Introduction to Acoustics

1.0
0.5
0.0
1.0
0.5
0.0
1.0
0.5
0.0
1.0
0.5
0.0
1.0
0.5
Bn
0.0
0
1
2
3
4
5
6
7
8
Nonlinear Acoustics in Fluids 8.6 Lossless Finite-Amplitude Acoustic Waves 267
σ = 0.0
σ = 0.2
σ = 0.5
σ = 0.8
σ = 1.0
9
10
ν/ν0
Fig. 8.7 Growth of the harmonics upon propagation for
the Fubini solution (8.59) (source problem) at different
normalized distances σ up to the shock distance. [8.38]
The quantity t⊥ is called the shock formation time,
because at this time the wave develops a vertical tangent.
The solution is valid only up to this time t = t⊥. To
simplify the notation the dimensionless normalized time
σt may be introduced
σt = t
. (8.65)
t⊥
The Fubini solution then reads:
∞�
u(x, t)
n+1 Jn(nσt)
= 2 (−1) sin nk0(x − c0t) .
ua
nσt
n=1
When comparing t⊥ with x⊥, the relation
c0t⊥ = x⊥
(8.66)
(8.67)
1.0
0.8
0.6
0.4
0.2
Bn
0
0
u
ua
1.0
0.5
0
–0.5
–1.0
0 0.4 0.8
0.2
1.2
0.4
1.6 2.0
(t-t⊥)/T
0.6
0.8
n = 1
n = 2
n = 3
n = 4
n = 5
1.0
σ
Fig. 8.8 Growth of the first five spectral components Bn of
a plane wave as a function of the normalized distance σ for
the Fubini solution. The inset gives the waveform at σ = 1
for the source problem, T being the period of the sound
wave. [8.64]
is noted. This means that the shock distance is reached
in the shock formation time when the wave travels at
the linear sound speed c0. This is in agreement with
the earlier observation that the wave travels at the linear
sound speed regardless of the nonlinearity as long as
the wave stays continuous. In the case of the quadratic
approximation the range in space and time for which
this property holds can be quantified explicitly.
To give an impression of what the wave looks like
when having propagated the shock distance x⊥ in the
shock formation time t⊥, Fig.8.5 shows two wavelengths
at the shock formation time t⊥ and Fig. 8.6 shows
two periods at the shock distance x⊥.
A set of spectra of the waveform for different normalized
distances σ is plotted in Fig. 8.7 for the source
problem, where all harmonics have positive value. The
growth of the harmonics Bn (8.55) in the spectrum on
the way to the shock distance is visualized. Similar plots
have been given by Fubini Ghiron [8.65]. A plot of the
first five Fourier coefficients Bn as a function of σ is
given in Fig. 8.8. In the inset the waveform at the shock
distance x⊥ is plotted for two periods of the wave.
The solutions (8.59)and(8.66) are given for the particle
velocity u. This quantity is difficult to measure.
Instead, in experiments, pressure is the variable of
Part B 8.6