Addendum-02 Equations & Calculations

Addendum-02 

Equations & Calculations 

My ASNT Level III UT Study Notes 

2014-June.

Speaker: Fion Zhang 

2014/July/31

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

Trigonometry 

http://www.mathwarehouse.com/trigonometry/sine-cosine-tangent.php

Contents: 

1. Material Acoustic Properties 

2. Ultrasonic Formula 

3. Properties of Acoustic Wave 

4. Speed of Sound 

5. Attenuation 

6. What id dB 

7. Acoustic Impedance 

8. Snell’s Law 

9. S/N Ratio 

10. Near / Far Field 

11. Focusing & Focal Length 

12. Offsetting for Circular Specimen 

13. Quality “Q” Factors 

14. Inverse Law & Inverse Square Law 

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

1.0 Material Acoustic Properties 

Material 

Logitudinal wave 

Shear wave 

Z Acoustic 

mm/μs 

mm/μs 

Impedence 

Acrylic resin 

2.74 

1.44 

3.23 

(Perspex) 

Steel - SS 300 

5.613 

3.048 

44.6 

Series 

Steel - SS 400 

5.385 

2.997 

41.3 

Series 

Steel 1020 

5.893 

3.251 

45.4 

Steel 4340 

5.842 

3.251 

45.6 

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

2.0 Ultrasonic Formula 

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

Ultrasonic Formula

Ultrasonic Formula 

α = Transducer radius

3.0 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.

4.0 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 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

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:

V is the speed of sound 

Eleatic 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.

E/N/G

5.0 Attenuation 

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. 

http://www.ndt.net/article/v04n06/gin_ut2/gin_ut2.htm

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

Adsoprtion 

Scaterring 

Spreading 

Scaterrring

Attenuation can be determined by evaluating the multiple backwall 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 v is the velocity of sound in meters per second and Ut is in decibels 

per second (attenuation coefficient). 

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

α dimensions of are nepers/length (nepers constant).

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 2 (f ) 2

Which U t ? 

U 0 t , A 0 o 

U 1 t , A 1 o , α 1 

1 1

7.0 Acoustic Impedance 

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. 

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/applet_2_6/applet_2_6.htm

Reflection/Transmission Energy as a function of Z

Reflection and Transmission Coefficients (Pressure) 

• This difference in Z is commonly referred to as the impedance 

mismatch. 

• 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. 

• the transmission coefficient is calculated by simply subtracting the 

reflection coefficient from one. 

Ipedence 

mismatch 

Reflection coefficient

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.

Practice Makes Perfect 

Following are the data:

Q1: What is the percentage of initial incident sound wave that will reflected 

from the water/Aluminum interface when the sound first enter Aluminum? 

R= (Z 1 -Z 2 ) 2 / (Z 1 +Z 2 ) 2 = (0.149-1.72) 2 /(0.149+1.72) 2 

R= 0.707, Answer= 70.7%

Q2: What is the percentage of sound energy that will finally reenter the water 

after reflected from the backwall of Aluminum? (Do not consider material 

attenuation and other factors) 

Answer: 6% 

0.706 – initial Back wall 

0.2934 

0.207x 0.2934=0.0609 

Second Backwall echo 

0.2934x 0.706 = 

0.207

8.0 Snell’s Law 

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 

http://education-portal.com/academy/lesson/refraction-dispersion-definition-snells-law-index-of-refraction.html#lesson


5. For an ultrasonic beam with normal incidence, the reflection coefficient is 

given by: 

(a) [(Z 1 +Z 2 ) 2 ]/[(Z 1 -Z 2 ) 2 ] 

(b) (Z 1 +Z 2 )/(Z 1 -Z 2 ) 

(c) [(4) (Z 1 )(Z 2 )]/[(Z 1 +Z 2 ) 2 ] 

(d) [(Z 1 -Z 2 ) 2 ]/[Z 1 +Z 2 ) 2 ] 

6. For an ultrasonic beam with normal incidence the transmission coefficient 

is given by: 

(a) [(Z 1 +Z 2 ) 2 ]/[(Z 1 -Z 2 ) 2 ] 

(b) (Z 1 +Z 2 )/(Z 1 -Z 2 ) 

(c) [(4) (Z 1 )(Z 2 )]/[(Z 1 +Z 2 ) 2 ] 

(d) [(Z 1 -Z 2 ) 2 ]/[Z 1 +Z 2 ) 2 ]

Practice Made Perfect 

7. Snell's law is given by which of the following: 

(a) (Sin A)/(Sin B) = VB/VA 

(b) (Sin A)/(Sin B) = VA/VB 

(c) (Sin A)/ VB = V(Sin B)/VA 

(d) (Sin A)[VA] = (Sin B)[ VB] 

