate the directionality of the elastic constants with 

respect to the wave type and direction of wave travel. In isotropic materials, 

the elastic constants are the same for all directions within the material. 

However, most materials are anisotropic and the elastic constants differ with 

each direction. For example, in a piece of rolled aluminum plate, the grains 

are elongated in one direction and compressed in the others and the elastic 

constants for the longitudinal direction are different than those for the 

transverse or short transverse directions. 

V longitudinal 

V transverse

Examples of approximate compressional sound velocities in materials are: 

Aluminum - 0.632 cm/microsecond 

1020 steel - 0.589 cm/microsecond 

Cast iron - 0.480 cm/microsecond. 

Examples of approximate shear sound velocities in materials are: 

Aluminum - 0.313 cm/microsecond 

1020 steel - 0.324 cm/microsecond 

Cast iron - 0.240 cm/microsecond. 

When comparing compressional and shear velocities, it can be noted that 

shear velocity is approximately one half that of compressional velocity. The 

sound velocities for a variety of materials can be found in the ultrasonic 

properties tables in the general resources section of this site.

Longitudinal Wave Velocity: V L 

The velocity of a longitudinal wave is described by the following equation: 

V L 

E 

μ 

P 

= Longitudinal bulk wave velocity 

= Young’s modulus of elasticity 

= Poisson ratio 

= Material density

Shear Wave Velocity: V S 

The velocity of a shear wave is described by the following equation: 

V s 

E 

μ 

P 

G 

= Shear wave velocity 

= Young’s modulus of elasticity 

= Poisson ratio 

= Material density 

= Shear modulus

2.4: Properties of Acoustic Plane Wave 

Wavelength, Frequency and Velocity 

Among the properties of waves propagating in isotropic solid materials are 

wavelength, frequency, and velocity. The wavelength is directly proportional 

to the velocity of the wave and inversely proportional to the frequency of the 

wave. This relationship is shown by the following equation.

The applet below shows a longitudinal and transverse wave. The direction of 

wave propagation is from left to right and the movement of the lines indicate 

the direction of particle oscillation. The equation relating ultrasonic 

wavelength, frequency, and propagation velocity is included at the bottom of 

the applet in a reorganized form. The values for the wavelength, frequency, 

and wave velocity can be adjusted in the dialog boxes to see their effects on 

the wave. Note that the frequency value must be kept between 0.1 to 1 MHz 

(one million cycles per second) and the wave velocity must be between 0.1 

and 0.7 cm/us.

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/applet_2_4/applet_2_4.htm


Java don’t work? Uninstalled → Reinstalled → Then → 

http://jingyan.baidu.com/article/9f63fb91d0eab8c8400f0e08.html

Java don’t work? 








As can be noted by the equation, a change in frequency will result in a 

change in wavelength. Change the frequency in the applet and view the 

resultant wavelength. At a frequency of .2 and a material velocity of 0.585 

(longitudinal wave in steel) note the resulting wavelength. Adjust the material 

velocity to 0.480 (longitudinal wave in cast iron) and note the resulting 

wavelength. Increase the frequency to 0.8 and note the shortened wavelength 

in each material. 

In ultrasonic testing, the shorter wavelength resulting from an increase in 

frequency will usually provide for the detection of smaller discontinuities. This 

will be discussed more in following sections. 

Keywords: 

the shorter wavelength resulting from an increase in frequency will usually 

provide for the detection of smaller discontinuities

The velocities sound waves 

The velocities of the various kinds of sound waves can be calculated from the 

elastic constants of the material concerned, that is the modulus of elasticity E 

(measured in N/m2), the density p in kg/m 3 , and Poisson's ratio μ (a 

dimensionless number). 

for longitudinal waves: 

for transverse waves:

The two velocities of sound are linked by the following relation: 

For all solid materials Poisson's ratio μ lies between 0 and 0.5, so that the 

numerical value of the expression 

always lies between 0 and 0.707. In steel and aluminum, μ= 0.28 and 0.34, 

respectively, 

-- 

= 0.55 and 0.49 respectIvely.

2.5: Wavelength and Defect Detection 

2.5.1 Sensitivity & Resolution 

In ultrasonic testing, the inspector must make a decision about the frequency 

of the transducer that will be used. As we learned on the previous page, 

changing the frequency when the sound velocity is fixed will result in a 

change in the wavelength of the sound. 

The wavelength of the ultrasound used has a significant effect on the 

probability of detecting a discontinuity. A general rule of thumb is that a 

discontinuity must be larger than one-half the wavelength to stand a 

reasonable chance of being detected.

Sensitivity and resolution are two terms that are often used in ultrasonic 

inspection to describe a technique's ability to locate flaws. Sensitivity is the 

ability to locate small discontinuities. Sensitivity generally increases with 

higher frequency (shorter wavelengths). Resolution is the ability of the system 

to locate discontinuities that are close together within the material or located 

near the part surface. Resolution also generally increases as the frequency 

increases.

Keywords: 

• Discontinuity must be larger than one-half the wavelength to stand a 

reasonable chance of being detected. 

• Sensitivity is the ability to locate small discontinuities. Sensitivity generally 

increases with higher frequency (shorter wavelengths). 

• Resolution is the ability of the system to locate discontinuities that are 

close together within the material or located near the part surface. 

• Resolution also generally increases as the frequency increases, pulse 

length decrease, bandwidth increase (highly damp)

2.5.2 Grain Size & Frequency Selection 

The wave frequency can also affect the capability of an inspection in adverse 

ways. Therefore, selecting the optimal inspection frequency often involves 

maintaining a balance between the favorable and unfavorable results of the 

selection. Before selecting an inspection frequency, the material's grain 

structure and thickness, and the discontinuity's type, size, and probable 

location should be considered. 

As frequency increases, sound tends to scatter from large or course grain 

structure and from small imperfections within a material. Cast materials often 

have coarse grains and other sound scatters that require lower frequencies to 

be used for evaluations of these products. 

(1) Wrought and (2) forged products with directional and refined grain 

structure can usually be inspected with higher frequency transducers.

Keywords: 

• Coarse grains →Lower frequency to avoid scattering and noise, 

• Fine grains →Higher frequency to increase sensitivity & resolution.

Since more things in a material are likely to scatter a portion of the sound 

energy at higher frequencies, the penetrating power (or the maximum depth 

in a material that flaws can be located) is also reduced. Frequency also has 

an effect on the shape of the ultrasonic beam. Beam spread, or the 

divergence of the beam from the center axis of the transducer, and how it is 

affected by frequency will be discussed later. 

It should be mentioned, so as not to be misleading, that a number of other 

variables will also affect the ability of ultrasound to locate defects. These 

include the pulse length, type and voltage applied to the crystal, properties of 

the crystal, backing material, transducer diameter, and the receiver circuitry of 

the instrument. These are discussed in more detail in the material on signalto-noise 

ratio.

Coarse grains →Lower frequency to avoid scattering and noise, 

Fine grains →Higher frequency to increase sensitivity & resolution. 

http://www.cnde.iastate.edu/ultrasonics/grain-noise

Detectability variable: 

• pulse length, 

• type and voltage applied to the crystal, 

• properties of the crystal, 

• backing material, 

• transducer diameter, and 

• the receiver circuitry of the instrument.

Keywords: 

• Higher the frequency, greater the scattering, thus less penetrating. 

• Higher the frequency better sensitivity and better resolution 

• If the grain size is 1/10 the wavelength, the ultrasound will be significantly 

scattered.

Q7: When a material grain size is on the order of ______ wavelength or 

larger, excessive scattering of the ultrasonic beam affect test result. 

A. 1 

B. ½ 

C. 1/10 

D. 1/100

2.5.3 Further Reading 

Detectability variable: 

• pulse length, 

• type and voltage applied to the crystal, 

• properties of the crystal, 

• backing material, 

• transducer diameter (focal length → Cross sectional area), and 

• the receiver circuitry of the instrument. 

