Introduction to Acoustics

water, where the deep sound channel behaves as
a waveguide, the fastest modes are the higher-order
modes (Fig. 5.24), a property of refraction-dominated
modal propagation. Since modal wavenumbers kn are
frequency dependent, the modes can be plotted in
a frequency–wavenumber space that is the Fourier transform
of the time-range representation of the dispersed
field in Fig. 5.25. As already seen in Fig. 5.21,Fig.5.26
shows that modes have a cut-off frequency and that there
exist a finite number of propagating modes at a given
frequency.
Dispersion and The Waveguide Invariant [5.3]
There is actually a fairly robust parameter, the waveguide
invariant, that describes dispersion over a (sometimes
large) interval of a group of modes. The waveguide invariant
β has two important interpretations; first, it is
related the local change in the modal group velocity
with respect to the change in phase velocity,
1
β =−∂Sg , (5.54)
∂Sp
where Sg and Sp and group and phase slowness, where
slowness is the inverse of speed.
It turns out that β often has a rather robust value
for certain circumstances. For example, it is unity for
many shallow-water situations that are dominated by
bottom reflection; on the other hand it is negative for
refraction-dominated propagation.
Figure 5.27 shows a calculation for a Pacific deepwater
case. Note from the definition that β is negative
up to a phase speed of about 1540; this region is one of
refraction such as deep sound channel and convergence
zone propagation. Beyond 1540, β is positive; this is the
bottom bounce region dominated by reflections rather
than refraction.
The invariant also relates the change in range in the
locations of the interference peaks of a transmission loss
curve to a change in frequency
∆r
r
1 ∆ω
= . (5.55)
β ω
In Fig. 5.28, weshowasetofTL curves for different
frequencies; the interference peaks are shifted
according to the invariant formula. Another way of representing
this shift is through the striations, where TL
is the third dimension of a frequency–range plot as
shown in Fig. 5.29, which was derived from shallowwater
transmission loss data. If one represents range
as the product of the velocity of the radiator and time,
then, the range axis can be replaced by time and one
Sound speed (m/s)
1600
1580
1560
1540
1520
1500
1480
1460
1440
1420
75
1
1
2
2
Underwater Acoustics 5.4 Sound Propagation Models 171
3
3
4
4
80 85 90 95 100 105 110 115 120 125
Frequency (Hz)
5
5 6
Fig. 5.21 Frequency dependence of the group speed (lower curves
1–6) and phase speed (upper curves 1–6) of modes 1–6 in a Pekeris
waveguide with a 100 m water depth, a bottom sound speed cb and
density of 1600 m/s and 1800 kg/m 3 . Sound speed in water cω is
constant and equal to 1500 m/s. The bold vertical lines show the
cut-off frequencies of modes 4–6, respectively
has a frequency–time plot, often called a spectrogram.
The TL curve is a slice through the spectrogram for
a given frequency and time converted to the appropriate
range.
a) b)
1
0.5
0
–0.5
–1
0 0.08
–1
13.4
Time (s)
1
0.5
0
–0.5
13.5 13.6 13.7 13.8 13.9 14 14.1 14.2
Time (s)
Fig. 5.22a,b Acoustic dispersion in a shallow water waveguide. The
right panel (a) corresponds to the waveguide response at R = 20 km
to the emitted signal (central frequency = 100 Hz, 50 Hz bandwidth)
displayed in the left panel. (b) Source depth is at 40 m and the receiver
depth at 60 m. Waveguide parameters are the same as in Fig. 5.21
6
Part A 5.4