Introduction to Acoustics

number, Pr = µcp/κ. The equation is an approximation
to a more exact second-order equation that, however,
does not lend itself to an exact solution as does the
Burgers equation above. In the context of acoustics this
relation was first derived by Mendousse, although for
viscous losses only [8.70]. The derivations make use of
a careful comparison of the order of magnitude of derivatives
retaining the leading terms. The above form of the
Burgers equation is best suited to initial-value problems.
Source problems are best described when transforming
(8.75) to new coordinates (x,τ) with the retarded time
τ = t − x/c0:
∂u β
−
∂x c2 u
0
∂u δ
=
∂τ 2c3 ∂
0
2u . (8.77)
∂τ2 The equation can be normalized to a form with only
one parameter:
∂W
∂σ
− W ∂W
∂ϕ
= 1
Γ
∂2W , (8.78)
∂ϕ2 where W = u/u0, σ = x/x⊥, ϕ = ωτ = ωt − k0x,
Γ = βM0k0/α = 2πβM0/αλ, with α being the damping
constant for linear waves:
α = δk2 0
. (8.79)
2c0
Γ is called the Gol’dberg number after Gol’dberg
[8.71] who introduced this normalization (Blackstock
[8.72]). It can be written as
Γ = 1/x⊥
α
Nonlinear Acoustics in Fluids 8.7 Thermoviscous Finite-Amplitude Acoustic Waves 269
(8.80)
where 1/x⊥ = βM0k0 is the strength of the nonlinearity
and α is the strength of the damping. The Gol’dberg
number is therefore a measure of the importance of nonlinearity
in relation to damping. For Γ>1 nonlinearity
takes over and for Γ 0 ,
or in terms of the variable W(σ,ϕ)
W(0,ϕ) = 0 for ϕ ≤ 0 ,
W(0,ϕ) = sin ϕ for ϕ>0 .
(8.87)
To solve the heat conduction equation the boundary
condition is needed for ζ(σ,ϕ). To this end the Hopf–
Cole transformation (8.81)isreversed:
�
Γ
ζ(σ,ϕ) = exp
2
�ϕ
−∞
W(σ,ϕ ′ )dϕ ′
�
. (8.88)
Insertion of W(0,ϕ) yields as the boundary condition
for ζ:
ζ(0,ϕ) = 1 for ϕ≤ 0
ζ(0,ϕ) = e 1 2 Γ (1−cos ϕ)
for ϕ> 0
(8.89)
and insertion into (8.85) yields the solution in terms of
ζ. Using ¯σ = √ 4σ/Γ and q = (ϕ ′ − ϕ)/ ¯σ the solution
for the vibrating piston in terms of ζ reads
ζ(σ,ϕ) = 1
−ϕ/ � ¯σ
√ e
π
−q2
−∞
+ 1
√ π e 1 2 Γ
�∞
dq
−ϕ/ ¯σ
e − 1 2 Γ cos( ¯σq+ϕ) e −q2
dq .
(8.90)
Part B 8.7