Investigating on: Sonic pulse volume ∝ pulse length, transducer Φ

Pulse Length: 

A sound pulse traveling through a 

metal occupies a physical 

volume. This volume changes 

with depth, being smallest in the 

focal zone. The pulse volume, a 

product of a pulse length L and a 

cross-sectional area A, can be 

fairly easily measured by 

combining ultrasonic A-scans and 

C-scans, as will be seen shortly. 

For many cases of practical interest, the inspection simulation models predict 

that S/N (signal to noise ratio) is inversely proportional to the square root of the 

pulse volume at the depth of the defect. This is known as the “pulse volume 

rule-of-thumb” and has become a guiding principle for designing 

inspections. Generally speaking, it applies when both the grain size and the 

lateral size of the defect are smaller than the sound pulse diameter. 

http://www.cnde.iastate.edu/ultrasonics/grain-noise

Determining cross sectional area using reflector- A Scan (6db drop)

Determining cross sectional area using reflector- C Scan

“Sonic pulse volume” and S/N (defect resolution)

Pulse volume rule-of-thumb: 

Competing grain noise ∝√(pulse volume)

2.6: Attenuation of Sound Waves 

2.6.1 Material Attenuation: 

Attenuation by definition is the rate of decrease of sound energy when a 

ultrasound wave id propagating in a medium. The sound attenuation in 

material depends on heat treatment, grain size, viscous friction, crystal 

stricture (anisotropy or isotropy), porosity, elastic hysteresis, hardness, 

Young’s modulus, etc. 

Sound attenuations are affected by; (1) Geometric beam spread, (2) 

Absorption, (3) Scattering. 

Material attenuation affects item (2) & (3).

When sound travels through a medium, its intensity diminishes with distance. 

In idealized materials, sound pressure (signal amplitude) is only reduced by 

the (1) spreading of the wave. Natural materials, however, all produce an 

effect which further weakens the sound. This further weakening results from 

(2) scattering and (3) absorption. Scattering is the reflection of the sound in 

directions other than its original direction of propagation. Absorption is the 

conversion of the sound energy to other forms of energy. The combined 

effect of scattering and absorption (spreading?) is called attenuation. 

Ultrasonic attenuation is the decay rate of the wave as it propagates through 

material. 

Attenuation of sound within a material itself is often not of intrinsic interest. 

However, natural properties and loading conditions can be related to 

attenuation. Attenuation often serves as a measurement tool that leads to the 

formation of theories to explain physical or chemical phenomenon that 

decreases the ultrasonic intensity.

Absorption: 

Sound attenuations are affected by; (1) Geometric beam spread, (2) Absorption, 

(3) Scattering. 

Absorption processes 

1. Mechanical hysteresis 

2. Internal friction 

3. Others (?) 

For relatively non-elastic material, these soft and pliable material include lead, 

plastid, rubbers and non-rigid coupling materials; much of the energy is loss as 

heat during sound propagation and absorption is the main reason that the 

testing of these material are limit to relatively thin section/

Scattering: 

Grain Size and Wave Frequency 

The relative impact of scattering source of a material depends upon their 

grain sizes in comparison with the Ultrasonic sound wave length. As the 

scattering size approaches that of a wavelength, scattering by the grain is a 

concern. The effects from such scattering could be compensated with the use 

of increasing wavelength ultrasound at the cost of decreasing sensitivity and 

resolution to detection of discontinuities. 

Other effect are anisotropic columnar grain with different elastic behavior at 

different grain direction. In this case the internal incident wave front becomes 

distorted and often appear to change direction (propagate better in certain 

preferred direction) in respond to material anisotropy.

Anisotropic Columnar Grains 

with different elastic behavior at different grain direction.

Spreading/ Scattering / adsorption (reflection is a form of scattering) 

Adsorption 

Scattering 

Spreading 

Scatterbrain

The amplitude change of a decaying plane wave can be expressed as: 

In this expression A o is the unattenuated amplitude of the propagating wave 

at some location. The amplitude A is the reduced amplitude after the wave 

has traveled a distance z from that initial location. The quantity α is the 

attenuation coefficient of the wave traveling in the z-direction. The α 

dimensions of are nepers/length, where a neper is a dimensionless 

quantity. The term e is the exponential (or Napier's constant) which is equal 

to approximately 2.71828.

The units of the attenuation value in Nepers per meter (Np/m) can be 

converted to decibels/length by dividing by 0.1151. Decibels is a more 

common unit when relating the amplitudes of two signals.

Attenuation is generally proportional to the square of sound frequency. 

Quoted values of attenuation are often given for a single frequency, or an 

attenuation value averaged over many frequencies may be given. Also, the 

actual value of the attenuation coefficient for a given material is highly 

dependent on the way in which the material was manufactured. Thus, quoted 

values of attenuation only give a rough indication of the attenuation and 

should not be automatically trusted. Generally, a reliable value of attenuation 

can only be obtained by determining the attenuation experimentally for the 

particular material being used. 

Attenuation ∝ Frequency (f ) 2

Attenuation can be determined by evaluating the multiple back wall reflections 

seen in a typical A-scan display like the one shown in the image at the bottom. 

The number of decibels between two adjacent signals is measured and this 

value is divided by the time interval between them. This calculation produces 

a attenuation coefficient in decibels per unit time Ut. This value can be 

converted to nepers/length by the following equation. 

Where v is the velocity of sound in meters per 

second and Ut is in decibels per second.

Amplitude at distance Z 

where: 

Where v is the velocity of sound in meters per 

second and Ut is in decibels per second.

A o 

Ut 

A

2.6.2 Factors Affecting Attenuation: 

1. Testing Factors 

• Testing frequency 

• Boundary conditions 

• Wave form geometry 

2. Base Material Factors 

• Material type 

• Chemistry 

• Integral constituents (fiber, voids, water content, inclusion, anisotropy) 

• Forms (casting, wrought) 

• Heat treatment history 

• Mechanical processes(Hot or cold working; forging, rolling, extruding, 

TMCP, directional working)

2.6.3 Frequency selection 

There is no ideal frequency; therefore, frequency selection must be made with 

consideration of several factors. Frequency determines the wavelength of the 

sound energy traveling through the material. Low frequency has longer 

wavelengths and will penetrate deeper than higher frequencies. To penetrate 

a thick piece, low frequencies should be used. Another factor is the size of the 

grain structure in the material. High frequencies with shorter wavelengths 

tend to reflect off grain boundaries and become lost or result in ultrasonic 

noise that can mask flaw signals. Low frequencies must be used with coarse 

grain structures. However, test resolution decreases when frequency is 

decreased. Small defects detectable at high frequencies may be missed at 

lower frequencies. In addition, variations in instrument characteristics and 

settings as well as material properties and coupling conditions play a major 

role in system performance. It is critical that approved testing procedures be 

followed.

2.6.4 Further Reading on Attenuation

Q94: In general, which of the following mode of vibration would have the 

greatest penetrating power in a coarse grain material if the frequency of 

the wave are the same? 

a) Longitudinal wave 

b) Shear wave 

c) Transverse wave 

d) All the above modes would have the same penetrating power 

Q: The random distribution of crystallographic direction in alloys with large 

crystalline structures is a factor in determining: 

A. Acoustic noise levels 

B. Selection of test frequency 

C. Scattering of sound 

D. All of the above

Q168: Heat conduction, viscous friction, elastic hysteresis, and scattering are 

four different mechanism which lead to: 

A. Attenuation 

B. Refraction 

C. Beam spread 

D. Saturation

Q7: When the material grain size is in the order of ____ wavelength or larger, 

excessive scattering of the ultrasound beam may affect test result: 

A. 1 

B. ½ 

C. 1/10 

D. 1/100

2.7: Acoustic Impedance 

Acoustic impedance is a measured of resistance of sound propagation 

through a part. 

From the table air has lower acoustic impedance than steel and for a given 

energy Aluminum would travel a longer distance than steel before the same 

amount of energy is attenuated.

Transmission & Reflection Animation: 

http://upload.wikimedia.org/wikipedia/commons/3/30/Partial_transmittance.gif

Sound travels through materials under the influence of sound pressure. 

Because molecules or atoms of a solid are bound elastically to one another, 

the excess pressure results in a wave propagating through the solid. 

