Introduction to Acoustics

62 Part A Propagation of Sound
Part A 3.10
Analysis that makes use of the boundary condition
(3.291) leads to the identification
Ê(θI,ω) = ξ(ω)cosθI − 1
ξ(ω)cosθI + 1
for the reflection coefficient, with the abbreviation
(3.296)
ξ(ω) = ZS
, (3.297)
ρc
which represents the ratio of the specific acoustic
impedance of the surface to the characteristic impedance
of the medium.
3.10.2 Reflection at an Interface
The above relations also apply, with an appropriate identification
of the quantity ZS, to sound reflection [3.65]
at an interface between two fluids with different sound
speeds and densities. Translational symmetry requires
that the disturbance in the second fluid have the same
apparent phase velocity (ω/kx) (trace velocity) along
the x-axis as does the disturbance in the first fluid. This
requirement is known as the trace velocity matching
principle [3.4,30] and leads to the observation that kx is
the same in both fluids. One distinguishes two possibilities:
the trace velocity is higher than the sound speed c2
or lower than c2.
For the first possibility, one has the inequality
c2 < c1
, (3.298)
sin θI
and a propagating plane wave (transmitted wave) is
excited in the second fluid, with complex pressure am-
ρI,cI
nI
ρII,cII
θI
y
θII
Fig. 3.19 Reflection of a plane wave at an interface between
two fluids
nII
x
plitude
ˆptrans = Ì(ω, θI) ˆ
f e ikx x e ik2 y cos θII , (3.299)
where k2 = ω/c2 is the wavenumber in the second fluid
and θII (angle of refraction) is the angle at which the
transmitted wave is propagating. The trace velocity
matching principle leads to Snell’s law,
sin θI
= sin θII
. (3.300)
c1
c2
The change in propagation direction from θI to θII is the
phenomenon of refraction.
The requirement that the pressure be continuous
across the interface yields the relation
1 + Ê = Ì , (3.301)
while the continuity of the normal component of the
fluid velocity yields
cos θI cos θII
(1 − Ê) = Ì . (3.302)
ρ1c1
ρ2c2
From these one derives the reflection coefficient
Ê = ZII − ZI
, (3.303)
ZII + ZI
which involves the two impedances defined by
ZI = ρ1c1
, (3.304)
cos θI
ZII = ρ2c2
. (3.305)
cos θII
The other possibility, which is the opposite of that in
(3.298), can only occur when c2 > c1 and, moreover,
only if θI is greater than the critical angle
θcr = arcsin(c1/c2) . (3.306)
In this circumstance, an inhomogeneous plane wave,
propagating in the x-direction, but dying out exponentially
in the +y-direction, is excited in the second
medium. Instead of (3.299), one has the transmitted
pressure given by
ˆptrans = Ì(ω, θI) fˆ e ikx x −βk2 y
e , (3.307)
with
β =[(c2/c1) 2 sin 2 θI − 1] 1/2 . (3.308)
The previously stated equations governing the reflection
and transmission coefficients are still applicable,
provided one replaces cos θII by iβ. This causes the magnitude
of the reflection coefficient Ê to become unity,