Introduction to Acoustics

egion in the form
p(r, z) = ψ(r, z)H(r) , (5.59)
and we define K 2 (r, z) ≡ K 2 0 n2 , n therefore being an index
of refraction c0/c, wherec0 is a reference sound
speed. Substituting (5.59) into(5.34) and taking K 2 0 as
the separation constant we end up with a Bessel equation
for H that has a Hankel function as the outgoing solution.
If we use the asymptotic form of the Hankel function,
H 1 0 (K0r), and invoke the paraxial (narrow-angle)
approximation,
∂2ψ ∂ψ
≪ 2K0 , (5.60)
∂r2 ∂r
we obtain the parabolic equation (in r),
∂2ψ ∂ψ
+ 2iK0
∂z2 ∂r + K 2 0 (n2 − 1)ψ = 0 , (5.61)
wherewenotethatnis a function of range and depth. We
use a marching solution to solve the parabolic equation.
There has been an assortment of numerical solutions
but the one that still remains a standard is the so-called
split-step range-marching algorithm,
� �
iK0
ψ (r + ∆r, z) = exp
F −1
120
0.25
×
��
exp
2 (n2 − 1)∆r
�
− i∆r
s
2K0
2
��
F [ψ(r, z)]
0.3 0.35 0.4 0.45 0.5 0.55
Wavenumber (1/m)
�
.
(5.62)
a) Frequency (Hz)
b)
110
100
90
80
120
110
100
90
80
–0.04
Range (km)
20
Underwater Acoustics 5.4 Sound Propagation Models 173
–0.03 –0.02 –0.01 0
Wavenumber (1/m)
Fig. 5.26a,b Frequency–wavenumber representation of the dispersed field in the waveguide. (a) Obtained from a twodimensional
FFT of the range–time plot in Fig. 5.25. The wavenumbers of propagating modes are bounded by the upper
and lower sound speeds in waveguide, ω/cω and ω/cb. (b) A rotated version of (a), so that the sound speed in water
appears infinite. This representation is easier to see the mode separation and the mode cut-off frequency. The color scale
is in dB
15
10
5
–0.1
0 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Retarded time (s)
Fig. 5.25 Range-versus-time representation of the field intensity for
the same waveguide and source–receiver depth as in Fig. 5.21. The
figure clearly shows that modes travel at different speed. The retarded
time t is t − R/cω. The color scale is in dB with a 0 dB
reference at range 5 km
The Fourier transforms F, are performed using FFTs.
Equation (5.62) is the solution for n constant, but the
error introduced when n (profile or bathymetry) varies
with range and depth can be made arbitrarily small by
increasing the transform size and decreasing the rangestep
size. It is possible to modify the split-step algorithm
to increase its accuracy with respect to higher-angle
propagation.
0
–5
–10
–15
–20
–25
0
–5
–10
–15
–20
–25
Part A 5.4