The acoustic impedance (Z) of a material is defined as the product of its 

density (p) and acoustic velocity (V). 

Z = pV 

Acoustic impedance is important in: 

1. the determination of acoustic transmission and reflection at the boundary 

of two materials having different acoustic impedances. 

2. the design of ultrasonic transducers. 

3. assessing absorption of sound in a medium.

The following applet can be used to calculate the acoustic impedance for any 

material, so long as its density (p) and acoustic velocity (V) are known. The 

applet also shows how a change in the impedance affects the amount of 

acoustic energy that is reflected and transmitted. The values of the reflected 

and transmitted energy are the fractional amounts of the total energy incident 

on the interface. Note that the fractional amount of transmitted sound energy 

plus the fractional amount of reflected sound energy equals one. The 

calculation used to arrive at these values will be discussed on the next page. 


Reflection/Transmission Energy as a function of Z

Reflection/Transmission Energy as a function of Z

Q2.8: The acoustic impedance of material used to determined: 

A. Angle of refraction at the interface 

B. Attenuation of material 

C. Relative amount of sound energy coupled through and reflected at an 

interface 

D. Beam spread within the material

2.8: Reflection and Transmission Coefficients (Pressure) 

Ultrasonic waves are reflected at boundaries where there is a difference in 

acoustic impedances (Z) of the materials on each side of the boundary. (See 

preceding page for more information on acoustic impedance.) This difference 

in Z is commonly referred to as the impedance mismatch. The greater the 

impedance mismatch, the greater the percentage of energy that will be 

reflected at the interface or boundary between one medium and another. 

The fraction of the incident wave intensity that is reflected can be derived 

because particle velocity and local particle pressures must be continuous 

across the boundary.

When the acoustic impedances of the materials on both sides of the boundary 

are known, the fraction of the incident wave intensity that is reflected can be 

calculated with the equation below. The value produced is known as the 

reflection coefficient. Multiplying the reflection coefficient by 100 yields the 

amount of energy reflected as a percentage of the original energy.

Since the amount of reflected energy plus the transmitted energy must equal 

the total amount of incident energy, the transmission coefficient is calculated 

by simply subtracting the reflection coefficient from one. 

Formulations for acoustic reflection and transmission coefficients (pressure) 

are shown in the interactive applet below. Different materials may be 

selected or the material velocity and density may be altered to change the 

acoustic impedance of one or both materials. The red arrow represents 

reflected sound and the blue arrow represents transmitted sound. 


Reflection Coefficient:

Note that the reflection and transmission coefficients are often expressed in 

decibels (dB) to allow for large changes in signal strength to be more easily 

compared. To convert the intensity or power of the wave to dB units, take the 

log of the reflection or transmission coefficient and multiply this value times 

10. However, 20 is the multiplier used in the applet since the power of sound 

is not measured directly in ultrasonic testing. The transducers produce a 

voltage that is approximately proportionally to the sound pressure. The power 

carried by a traveling wave is proportional to the square of the pressure 

amplitude. Therefore, to estimate the signal amplitude change, the log of the 

reflection or transmission coefficient is multiplied by 20.

Using the above applet, note that the energy reflected at a water-stainless 

steel interface is 0.88 or 88%. The amount of energy transmitted into the 

second material is 0.12 or 12%. The amount of reflection and transmission 

energy in dB terms are -1.1 dB and -18.2 dB respectively. The negative sign 

indicates that individually, the amount of reflected and transmitted energy is 

smaller than the incident energy.

If reflection and transmission at interfaces is 

followed through the component, only a small 

percentage of the original energy makes it back 

to the transducer, even when loss by attenuation 

is ignored. For example, consider an immersion 

inspection of a steel block. The sound energy 

leaves the transducer, travels through the water, 

encounters the front surface of the steel, 

encounters the back surface of the steel and 

reflects back through the front surface on its way 

back to the transducer. At the water steel 

interface (front surface), 12% of the energy is 

transmitted. At the back surface, 88% of the 

12% that made it through the front surface is 

reflected. This is 10.6% of the intensity of the 

initial incident wave. As the wave exits the part 

back through the front surface, only 12% of 10.6 

or 1.3% of the original energy is transmitted back 

to the transducer.

Incident Wave other than Normal? – Oblique Incident 

http://www.slideshare.net/crisevelise/fundamentals-ofultrasound?related=1&utm_campaign=related&utm_medium=1&utm_sourc 

e=29

Incident Wave other than Normal? – Oblique Incident

Q: The figure above shown the partition of incident and reflected wave at 

water-Aluminum interface at an incident angle of 20, the reflected and 

transmitted wave are: 

A. 60% and 40% 

B. 40% and 60% 

C. 1/3 and 2/3 

D. 80% and 20% 

Note: if normal incident the reflected 70% Transmitted 30%

Further Reading (Olympus Technical Note) 

The boundary between two materials of different acoustic impedances is 

called an acoustic interface. When sound strikes an acoustic interface at 

normal incidence, some amount of sound energy is reflected and some 

amount is transmitted across the boundary. The dB loss of energy on 

transmitting a signal from medium 1 into medium 2 is given by: 

dB loss of transmission = 10 log 10 [ 4Z 1 Z 2 / (Z 1 +Z 2 ) 2 ] 

The dB loss of energy of the echo signal in medium 1 reflecting from an 

interface boundary with medium 2 is given by: 

dB loss of Reflection = 10 log 10 [ (Z 1 -Z 2 ) 2 / (Z 1 +Z 2 ) 2 ]

For example: The dB loss on transmitting from water (Z = 1.48) into 1020 

steel (Z = 45.41) is -9.13 dB; this also is the loss transmitting from 1020 steel 

into water. The dB loss of the backwall echo in 1020 steel in water is -0.57 

dB; this also is the dB loss of the echo off 1020 steel in water. The waveform 

of the echo is inverted when Z2

Further Reading: Reflection & Transmission for Normal Incident 

http://www.slideshare.net/crisevelise/fundamentals-ofultrasound?related=1&utm_campaign=related&utm_me 

dium=1&utm_source=29

Q6: For an ultrasonic beam with normal incidence the transmission coefficient 

is given by: 

http://webpages.ursinus.edu/lriley/courses/p212/lectures/node19.html#eq:acousticR 

http://sepwww.stanford.edu/sep/prof/waves/fgdp8/paper_html/node2.html

2.9: Refraction and Snell's Law

Refraction and Snell's Law 

When an ultrasonic wave passes through an 

interface between two materials at an oblique 

angle, and the materials have different indices 

of refraction, both reflected and refracted waves 

are produced. This also occurs with light, which 

is why objects seen across an interface appear 

to be shifted relative to where they really are. 

For example, if you look straight down at an 

object at the bottom of a glass of water, it looks 

closer than it really is. A good way to visualize 

how light and sound refract is to shine a 

flashlight into a bowl of slightly cloudy water 

noting the refraction angle with respect to the 

incident angle.

V s1 

Only If this medium support shear wave i.e. Solid 

V L1 

V L1 

V S2 

V L2

Refraction takes place at an interface due to the different velocities of the 

acoustic waves within the two materials. The velocity of sound in each 

material is determined by the material properties (elastic modulus and density) 

for that material. In the animation below, a series of plane waves are shown 

traveling in one material and entering a second material that has a higher 

acoustic velocity. Therefore, when the wave encounters the interface between 

these two materials, the portion of the wave in the second material is moving 

faster than the portion of the wave in the first material. It can be seen that this 

causes the wave to bend. 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Graphics/Flash/waveRefraction.swf

http://www.ni.com/white-paper/3368/en/

Snell's Law describes the relationship between the angles and the velocities 

of the waves. Snell's law equates the ratio of material velocities V1 and V2 to 

the ratio of the sine's of incident (Ɵ 1 °) and refracted (Ɵ 2 °) angles, as shown in 

the following equation. 

Where: 

V L1 is the longitudinal wave velocity 

in material 1. 

V L2 is the longitudinal wave velocity 

in material 2.

Note that in the diagram, there is a reflected longitudinal wave (V L1' ) shown. 

