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The M-mode image basically corresponds to a displace-
ment-time graph, so that the speed of movement can
be determined from the rise time.
4.11. Transmission coefficient and transverse
speed of sound
With an experiment set-up like that shown in Figure 2
(transducer in transmission mode attached to a water-
filled trough containing a rotatable plate with a speci-
fied thickness of 1 cm), by turning the plate it can be
demonstrated that when an ultrasonic wave passes
from a fluid to a solid body at a non-perpendicular
angle, both longitudinal and transverse waves are ex-
cited.
Since transverse sound waves are produced by shear-
ing and their speed is lower than that of longitudinal
waves, the following regions arise (example with
acrylic):
Angle of incidence 0°: only a peak for longitudinal waves
possibly with multiple reflections
Small angle of incidence (<=10°): multiple reflections
vanish, amplitude decreases
Angle of 10° - 30°: peaks for both longitudinal and
transverse waves
Angle >30°: only transverse waves remain with ampli-
tude maxima at an angle of incidence of about 40°.
Amplitude becomes smaller at increasing angles
The amplitude in transmission mode or, by measuring
the transmission in the absence of the plate, the am-
plitude transmission coefficient can now be calculated
for both longitudinal and transverse waves (see follow-
ing diagram).
Since the transmission of transverse waves through the
plate is greatest at a transmission angle of 45°, the
maximum in the transverse amplitude curve may be
used to determine the angle of incidence Φ and thus
the transverse speed of sound by means of the follow-
ing equation
1
2
=
C
T
sin φ
( )
where c
is the speed of sound in water (1480 m/s).
F
Fig. 17 shows the measurement results for a test body
composed of 1 cm thick acrylic. For an amplitude
maximum at 40°, equation (7) gives the speed of trans-
verse waves to be approximately 1600 m/s. The pub-
lished value is 1450 m/s. Determining the angle more
precisely would certainly lead to greater accuracy in
this case.
Separation into longitudinal and transverse amplitudes
is possible due to the differences in time of travel re-
sulting from the large differences in speed of longitu-
dinally and transversely propagated waves. Even with
a plate of only 1 cm thickness, the transverse waves
are sufficiently delayed to be measured (see following
illustration).
The speed of transverse waves allows the shear modu-
lus (torsion modulus) G to be calculated:
G
=
C
T
ρ
The modulus of elasticity E (Young's modulus) for the
body can be calculated from the longitudinal speed of
sound if the cross-sectional contraction coefficient ( υ -
-Poisson number) is known:
E
=
C
L
ρ
(
1
When cross-sectional expansion is negligible (for thin
rods):
E
=
C
L
ρ
where ρ is always the density of the body
19
c
F
−
υ
1
υ
υ
+
) −
(
)
1 2
(7)
(8)
(9)
(10)
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