8. Snell's law is used to calculate: 

(a) Angle of beam divergence 

(b) Angle of diffraction 

(c) Angle of refraction 

(d) None of the above


9. Calculate the refracted shear wave angle in steel [VS = 0.323cm/microsec] 

for an incident longitudinal wave of 37.9 degrees in Plexiglas [VL = 0.267cm/ 

microsec] 

(a) 26 degrees 

(b) 45 degrees 

(c) 48 degrees 

(d) 64 degrees 

10. Calculate the refracted shear wave angle in steel [VS = 0.323cm/microsec] 

for an incident longitudinal wave of 45.7 degrees in Plexiglas [VL = 0.267cm/ 

microsec] 


(b) 45.7 degrees 


(d) 70 degrees


11. Calculate the refracted shear wave angle in aluminium [VS = 0.31cm/ 

microsec] for an incident longitudinal wave of 43.5 degrees in Plexiglas [VL = 

0.267cm/microsec] 




(d) 68 degrees 

12. Calculate the refracted shear wave angle in aluminium [VS = 

0.31cm/microsec] for an incident longitudinal wave of 53 degrees in Plexiglas 

[VL = 0.267cm/microsec] 




(d) 68 degrees

9.0 S/N Ratio 

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

ratio (S/N) of a defect: 

FOM: Factor of merits at center frequency

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

ratio (S/N) of a defect:

Sound Volume: Area x pulse length Δt 

Material properties 

Flaw geometry at center frequency: 

Figure of merit FOM and 

amplitudes responds

10. Near/ Far Fields 

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

where α is the radius of the transducer and λ the wavelength. 

For beam edges at null condition K=1.22

Modified Near Zone 

T Perspex 

Modified Z f

Example: Calculate the modified Near Zone for; 

• 5 MHz shear wave transducer 

• 10mm crystal 

• 10 mm perspex wedge 

Perspex L-wave: 2730 m/s 

Steel S-wave: 3250 m/s 

Steel L-wave: 5900 m/s 

Modified NZ= (0.01 2 x f) / (4v) – 0.01(2730/3250) 

=0.0300m 

= 30mm

Apparent Near Zone distance

11.0 Focusing & Focal Length 

http://www.olympus-ims.com/en/ndt-tutorials/flaw-detection/beam-characteristics/

The focal length F is determined by following equation; 

Where: 

F = Focal Length in water 

R = Curvature of the focusing len 

n = Ration of L-velocity of epoxy to L-velocity of water

12.0 Offset of Normal probe above circular object 

θ 1 

V 1 

θ 1R 

θ 2 V 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) Longitudinal 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) 

Question (b)

13.0 “Q” Factor 

3dB down

14.0 Inverse Law and Inverse Square Law 

For a small reflector where the size of reflector is smaller than the beam width, 

the echoes intensity from the same reflector varies inversely to the square of 

the distance. 

5cm 

7.5cm 

75% FSH 33% FSH

Inverse Square Law 

http://www.cyberphysics.co.uk/general_pages/inverse_square/inverse_square.htm

Inverse Law: 

For large reflector, reflector greater than the beam width, e.g. backwall 

echoes from the same reflector at different depth; the reflected signal 

amplitude varies inversely with the distance. 

10cm 

7.5cm

DGS Distance Gain Sizing 

Y-axis shows the 

Gain 

size of reflector is given as 

a ratio between the size of 

the disc and the size of 

the crystal. 

X-axis shows the Distance from the probe in # of Near Field

– Distance Gain Size is a method of setting sensitivity or assessing the signal 

from an unknown reflector based on the theoretical response of a flatbottomed 

hole reflector perpendicular to the beam axis. (DGS does not size 

the flaw, but relate it with a equivalent reflector) The DGS system was 

introduced by Krautkramer in 1958 and is referred to in German as AVG. A 

schematic of a general DGS diagram is shown in the Figure. The Y-axis 

shows the Gain and X-axis shows the Distance from the probe. In a general 

DGS diagram the distance is shown in units of Near Field and the scale is 

logarithmic to cover a wide range.

The blue curves plotted show how the amplitudes obtained from different 

sizes of disc shaped reflector (equivalent to a FBH) decrease as the distance 

between the probe and the reflector increases.

In the general diagram the size of reflector is given as a ratio between the 

size of the disc and the size of the crystal. The red curve shows the response 

of a backwall reflection. The ratio of the backwall to the crystal is infinity (∞). 

Specific DGS curves for individual probes can be produced and so both the 

distance axis and the reflector sizes can be in mm. 

If the sensitivity for an inspection is specified to be a disc reflector of a given 

size, the sensitivity can be set by putting the reflection from the backwall of a 

calibration block or component to the stated %FSH. The gain to be added can 

be then obtained by the difference on the Y-axis between the backwall curve 

at the backwall range and the curve of the disc reflector of the given size at 

the test range. If the ranges of the backwall and the disc reflector are different, 

then attenuation shall be accounted for separately. Alternatively, the curves 

can be used to find the size of the disc shaped reflector which would give the 

same size echo as a response seen in the flaw detector screen.