This wave is reflected at the same angle as the incident wave because the 

two waves are traveling in the same material, and hence have the same 

velocities. This reflected wave is unimportant in our explanation of Snell's Law, 

but it should be remembered that some of the wave energy is reflected at the 

interface. In the applet below, only the incident and refracted longitudinal 

waves are shown. The angle of either wave can be adjusted by clicking and 

dragging the mouse in the region of the arrows. Values for the angles or 

acoustic velocities can also be entered in the dialog boxes so the that applet 

can be used as a Snell's Law calculator.

Snell Law 


Snell Law

When a longitudinal wave moves from a slower to a faster material, there is 

an incident angle that makes the angle of refraction for the wave 90 o . This is 

know as the first critical angle. The first critical angle can be found from 

Snell's law by putting in an angle of 90° for the angle of the refracted ray. At 

the critical angle of incidence, much of the acoustic energy is in the form of an 

inhomogeneous compression wave, which travels along the interface and 

decays exponentially with depth from the interface. This wave is sometimes 

referred to as a "creep wave." Because of their inhomogeneous nature and 

the fact that they decay rapidly, creep waves are not used as extensively as 

Rayleigh surface waves in NDT. However, creep waves are sometimes more 

useful than Rayleigh waves because they suffer less from surface 

irregularities and coarse material microstructure due to their longer 

wavelengths.

Snell Law

Refraction and mode conversion occur 

because of the change in L-wave 

velocity as it passes the boundary from 

one medium to another. The higher the 

difference in the velocity of sound 

between two materials, the larger the 

resulting angle of refraction. L-waves 

and S-waves have different angles of 

refraction because they have dissimilar 

velocities within the same material. 

s the angle of the ultrasonic transducer 

continues to increase, L-waves move 

closer to the surface of the UUT. 

The angle at which the L-wave is parallel with the surface of the UUT is 

referred to as the first critical angle. This angle is useful for two reasons. Only 

one wave mode is echoed back to the transducer, making it easy to interpret 

the data. Also, this angle gives the test system the ability to look at surfaces 

that are not parallel to the front surface, such as welds.

Example: Snell’s Law 

L-wave and S-wave refraction angles are calculated using Snell’s law. You 

also can use this law to determine the first critical angle for any combination 

of materials. 

Where: 

Ɵ 2 ° = angle of the refracted beam in the UUT 

Ɵ 1 ° = incident angle from normal of beam in the wedge or liquid 

V 1 = velocity of incident beam in the liquid or wedge 

V 2 = velocity of refracted beam in the UUT

For example, calculate the first critical angle for a transducer on a plastic 

wedge that is examining aluminum. 

V 1 = 0.267 cm/µs (for L-waves in plastic) 

V 2 = 0.625 cm/µs (for L-waves in aluminum) 

Ɵ 2 ° = 90 degree (angle of L-wave for first critical angle) 

Ɵ 1 ° = unknown 

The plastic wedge must have a minimum angle of 25.29 ° to transmit only S- 

waves into the UUT. When the S-wave angle of refraction is greater than 90°, 

all ultrasonic energy is reflected by the UUT.

Snell Law: First critical angle

Snell Law: 1 st / 2 nd Critical Angles

Q155 Which of the following can occur when an ultrasound beam reaches the 

interface of 2 dissimilar materials? 

a) Reflection 

b) refraction 

c) mode conversion 

d) all of the above

Q. Both longitudinal and shear waves may be simultaneously generated in a 

second medium when the angle of incidence is: 

a) between the normal and the 1st critical angle 

b) between the 1st and 2nd critical angle 

c) past the second critical angle 

d) only at the second critical angle

Q: When angle beam contact testing a test piece, increasing the incident 

angle until the second critical angle is reached results in: 

A. Total reflection of a surface wave 

B. 45 degree refraction of the shear wave 

C. Production of a surface wave 

D. None of the above

Typical angle beam assemblies make use of mode conversion and Snell's 

Law to generate a shear wave at a selected angle (most commonly 30°, 45°, 

60°, or 70°) in the test piece. As the angle of an incident longitudinal wave 

with respect to a surface increases, an increasing portion of the sound energy 

is converted to a shear wave in the second material, and if the angle is high 

enough, all of the energy in the second material will be in the form of shear 

waves. There are two advantages to designing common angle beams to take 

advantage of this mode conversion phenomenon. 

• First, energy transfer is more efficient at the incident angles that generate 

shear waves in steel and similar materials. 

• Second, minimum flaw size resolution is improved through the use of 

shear waves, since at a given frequency, the wavelength of a shear wave 

is approximately 60% the wavelength of a comparable longitudinal wave.

Snell Law: 

http://techcorr.com/services/Inspection-and-Testing/Ultrasonic-Shear-Wave.cfm

Depth & Skip

More on Snell Law 

Like light, when an incident ultrasonic wave encounters an interface to an 

adjacent material of a different velocity, at an angle other than normal to the 

surface, then both reflected and refracted waves are produced. 

Understanding refraction and how ultrasonic energy is refracted is especially 

important when using angle probes or the immersion technique. It is also the 

foundation formula behind the calculations used to determine a materials first 

and second critical angles. 

First Critical Angle 

Before the angle of incidence reaches the first critical angle, both longitudinal 

and shear waves exist in the part being inspected. The first critical angle is 

said to have been reached when the longitudinal wave no longer exists within 

the part, that is, when the longitudinal wave is refracted to greater or equal 

than 90°, leaving only a shear wave remaining in the part.

Second Critical Angle 

The second critical angle occurs when the angle of incidence is at such an 

angle that the remaining shear wave within the part is refracted out of the part. 

At this angle, when the refracted shear wave is at 90° a surface wave is 

created on the part surface 

Beam angles should always be plotted using the appropriate industry 

standard, however, knowing the effect of velocity and angle on refraction will 

always benefit an NDT technician when working with angle inspection or the 

immersion technique. 

The above calculator uses the following equation: 

ultrasonic snells law formula 

Where: 

A1 = The angle of incidence. 

V1 = The incident material velocity 

A2 = The angle of refraction 

V2 = The refracted material velocity

http://www.ndtcalc.com/calculators.html

2.10: Mode Conversion 

When sound travels in a solid material, one form of wave energy can be 

transformed into another form. For example, when a longitudinal waves hits 

an interface at an angle, some of the energy can cause particle movement in 

the transverse direction to start a shear (transverse) wave. Mode conversion 

occurs when a wave encounters an interface between materials of different 

acoustic impedances and the incident angle is not normal to the interface. 

From the ray tracing movie below, it can be seen that since mode conversion 

occurs every time a wave encounters an interface at an angle, ultrasonic 

signals can become confusing at times.

Mode Conversion 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Graphics/Flash/ModeConversion/ModeConv.swf

In the previous section, it was pointed out 

that when sound waves pass through an 

interface between materials having different 

acoustic velocities, refraction takes place at 

the interface. The larger the difference in 

acoustic velocities between the two 

materials, the more the sound is refracted. 

Notice that the shear wave is not refracted 

as much as the longitudinal wave. This 

occurs because shear waves travel slower 

than longitudinal waves. Therefore, the 

velocity difference between the incident 

longitudinal wave and the shear wave is not 

as great as it is between the incident and 

refracted longitudinal waves. 

Also note that when a longitudinal wave is reflected inside the material, the 

reflected shear wave is reflected at a smaller angle than the reflected 

longitudinal wave. This is also due to the fact that the shear velocity is less 

than the longitudinal velocity within a given material.

Snell's Law holds true for shear waves as well as longitudinal waves and can 

be written as follows 

= 

Where: 

VL1 is the longitudinal wave velocity in material 1. 

VL2 is the longitudinal wave velocity in material 2. 

VS1 is the shear wave velocity in material 1. 

VS2 is the shear wave velocity in material 2.

Snell's Law

In the applet below, the shear (transverse) wave ray path has been added. 

