Introduction to Acoustics

172 Part A Propagation of Sound
Part A 5.4
0
20
40
60
80
Depth (m)
100
13.4
13.6 13.8 14
14.2
Time (s)
Fig. 5.23 Depth-versus-time representation of the field intensity after
a R = 20 km propagation in a shallow-water waveguide. The
waveguide characteristics are the same as in Fig. 5.21. Source depth
is 40 m. The color scale is in dB with a 0 dB source level amplitude
at the source
5000
1336
Depth (m)
Doppler Shift in a Waveguide
The theory of Doppler shifts involving either a moving
source and/or receiver is well known in acoustics, particularly
in free space. However, in a waveguide, even if
we limit ourselves to horizontal motion, the results are
slightly more complex. The individual paths associated
with modal propagation, as per Fig. 5.7a, all have finite
and different angles with respect to the horizontal. Thus,
each mode has a different Doppler shift. The waveguide
0
500
1000
1500
2000
2500
3000
3500
4000
4500
1338 1340 1342 1344
Time (s)
–70
–75
–80
–85
–90
–95
–125
–130
–135
–140
–145
–150
Fig. 5.24 Depth-versus-time representation of the field intensity after
a R = 2000 km propagation in a deep-water waveguide. We used
the Munk profile as a depth-dependent sound-speed profile. Source
depth is 900 m, source frequency is 22.5 Hz with a 15 Hz bandwidth.
The color scale is in dB with a 0 dB source level amplitude at the
source
theory for both source and/or receiver has been worked
out (see [5.33], in which there is also a review of the
pertinent literature). We return to (5.50) for a harmonic
source of angular frequency ω, which therefore results
in an additional, identical (when there is no motion) factor
for each term of e−iωt in the time domain. We now
consider a source with a frequency spectrum S(ω). For
constant, horizontal velocities, the normal-mode field
results in the receiver reference frame are still valid
ψ(r0 + vrt, z,ω) = i
4ρ(zs)
× �
S(Ωn)un(zs)un(z)H 1 0 (knr),
n
(5.56)
but with Doppler shifted modal wavenumbers
�
kn → kn 1 + vr
�
cos θr , (5.57)
vgn
and Doppler-shifted frequencies (now for each modal
term in the summation) of
Ωn = ω − kn (vs cos θs − vr cos θr) , (5.58)
where vgn is the group velocity of the n-th mode,
vs cos θs is the radial speed of the source, and vr cos θr
is the radial speed of the receiver. Here, radial refers to
the projection of the velocities onto the horizontal line
between the source and receiver. Note that this latter expression,
when multiplied through by the wavenumber,
shows that the frequency shift is proportional to the ratio
of speeds to the modal phase speed, as opposed to
the wavenumber shift which involves the group speed.
5.4.5 Parabolic Equation (PE) Model
The PE method was introduced into ocean acoustics and
made viable with the development of the Tappert splitstep
algorithm, which utilized fast Fourier transforms at
each range step [5.6]. Subsequent numerical developments
greatly expanded the applicability and accuracy
of the parabolic equation method.
Standard PE Split-Step Algorithm
The PE method is presently the most practical
and all-encompassing wave-theoretic range-dependent
propagation model. In its simplest form, it is a far-field
narrow-angle (≈±20 ◦ ) with respect to the horizontal
– adequate for many underwater propagation problems
– approximations to the wave equation. Assuming
azimuthal symmetry about a source, we express the solution
of (5.34) in cylindrical coordinates in a source-free