Introduction to Acoustics

They are related via the impedance ϱ0c0 (8.70):
(p − p0)(x, t) = 4αϱ0c 3 0
βω e−αx sin(ωt − k0x) (8.100)
= 2δϱ0ω
e
β
−αx sin(ωt − k0x) . (8.101)
Good approximations for Γ ≫ 1 have been given by
Fay [8.78]:
∞�
u(x, t)
2/Γ
=
sinh[n(1 + x/x⊥)/Γ ] sin n(ωt − k0x) ,
ua
n=1
and Blackstock [8.72]:
u(x, t)
ua
(8.102)
= 2
∞� 1 − (n/Γ
Γ
n=1
2 )coth[n(1 + x/x⊥)/Γ ]
sinh[n(1 + x/x⊥)/Γ ]
×sinn(ωt− k0x) . (8.103)
With an error of less than 1% at Γ = 50 Fay’s
solution is valid for σ>3.3 and Blackstock’s solution
for σ>2.8, rapidly improving with σ. Thegap
in σ from about one to three between the Fubini and
Fay solution has been closed by Blackstock. He connected
both solutions using weak shock theory giving the
Fubini–Blackstock–Fay solution ([8.77], see also [8.1]).
Figure 8.9 shows the first three harmonic components of
the Fubini–Blackstock–Fay solution from [8.79]. Similar
curves have been given by Cook [8.80]. He developed
a numerical scheme for calculating the harmonic content
of a wave as it propagates by including the losses
in small spatial steps linearly for each harmonic component.
As there are occurring only harmonic waves that
do not break the growth in each small step is given by
the Fubini solution.
In the limit Γ →∞the Fay solution reduces to
∞�
u(x, t) 2
=
sin n(ωt − k0x) , (8.104)
ua 1 + x/x⊥
n=1
a solution that is also obtained by weak shock theory.
8.8 Shock Waves
The characteristics of Fig. 8.3 must cross. According
to the geometrical construction the profile of the wave
then becomes multivalued. This is easily envisaged with
water surface waves. With pressure or density waves,
however, there is only one pressure or one density at one
1.0
0.8
0.6
0.4
0.2
Bn
0
0
1
n = 1
n = 2
n = 3
2
Nonlinear Acoustics in Fluids 8.8 Shock Waves 271
3
4
5
6
7
8
9
10
11
σ
Fig. 8.9 Growth and decay of the first three harmonic
components Bn of a plane wave as a function of the normalized
distance σ according to the Fubini–Blackstock–Fay
solution (after Blackstock [8.77])
When Fay’s solution (8.102) is taken and
σ ≫ Γ (farfield), the old-age region of the wave
is reached. Then sinh[n(1 + x/x⊥)/Γ ]� 1 2 (enx/x⊥Γ −
e −nx/x⊥Γ ) � 1 2 e nx/x⊥Γ = 1 2 e nαx and
u(x, t) = 4αc2 0
βω ,
∞�
e −nαx sin n(ωt − k0x) (8.105)
n=1
is obtained similarly as for the fundamental (8.98).
Additionally all harmonics that behave like the fundamental,
i. e. that are not dependent on the initial peak
particle velocity ua, are obtained . Moreover, they do
not decay (as linear waves do) proportionally to e −n2 αx
but only proportionally to e −nαx .
In the range σ>1 shock waves may develop. These
are discussed in the next section.
place. The theory is therefore oversimplified and must
be expanded with the help of physical arguments and
the corresponding mathematical formulation. Shortly
before overturning, the gradients of pressure and density
become very large, and it is known that damping
Part B 8.8