The ray paths of the waves can be adjusted by clicking and dragging in the 

vicinity of the arrows. Values for the angles or the wave velocities can also be 

entered into the dialog boxes. It can be seen from the applet that when a 

wave moves from a slower to a faster material, there is an incident angle 

which makes the angle of refraction for the longitudinal wave 90 degrees. As 

mentioned on the previous page, this is known as the first critical angle and 

all of the energy from the refracted longitudinal wave is now converted to a 

surface following longitudinal wave. This surface following wave is sometime 

referred to as a creep wave and it is not very useful in NDT because it 

dampens out very rapidly.

Reflections

Creep wave

V S1 

V S2

Beyond the first critical angle, only the shear wave propagates into the 

material. For this reason, most angle beam transducers use a shear wave so 

that the signal is not complicated by having two waves present. In many 

cases there is also an incident angle that makes the angle of refraction for the 

shear wave 90 degrees. This is known as the second critical angle and at this 

point, all of the wave energy is reflected or refracted into a surface following 

shear wave or shear creep wave. Slightly beyond the second critical angle, 

surface waves will be generated. 

Keywords: 

■ 

■ 

Longitudinal creep wave 

Shear creep wave

Snell Law- 1 st & 2 nd Critical Angles

Note that the applet defaults to compressional velocity in the second material. 

The refracted compressional wave angle will be generated for given 

materials and angles. To find the angle of incidence required to generate a 

shear wave at a given angle complete the following: 

1. Set V1 to the longitudinal wave velocity of material 1. This material could 

be the transducer wedge or the immersion liquid. 

2. Set V2 to the shear wave velocity (approximately one-half its 

compressional velocity) of the material to be inspected. 

3. Set Q2 to the desired shear wave angle. 

4. Read Q1, the correct angle of incidence.

Transverse wave can be introduced into the test material by various methods: 

1. Inclining the incident L-wave at an angle beyond the first critical angle, yet 

short of second critical angle using a wedge. 

2. In immersion method, changing the angle of the normal search unit 

manipulator, 

3. Off-setting the normal transducer from the center-line for round bar or pipe. 

for 45° refracted transverse wave, the rule 

of thumb is the offset d= 1/6 of rod diameter

Offset of Normal probe above circular object 

θ 1 

θ 1R 

θ 2

Calculate the offset for following conditions: 

Aluminum rod being examined is 6" diameter, what is the off set needed for (a) 

45 refracted shear wave (b) Logitudinal wave to be generated? 

(L-wave velocity for AL=6.3x10 5 cm/s, T-wave velocity for AL=3.1x10 5 cm/s, 

Wave velocity in water=1.5X10 5 cm/s) 

Question (a)

Refraction and mode conversion at non-perpendicular boundaries

Refraction and mode conversion at non-perpendicular boundaries 

http://static4.olympus-ims.com/data/Flash/HTML5/incident_angle/IncidentAngle.html?rev=5E62




Q1. From the above figures, if the incident angle is 50 Degree, what are the 

sound wave in the steel? 

Answer: 65 Degree Shear wave in steel. 

Q2. If 50 Degree longitudinal wave in steel is used what is the possible 

problem? 

Answer: If 50 degree Longitudinal wave is generated in steel, shear wave at 

28 degree is also generated and this may cause fault indications.

Q118: At the water-steel interface, the angle of incidence in water is 7 degree. 

The principle mode of vibration that exist in steel is: 

A. Longitudinal 

B. Shear 

C. Both A & B (Possible incorrect answer) 

D. Surface 

Hint: The keyword is “the principle mode”

Q: On Calculation: 

Incident angle= 7° 

Refracted longitudinal wave = 29.11° 

Refracted shear wave = 15.49°

Q72. In a water immersion test, ultrasonic energy is transmitted into steel at 

an incident angle of 14. What is the angle of refracted shear wave within 

the material? V s = 3.2 x 10 5 cm/s, V w = 1.5 x 10 5 cm/s 

a) 45° 

b) 23° 

c) 31° 

d) 13°

Q1. If you were requested to design a plastid shoe to generate Rayleigh wave 

in aluminum, what would be the incident angle of the ultrasonic energy? 

VA = 3.1 x 105 cm/s, Vp = 2.6 x 105 cm/s 

a) 37° 

b) 57° 

c) 75° 

d) 48°

Q53. The term used to determined the relative transmittance and reflectance 

of an ultrasonic energy at an interface is called: 

a) Acoustic attenuation 

b) Interface reflection 

c) Acoustic impedance ratio 

d) Acoustic frequency

2.11: Signal-to-Noise Ratio 

In a previous page, the effect that frequency and wavelength have on flaw 

detectability was discussed. However, the detection of a defect involves many 

factors other than the relationship of wavelength and flaw size. For example, 

the amount of sound that reflects from a defect is also dependent on the 

acoustic impedance mismatch between the flaw and the surrounding material. 

A void is generally a better reflector than a metallic inclusion because the 

impedance mismatch is greater between air and metal than between two 

metals. 

Often, the surrounding material has competing reflections. Microstructure 

grains in metals and the aggregate of concrete are a couple of examples. A 

good measure of detectability of a flaw is its signal-to-noise ratio (S/N). The 

signal-to-noise ratio is a measure of how the signal from the defect compares 

to other background reflections (categorized as "noise"). A signal-to-noise 

ratio of 3 to 1 is often required as a minimum.

The absolute noise level and the absolute strength of an echo from a "small" 

defect depends on a number of factors, which include: 

1. The probe size and focal properties. 

2. The probe frequency, bandwidth and efficiency. 

3. The inspection path and distance (water and/or solid). 

4. The interface (surface curvature and roughness). 

5. The flaw location with respect to the incident beam. 

6. The inherent noisiness of the metal microstructure. 

7. The inherent reflectivity of the flaw, which is dependent on its acoustic 

impedance, size, shape, and orientation. 

8. Cracks and volumetric defects can reflect ultrasonic waves quite differently. 

Many cracks are "invisible" from one direction and strong reflectors from 

another. 

9. Multifaceted flaws will tend to scatter sound away from the transducer.

The following formula relates some of the variables affecting the signal-tonoise 

ratio (S/N) of a defect:

Sound Volume: Area x pulse length 

Material properties 

Flaw geometry: Figure of merit 

FOM and amplitudes responds

Rather than go into the details of this formulation, a few fundamental 

relationships can be pointed out. The signal-to-noise ratio (S/N), and 

therefore, the detectability of a defect: 

1. Increases with increasing flaw size (scattering amplitude). The detectability 

of a defect is directly proportional to its size. 

2. Increases with a more focused beam. In other words, flaw detectability is 

inversely proportional to the transducer beam width. 

3. Increases with decreasing pulse width (delta-t). In other words, flaw 

detectability is inversely proportional to the duration of the pulse (∆t) 

produced by an ultrasonic transducer. The shorter the pulse (often higher 

frequency), the better the detection of the defect. Shorter pulses 

correspond to broader bandwidth frequency response. See the figure 

below showing the waveform of a transducer and its corresponding 

frequency spectrum.

Acoustic Volume: w x w y ∆t

Determining cross sectional area using reflector- A Scan (6db drop)

Determining cross sectional area using reflector- C Scan

“Sonic pulse volume” and S/N (defect resolution)

4. Decreases in materials with high density and/or a high ultrasonic velocity. 

The signal-to-noise ratio (S/N) is inversely proportional to material density 

and acoustic velocity. 

5. Generally increases with frequency. However, in some materials, such as 

titanium alloys, both the "A flaw " and the "Figure of Merit (FOM)" terms in the 

equation change at about the same rate with changing frequency. So, in 

some cases, the signal-to-noise ratio (S/N) can be somewhat independent 

of frequency.

Pulse Length

Pulse Length Affect Resolution

2.12: The Sound Fields 

2.12.1 Wave Interaction or Interference 

Before we move into the next section, the subject of wave interaction must 

be covered since it is important when trying to understand the performance 

of an ultrasonic transducer. On the previous pages, wave propagation was 

discussed as if a single sinusoidal wave was propagating through the 

material. However, the sound that emanates from an ultrasonic transducer 

does not originate from a single point, but instead originates from many 

points along the surface of the piezoelectric element. This results in a 

sound field with many waves interacting or interfering with each other.

