Introduction to Acoustics

212 Part B Physical and Nonlinear Acoustics
Part B 6.1
reflection coefficient given by
�2 Z2 + Z1
, (6.7)
�
Z2 − Z1
R =
where Z1 and Z2 are the impedances of the two media.
Since both values of Z must be positive, R must be less
than one. The fraction of the energy transmitted into the
second medium is given by T = 1 − R because 100% of
the energy must be divided between T and R.
Refraction
Refraction is a change of the direction of wave propagation
as the wave passes from one medium into another
across an interface. Bending occurs when the wave speed
is different in the two media. If there is an angle between
the normal to the plane of the boundary and the incident
wave, there is a brief time interval when part of the wave
is in the original medium (traveling at one velocity) and
part of the wave is in the second medium (traveling
at a different velocity). This causes the bending of the
waves as they pass from the first medium to the second.
(There is no bending at normal incidence.)
Reflection and refraction can occur simultaneously
when a wave impinges on a boundary between two
media with different wave propagation speeds. Some
of the energy of the wave is reflected back into the
original medium, and some of the energy is transmitted
and refracted into the second medium. This
means that a wave incident on a boundary can generate
two waves: a reflected wave and a transmitted
wave whose direction of propagation is determined by
Snell’s law.
a) b)
Wave velocity
Waves are here but
transverse particle motion
is suppressed
Waves pass through each other
and continue propagating
Wave velocity
Wave velocity
Wave velocity
Destructive
interference
All waves obey Snell’s law. For optical waves the
proper form of Snell’s law is:
n1 sin θ1 = n2 sin θ2 , (6.8)
where n1 and n2 are the refractive indices and θ1 and
θ2 are propagation directions. For acoustic waves the
proper form of Snell’s law is:
sin θ1
= sin θ2
Wave velocity Wave velocity
Addition of
wave amplitudes
Waves pass through each other
and continue propagating
Wave velocity
Wave velocity
Fig. 6.6a,b Two waves passing through each other exhibiting (a) destructive and (b) constructive interference
v1
, (6.9)
v2
where v1 is the wave velocity in medium 1 and v2 is the
wave velocity in medium 2. These two forms are very
similar since the refractive index is n = c/Cm,wherec is
the velocity of light in a vacuum and Cm is the velocity
of light in the medium under consideration.
Interference
Spatial Interference. Interference is a phenomenon that
occurs when two (or more) waves add together. Consider
two identical transverse waves traveling to the right
(one after the other) down a string towards a boundary
at the end. When the first wave encounters the
boundary, it reflects and travels in the leftward direction.
When it encounters the second, rightward moving
wave the two waves add together linearly (in accordance
with the principle of superposition). The displacement
amplitude at the point in space where two waves combine
is either greater than or less than the displacement
amplitude of each wave. If the resultant wave has an
amplitude that is smaller than that of either of the
original two waves, the two waves are said to have
Constructive
interference