Introduction to Acoustics

232 Part B Physical and Nonlinear Acoustics
Part B 6.4
a) Incident,
reflected,
and refracted
waves
Liquid
Solid
b) Incident, reflected,
and surface
waves
I
I
Liquid
Solid
αi
αi
αr
γ
�
αr
R
S
Surface wave
Fig. 6.24 Sound waves at a liquid–solid interface
process can be used to receive surface acoustic waves
and convert them into an electrical signal, which can
then be amplified.
Another type of surface wave is possibly more illustrative
of the connection between surface acoustic waves
and physical acoustics. This is the surface acoustic wave
generated when the trace velocity of an incident wave is
equal to the velocity of the surface acoustic wave. This
occurs when a longitudinal wave is incident on a solid
from a liquid. This is analogous to the optical case of total
internal reflection [6.28], but new information comes
from the acoustic investigation.
The interface between a liquid and a solid is shown
in Fig. 6.24, in which the various waves and their angles
are indicated. The directions in which the various waves
propagate at a liquid–solid interface can be calculated
from Snell’s law, which for this situation can be written
sin θi
v
= sin θr
v
= sin θL
vL
= sin θS
R
L
, (6.66)
vS
where the velocity of the longitudinal wave in the liquid,
that in the solid, and the velocity of the shear wave in the
solid are, respectively, v, vL, andvS. The propagation
directions of the various waves are indicated in Fig. 6.24.
Since much of the theory has been developed in
connection with geology, the theoretical development
of Ergin [6.29] can be used directly. Ergin has shown
that the energy reflected at an interface is proportional to
the square of the amplitude reflection coefficient, which
can be calculated directly [6.29]. The energy reflection
coefficient is given by
where
RE =
� cos β − A cos α (1 − B)
cos β + A cos α (1 − B)
A = ρ1VL
and
ρV
B = 2sinγsin 2γ
�2 , (6.67)
�
cos γ − vS
�
cos β . (6.68)
vL
The relationship among the angles α, β and γ can be
determined from Snell’s law as given in (6.66). The
book by Brekhovskikh [6.30] is also a good source of
information on this subject.
Typical plots of the energy reflection coefficient as
a function of incident angle are given in Fig. 6.25 in
a) νL > νS > ν
Energy ratio
0.1
0.5
0 α CL
b) νL > ν > νS
Energy ratio
α CS
90
Incident angle (deg)
0.1
0.5
0 α CL
90
Incident angle (deg)
Fig. 6.25 Behavior of energy reflected at a liquid–solid
interface