Transducer cut-out 

http://ichun-chen.com/ultrasonic-transducer

When waves interact, they superimpose on each other, and the amplitude of 

the sound pressure or particle displacement at any point of interaction is the 

sum of the amplitudes of the two individual waves. First, let's consider two 

identical waves that originate from the same point. When they are in phase 

(so that the peaks and valleys of one are exactly aligned with those of the 

other), they combine to double the displacement of either wave acting alone. 

When they are completely out of phase (so that the peaks of one wave are 

exactly aligned with the valleys of the other wave), they combine to cancel 

each other out. When the two waves are not completely in phase or out of 

phase, the resulting wave is the sum of the wave amplitudes for all points 

along the wave.

UT Transducer

UT Transducer 

http://www.fhwa.dot.gov/publications/research/infrastructure/structures/04042/index.cfm#toc

UT Transducer- Surface creep wave transducer



Wave Interaction 

Complete in-phase Complete out of-phase not in-phase

When the origins of the two interacting waves are not the same, it is a little 

harder to picture the wave interaction, but the principles are the same. Up 

until now, we have primarily looked at waves in the form of a 2D plot of wave 

amplitude versus wave position. However, anyone that has dropped 

something in a pool of water can picture the waves radiating out from the 

source with a circular wave front. If two objects are dropped a short distance 

apart into the pool of water, their waves will radiate out from their sources and 

interact with each other. At every point where the waves interact, the 

amplitude of the particle displacement is the combined sum of the amplitudes 

of the particle displacement of the individual waves. 

With an ultrasonic transducer, the waves propagate out from the transducer 

face with a circular wave front. If it were possible to get the waves to 

propagate out from a single point on the transducer face, the sound field 

would appear as shown in the upper image to the right. Consider the light 

areas to be areas of rarefaction and the dark areas to be areas of 

compression.

With an ultrasonic transducer, the waves propagate out from the transducer 

face with a circular wave front. If it were possible to get the waves to 

propagate out from a single point on the transducer face, the sound field 

would appear as shown in the upper image to the right. Consider the light 

areas to be areas of rarefaction and the dark areas to be areas of 

compression.

However, as stated previously, sound waves originate from multiple points 

along the face of the transducer. The lower image to the right shows what the 

sound field would look like if the waves originated from just two points. It can 

be seen that where the waves interact, there are areas of constructive and 

destructive interference. The points of constructive interference are often 

referred to as nodes. 

The points of constructive interference 

are often referred to as nodes

29. It is possible for a discontinuity smaller than the transducer to produce 

indications of fluctuating amplitude as the search unit is moved laterally if 

testing is being performed in the: 

(a) Fraunhofer zone 

(b) Near field 

(c) Snell field 

(d) Shadow zone

Q5: Acoustic pressure along the beam axis moving away from the probe has 

various maxima and minima due to interference. At the end of the near field 

pressure is: 

a) a maximum 

b) a minimum 

c) the average of all maxima and minima 

d) none of the above 

Q4: For a plane wave, sound pressure is reduced by attenuation in a 

_______ fashion. 

a) linear 

b) exponential 

c) random 

d) none of the above

2.12.2 Variations in sound intensity 

Intensity 

Distance

Of course, there are more than two points of origin along the face of a 

transducer. The image below shows five points of sound origination. It can be 

seen that near the face of the transducer, there are extensive fluctuations or 

nodes and the sound field is very uneven. In ultrasonic testing, this in known 

as the near field (near zone) or Fresnel zone. The sound field is more 

uniform away from the transducer in the far field, or Fraunhofer zone, where 

the beam spreads out in a pattern originating from the center of the 

transducer. It should be noted that even in the far field, it is not a uniform 

wave front. However, at some distance from the face of the transducer and 

central to the face of the transducer, a uniform and intense wave field 

develops.

The sound wave exit from a transducer can be separated into 2 zones or 

areas; The Near Field (Fresnel) and the Far Field (Fraunhofer).

2.12.3 Fresnel & Fraunhofer Zone 

Fresnel Field, the Near Field are region directly adjacent to the transducer 

and characterized as a collection of symmetrical high and low pressure 

regions cause by interference wave fronts emitting from the continuous or 

near continuous sound sources. 

http://blog.3bscientific.com/science_education_insight/2013/04/3b-scientific-makes-waves-with-new-physics-education-kit.html

The Near Field (Fresnel) and the Far Field (Fraunhofer).

The Near Field (Fresnel)– Wave Interference (Maxima & Minima) 

The sound field of a transducer is divided into two zones; the near field and 

the far field. The near field is the region directly in front of the transducer 

where the echo amplitude goes through a series of maxima and minima and 

ends at the last maximum, at distance N from the transducer.

Amplitude ← 

Near Field Effect: Because of the variations within the near field it can be 

difficult to accurately evaluate flaws using amplitude based techniques. 

Near Field Y o 

+ 

Far Field 

Distance from Transducer face →

Fresnel / Fraunhofer Zone

Near field (near zone) 

or Fresnel zone 

far field (far zone) 

or Fraunhofer zone 

Z f

Near field (near zone) 

or Fresnel zone 

Crystal Focus 

Accoustical axis 

far field (far zone) 

or Fraunhofer zone 

Angle of divergence 

D 0 

6 

N 

Near field 

Far field 

Z f

Near/ Far Fields 

http://miac.unibas.ch/PMI/05-UltrasoundImaging.html

Near/ Far Fields

where α is the radius of the 

transducer and λ the wavelength.

where D is the diameter of the transducer 

and λ the wavelength. 

K= is the spread factor 

K=1.22 for null edges 

K=1.08 for 20dB down point (10% of peak) 

K=0.88 for 10dB down point (32% of peak) 

K=0.7(0.56?) for 6dB down point (50% of peak) 

Source for K, ASNT Study Guide UT by Matthew J Golis

The curvature and the area over which the sound is being generated, the 

speed that the sound waves travel within a material and the frequency of the 

sound all affect the sound field. Use the Java applet below to experiment with 

these variables and see how the sound field is affected. 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/appletUltrasoundPropagation/Applet.html

Fresnel & Fraunhofer Zone 

10dB, K-0.88 

6dB, K=0.7? Or 0.56?

Fresnel & Fraunhofer Zone

Fresnel & Fraunhofer Zone

Fresnel & Fraunhofer Zone 

http://static1.olympus-ims.com/data/Flash/HTML5/beamSpread/BeamSpread.html?rev=6C43 

http://www.olympus-ims.com/en/ndt-tutorials/transducers/wave-front/

Q: Where does beam divergence occur? 

A. Near field 

B. Far field 

C. At the crystal 

D. None of the above

Q4: A transducer has a near field in water of 35 mm. When used in contact 

on steel the near zone will be about: 

a) 47 mm 

b) 35 mm 

c) 18 mm 

d) 9 mm 

Q8: A rectangular probe, 4 mm X 8 mm, will have its maximum half angle of 

divergence: 

a) in the 4 mm direction 

b) in the 8 mm direction 

c) in no particular orientation 

d) constant in all directions

Q160 Beam divergence is a function of the dimensions of the crystal and the 

wavelength of the beam transmitted through a medium, and it: 

A. increase if the frequency or the crystal diameter is decrease 

B. Decrease if the frequency or the crystal diameter is decrease 

C. increase if the frequency is increase and the diameter is decrease 

D. decrease if the frequency is increase and the crustal diameter is decrease 

Q52: What is the transducer half-angle beam spread of a 1.25cm diameter 

2.25 MHz transducer in water (VL= 1.5 x 10 5 cm/s)? 

A. 2.5 degree 

B. 3.75 degree 

C. 37.5 degree 

D. 40.5 degree

2.12.4 Dead Zone 

In ultrasonic testing, the interval following the initial pulse where the 

transducer ring time of the crystal that prevents detection or interpretation of 

reflected energy (echoes). In contact ultrasonic testing, the area just below 

the surface of a test object that can not be inspected because of the 

transducer is still ringing down and not yet ready to receive signals. The dead 

is minimized by the damping medium behind the crystal. The dead zone 

increase when the probe frequency decrease and it only found in single 

crystal contact techniques.