20-4dB=16dB (deduced) 

Δ Flaw =30-16=14dB 

20dB 

(measured) 

Data: 

Probe frequency: 5MHz 

Diameter: 10mm compression probe 

Plate thickness: 100mm steel 

Defect depth: 60mm deep 

Gain for flaw to FSH: 30dB 

BWE at 100mm: 20dB

Example: If you has a signal at a certain depth, you can compare the signal of 

the flaw to what the back wall echo (BWE) from the same depth and estimate 

the FBH that would give such a signal at the same depth. The defect can then 

be size according to a FBH equivalent. 

Data: 

Probe frequency: 5MHz 

Diameter: 10mm compression probe 

Plate thickness: 100mm steel 

Defect depth: 60mm deep 

Gain for flaw to FSH: 30dB 

BWE at 100mm: 20dB 

------------------------------------------------------------------------- 

Near field: 21mm, flaw location= 3xNear Field 

From the chart BWE at 60mm will be 20-4dB=16dB 

Flaw signal Gain is 30dB-16dB= 14dB 

Used the flaw signal Gain and locate the equivalent reflector size is between 

0.4 to 0.48 of the probe diameter, say 0.44 x10mm = 4.4mm equivalent 

reflector size.

http://www.olympus-ims.com/en/atlas/dgs/

More on DGS/AVG by Olympus 

http://www.olympus-ims.com/en/ndt-tutorials/flaw-detection/dgs-avg/ 

DGS is a sizing technique that relates the amplitude of the echo from a 

reflector to that of a flat bottom hole at the same depth or distance. This is 

known as Equivalent Reflector Size or ERS. DGS is an acronym for Distance- 

Gain-Size and is also known as AVG from its German name, Abstand 

Verstarkung Grosse. Traditionally this technique involved manually 

comparing echo amplitudes with printed curves, however contemporary 

digital flaw detectors can draw the curves following a calibration routine and 

automatically calculate the ERS of a gated peak. The generated curves are 

derived from the calculated beam spreading pattern of a given transducer, 

based on its frequency and element diameter using a single calibration point. 

Material attenuation and coupling variation in the calibration block and test 

specimen can be accounted for.

DGS is a primarily mathematical technique originally based on the ratio of a 

circular probe’s calculated beam profile and measurable material properties 

to circular disk reflectors. The technique has since been further applied to 

square element and even dual element probes, although for the latter, curve 

sets are empirically derived. It is always up to the user to determine how the 

resultant DGS calculations relate to actual flaws in real test pieces. 

An example of a typical DGS curve set is seen below. The uppermost curve 

represents the relative amplitude of the echo from a flat plate reflector in 

decibels, plotted at various distances from the transducer, and the curves 

below represent the relative amplitude of echoes from progressively smaller 

disk reflectors over the same distance scale.

As implemented in contemporary digital flaw detectors, DGS curves are 

typically plotted based on a reference calibration off a known target such as a 

backwall reflector or a flat bottom hole at a given depth. From that one 

calibration point, an entire curve set can be drawn based on probe and 

material characteristics. Rather than plotting the entire curve set, instruments 

will typically display one curve based on a selected reflector size (registration 

level) that can be adjusted by the user. 

In the example below, the upper curve represents the DGS plot for a 2 mm 

disk reflector at depths from 10 mm to 50 mm. The lower curve is a reference 

that has been plotted 6 dB lower. In the screen at left (figure 1), the red gate 

marks the reflection from a 2 mm diameter flat bottom hole at approximately 

20 mm depth. Since this reflector equals the selected registration level, the 

peak matches the curve at that depth. In the screen at right (Figure 2), a 

different reflector at a depth of approximately 26 mm has been gated. Based 

on its height and depth in relation to the curve the instrument calculated an 

ERS of 1.5 mm.

Figure1:

Figure2:

15.0 Pulse Repetitive Frequency/Rate and Maximum 

Testable Thickness 

Clock interval = 1/PRR 

Maximum testable length = ½ x Velocity x Clock interval 

Note: The Clock interval has neglected the time occupied by each pulse.

16.0 Immersion Testing of Circular Rod

Q4-12 

Answer: 

First calculate the principle offset d; ϴ = Sin-1(1483/3250 xSin45)=18.8 ° 

d=R.Sin18.8= 0.323 (Assume R=1). 

Wobbling ±10%; d’=0.355 ~ 0.290 

d’=0.355, ϴ = Sin-1(0.355)=20.8 ° 

giving inspection Φ = Sin-1(3250/1483xSin20.8)=51, 13.3% above 45 ° 

d’=0.290, ϴ = Sin-1(0.290)=16.9 ° 

giving inspection Φ = Sin-1(3250/1483xSin16.9)=39.6, 12% below 45 °

Maximum ϴ 

ϴ max = Sin -1 (ID/OD)

Addendum-02 Equations & Calculations

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