Dead Zone - The interval following the surface of a test object to the nearest 

inspectable depth. Any interval following a reflected signal where no direct 

echoes from discontinuities cannot be detected, due to characteristics of the 

equipment. 

dead zone after echo and dead zone after initial pulse, both are common 

phenomena. Actually the dead zone cannot be determined as a single figure 

without additional parameters, hence the echo can be recognized, however, 

signal quality is important. Useful parameters are linearity or signal in a nice 

ratio that can describe the echo amplitude quality within a dead zone. For this 

reason standards such as GE specifications are needed to check equipment 

capability. The appearance of inference effects, within the dead zone, has to 

be considered as well. 

Definition by: http://www.ndt.net/ndtaz/content.php?id=103

Dead Zone -The initial pulse is a technical necessity. It limits the detectability 

of near-surface discontinuities. Reflectors in the dead zone, the nonresolvable 

area immediately beneath the surface, cannot be detected (Figure 

8-10). The dead zone is a function of the width of the initial pulse which is 

influenced by the probe type, test instrument discontinuities and quality of the 

interface. 

The dead zone can be verified with an International Institute of Welding (IIW) 

calibration block. With the time base calibrated to 50 mm, and the transducer 

on position A (Figure 8-11), the extent of the dead zone can be inferred to be 

either less than or greater than 5 mm. With the probe at position B, the dead 

zone can be said to be either less than or greater than 10 mm. 

This is done by ensuring that the peak from the Perspex insert appears 

beyond the trailing edge of the initial pulse start. Excessive dead zones are 

generally attributable to a probe with excessive ringing in the crystal.

Dead Zone -The initial pulse is a technical necessity. It limits the detectability 

of near-surface discontinuities. Reflectors in the dead zone, the nonresolvable 

area immediately beneath the surface, cannot be detected. The 

dead zone is a function of the width of the initial pulse which is influenced by 

the probe type, test instrument discontinuities and quality of the interface. 

This is done by ensuring that the peak from the Perspex insert appears 

beyond the trailing edge of the initial pulse start. Excessive dead zones are 

generally attributable to a probe with excessive ringing in the crystal.

Dead Zone Illustration 

http://www.ndt.net/ndtaz/content.php?id=103

Dead Zone 

http://www.ni.com/white-paper/5369/en/

Q: On an A-scan display, the “dead zone” refers to: 

A. The distance contained within the near field 

B. The area outside the beam spread 

C. The distance covered by the front surface pulse with and recovery 

time 

D. The area between the near field and the far field

Q36: To eliminate the decrease in sensitivity close to a wall which is parallel 

to the beam direction, the transducer used should be: 

A. As small as possible 

B. As low frequency as possible 

C. Both A & B 

D. Large and with a frequency as high as possible 

Q45: The length of the near field for a 2.5cm diameter, 5MHz transducer 

placed in oil with V=1.4 x 10 5 cm/s is approximately: 

A. 0.028 cm 

B. 6.25 cm 

C. 22.3 cm 

D. 55.8 cm

2.13: Inverse Square Rule/ Inverse Rule 

Large Reflector, a reflector larger than the extreme edge of beam / 3D away 

from the Near Zone- Inverse Rule

Large Reflector Inverse Rule

Small Reflector, a reflector smaller than the extreme edge of beam / 3D away 

from the Near Zone – Inverse Square Rule

Small Reflector Inverse Square Rule

2.14: Resonance 

Another form wave interference occurred when the normal incidence and 

reflected plane wave interact within a narrow parallel interface. When the 

phase of the reflected wave match that of incoming incident wave, the 

amplitude of the superimposed wave doubling, creating a standing wave. 

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html#c3 

Resonance occurred when the thickness of the material is equal to half the 

wave length or multiple of it. It also occur when longitudinal wave travel thru 

a thin sheet of materials during immersion testing.

Fundamental Frequency 

The lowest resonant frequency of a vibrating object is called its fundamental 

frequency. 

Most vibrating objects have more than one resonant frequency and those 

used in musical instruments typically vibrate at harmonics of the fundamental. 

A harmonic is defined as an integer (whole number) multiple of the 

fundamental frequency. 

Vibrating strings, open cylindrical air columns, and conical air columns will 

vibrate at all harmonics of the fundamental. Cylinders with one end closed will 

vibrate with only odd harmonics of the fundamental. Vibrating membranes 

typically produce vibrations at harmonics, but also have some resonant 

frequencies which are not harmonics. It is for this class of vibrators that the 

term overtone becomes useful - they are said to have some non-harmonic 

overtones. 

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html

Thickness of Crystal at Fundamental Frequency 

Fundamental resonance frequency 

Harmonic resonance frequency 

= V/f 

= N. V/f (N = integer) 

Piezoelectric crystal will has the greatest sensitivity when it is driven at its 

fundamental frequency, this occurs when the thickness of the crystal is at ½ 

λ. 

If the thickness is given, the fundamental frequency could be calculated:

Transducers Piezoelectric Thickness: 

The resonant phenomenon occurred when piezoelectric are electrically 

excited at their characteristic (fundamental resonance) frequency. 

http://bme240.eng.uci.edu/students/09s/patelnj/Ultrasound_for_Nerves/Ultrasound_Background.html

Resonance UT Testing- The diagram below shown how resonance is used 

to measured thickness and detect defect. However pulse-echo methods have 

been refined to perform most of function of flaw detections and resonant 

instruments are rarely used.

Application Case#1: 

The specimen's geometry determines the number of its natural frequencies: a 

rod has few whilst a complex work-piece has many such frequencies. 

Typically, the information that can be obtained by acoustic resonance 

analysis includes cracks, structural properties, cavities, layer separation, 

chipping, density fluctuations etc. Damping behaviour depends firstly on the 

material, and secondly, on how the specimen is positioned during its 

excitation. In order to achieve high frequency resolution, signal duration 

("ringing duration") should be as long as possible (> 50 ms).

From the natural frequencies it is possible to calculate specimen-specific 

characteristics and assign them to quality attributes, e. g. pass / OK, cracked, 

material structure, hardness deviation / partly hardened etc. 

Application 

Acoustic resonance testing can be applied to all work pieces that "sound". 

Summary 

Resonance analysis is a qualitative method, i.e. it can differentiate between 

defective and non-defective parts, so that it is especially suitable for quality 

assurance in the series production cycle. It compares the actual oscillatory 

situation with the target one derived from a learning base. This learning base 

is established by using defined standard parts. The number of self-resonant 

frequencies is determined by the geometry of the object under test. For 

instance a bar has few resonant frequencies, while a complex lattice-type 

object has many natural resonances. After a systematic engineering 

approach, it is possible to compensate the influence of the production scatter. 

http://ndttechnologies.com/products/AcousticResonance.html

Application Case#2: 

Electromagnetic Acoustic Resonance Nondestructive Testing (NDT) 

Equipment Datasheets 

Coating Thickness Gauge -- DTG-500 

from OMEGA Engineering, Inc. 

Digital coating thickness gauge with a range of 0 to 40.0 mils (0 to 1000 

micrometers). SPECIFICATIONS. Display: 3-digit LCD with max readout of 

1999 counts. Range: 0 to 40 mils/0 to 1000 µm. Resolution: 0. 

Instrument Information 

Instrument Type: Coating Thickness 

Instrument Technology: Electromagnetic Acoustic Resonance 

Form Factor: Portable / Handheld / Mobile 

http://www.globalspec.com/specsearch/PartSpecs?partid={0DBF141D-6832-4F31-9AB8- 

B87F063BFDC4}&vid=99786&comp=2975

Q: The formula used to determine the fundamental resonance frequency is: 

A. F= V/T 

B. F= V/2T 

C. F= T/V 

D. F= VT 

Q: When maximum sensitivity is required from a transducer: 

A. A straight beam unit should be used 

B. A large diameter crystal should be used 

C. The piezoelectric element should be driven at its fundamental 

frequency 

D. The bandwidth of the transducer should be as large as possible

Q7: The resonance frequency of 2cm thick plate of Naval Brass (V=4.43 x 10 5 

cm/s) is: 

A. 0,903 MHz 

B. 0.443 MHz 

C. 0.222 MHz 

D. 0.111 MHz 

Q35: Resonance testing equipment generally utilized: 

A. Pulsed longitudinal; waves 

B. Continuous longitudinal waves 

C. Pulsed shear wave 

D. Continuous shear waves

2.15 Measurement of Sound

dB is a measures of ratio of 2 values in a logarithmic scale given by following 

equation: 

Unlike the SPL (standard pressure level) used in noise measurement, in UT testing, 

we do not know the exactly ultrasonic sound level energy generated by the probe 

(neither is it necessary). The used of the ratio of 2 values given by the above equation 

is used .

Ultrasonic Formula - Signal Amplitude Gain/Loss Expressed in dB 

The dB is a logarithmic unit that describes a ratio of two measurements. The 

equation used to describe the difference in intensity between two ultrasonic or 

other sound measurements is: 

where: ∆I is the difference in sound intensity expressed in decibels (dB), P1 

and P2 are two different sound pressure amplitude measurements, and the 

log is to base 10.

The Decibel 

The equation used to describe the difference in intensity between two 

ultrasonic or other sound measurements is: 

where: ∆I is the difference in sound intensity expressed in decibels (dB), P1 

and P2 are two different sound pressure measurements, and the log is to 

base 10. 

What exactly is a decibel? 

The decibel (dB) is one tenth of a Bel, which is a unit of measure that was 

developed by engineers at Bell Telephone Laboratories and named for 

Alexander Graham Bell. The dB is a logarithmic unit that describes a ratio of 

two measurements. The basic equation that describes the difference in 

decibels between two measurements is:

where: delta X is the difference in some quantity expressed in decibels, X1 

and X2 are two different measured values of X, and the log is to base 10. 

(Note the factor of two difference between this basic equation for the dB and 

the one used when making sound measurements. This difference will be 

explained in the next section.)

Why is the dB unit used? 

Use of dB units allows ratios of various sizes to be described using easy to 

work with numbers. For example, consider the information in the table.

From this table it can be seen that ratios from one up to ten billion can be 

represented with a single or double digit number. Ease to work with numbers 

was particularly important in the days before the advent of the calculator or 

computer. The focus of this discussion is on using the dB in measuring sound 

levels, but it is also widely used when measuring power, pressure, voltage 

and a number of other things.

Use of the dB in Sound Measurements 

Sound intensity is defined as the sound power per unit area perpendicular to 

the wave. Units are typically in watts/m2 or watts/cm2. For sound intensity, 

the dB equation becomes: 

However, the power or intensity of sound is generally not measured directly. 

Since sound consists of pressure waves, one of the easiest ways to quantify 

sound is to measure variations in pressure (i.e. the amplitude of the pressure 

wave). When making ultrasound measurements, a transducer is used, which 

is basically a small microphone. Transducers like most other microphones 

produced a voltage that is approximately proportionally to the sound pressure 

(P). The power carried by a traveling wave is proportional to the square of the 

amplitude. Therefore, the equation used to quantify a difference in sound 

intensity based on a measured difference in sound pressure becomes:

However, the power or intensity of sound is generally not measured directly. 

Since sound consists of pressure waves, one of the easiest ways to quantify 

sound is to measure variations in pressure (i.e. the amplitude of the pressure 

wave). When making ultrasound measurements, a transducer is used, which 

is basically a small microphone. Transducers like most other microphones 

produced a voltage that is approximately proportionally to the sound pressure 

(P). The power carried by a traveling wave is proportional to the square of the 

amplitude. 

I α P 2 , I α V 2 where I=intensity, P=amplitude, V=voltage 

Therefore, the equation used to quantify a difference in sound intensity based 

on a measured difference in sound pressure becomes: 

(The factor of 2 is added to the equation because the logarithm of the square of a 

quantity is equal to 2 times the logarithm of the quantity.)

Since transducers and microphones produce a voltage that is proportional to 

the sound pressure, the equation could also be written as: 

where: ∆I is the change in sound intensity incident on the transducer and 

V1 and V2 are two different transducer output voltages.

Revising the table to reflect the relationship between the ratio of the measured 

sound pressure and the change in intensity expressed in dB produces 

From the table it can be seen that 6 dB equates to 

a doubling of the sound pressure. Alternately, 

reducing the sound pressure by 2, results in a – 6 

dB change in intensity.

Sound Levels- Relative

Sound Levels- Relative dB

Practice:

“Absolute" Sound Levels 

Sound pressure level (SPL) or sound level is a logarithmic measure of the 

effective sound pressure of a sound relative to a reference value. It is 

measured in decibels (dB) above a standard reference level. The standard 

reference sound pressure in air or other gases is 20 µPa, which is usually 

considered the threshold of human hearing (at 1 kHz). 

http://en.wikipedia.org/wiki/DB_SPL#Sound_pressure_level

“Absolute" Sound Levels 

Whenever the decibel unit is used, it always represents the ratio of two values. 

Therefore, in order to relate different sound intensities it is necessary to 

choose a standard reference level. The reference sound pressure 

(corresponding to a sound pressure level of 0 dB) commonly used is that at 

the threshold of human hearing, which is conventionally taken to be 2×10 −5 

Newton per square meter, or 20 micropascals (20μPa). To avoid confusion 

with other decibel measures, the term dB(SPL) is used.

dB meter 

97.3dB against standards sound pressure level 

20log(P/20X10 -6 )=97.3 

Absolute level =10 97.3/20 x 20 X 10 -6 

=1.46564 N/M 2 

Actual Sound pressure → 

↖ Standard reference pressure 20 μMpa

Absolute: 

The standard reference sound pressure in air or other gases is 20 µPa, which 

is usually considered the threshold of human hearing (at 1 kHz). 

Actual Sound pressure → 

↖ Standard reference pressure 20 μMpa 

Absolute: 

Sound pressure level in dB as a ratio to 

standard reference in logarithmic scale. 

76db= 20log(P/20 μPa) 

Log(P/20 μPa)=3.8dB 

P= 10 3.8 x 20 μPa 

=126191 μPa 

http://www.ncvs.org/ncvs/tutorials/voiceprod/equation/chapter9/index.html

Exercise: 

Find the absolute sound level in μPa for the following measurement of air 

traffic noise.

Exercise: ANSWER 

Find the absolute sound level in μPa for the following measurement of air 

traffic noise. 

SPL= 95.8 dB= 20log(P/20x10 -6 ) 

log(P/20x10-6)= 95.8/20 

P= 10 95.8/20 x 20x10 -6 

P= 1.233 N/M 2 #

Practice: 

dB

Relative dB: Example Calculation 1 

Two sound pressure measurements are made using an ultrasonic 

transducer. The output voltage from the transducer is 600 mv for the first 

measurement and 100 mv for the second measurement. Calculate the 

difference in the sound intensity, in dB, between the two measurements? 

The sound intensity changed by -15.56dB. In other words, the sound 

intensity decreased by 15.56 dB

Example Calculation 2 

If the intensity between two ultrasonic measurements increases by 6 dB, and 

the first measurement produces a transducer output voltage of 30 mv, what 

was the transducer output voltage for the second measurement?

Example Calculation 3 

Consider the sound pressure difference between the threshold of human 

hearing, 0 dB, and the level of sound often produce at a rock concert, 120 dB. 

How much is the rock concert sound greater than that of the threshold of 

human hearing.

What is the absolute rock concert sound pressure?

2.16 Practice Makes Perfect

Practice Makes Perfect 

28. An advantage of using lower frequencies during ultrasonic testing is that: 

(a) Near surface resolution is improved 

(b) Sensitivity to small discontinuities is improved 

(c) Beam spread is reduced 

(d) Sensitivity to unfavorable oriented flaws is improved

Q104: If an ultrasonic wave is transmitted through an interface of two 

materials in which the first material has a higher acoustic impedance value 

but the same velocity value as the secong material, the angle of refraction 

will be: 

a) A greater than the incidence 

b) Less than the angle of incidence 

c) The same as the angle of incidence 

d) Beyond the critical angle.

学习总是开心事

UT Testing-Section 2 Physics of Ultrasound

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