292
6 Shocks and Surprises: Refining the Elementary Model
compared to unity. A suitable parameter in the present case is a Mach number
defined in the vicinity of the acoustic source of the system. This number serves as
a basis for the definition of ‘fast’ and ‘slow’ scales. The fast scale, which is a short
spatial scale, is a dimensionless delayed time θ that describes the wave propagation
locally. The slow scale is a long spatial scale, represented by the variable σ = x/L s ,
that represents the cumulative effects of nonlinear distortion and losses.
After some mathematical operations which will not be detailed here, the weakly
nonlinear lossless wave equation, applied to a function q which is one of the
dimensionless acoustic parameters, takes the form
∂q
∂σ
− q
∂q
∂θ
= 0.
(6.26)
For a sine wave input, the above equation has an exact solution for σ < 1 given by
Fubini (see, e.g. Pierce (1989)):
q(σ, θ) =
∞
n=1
Q n (σ ) sin(nθ ),
(6.27)
with
Q n (σ ) = 2
J n (nσ )
nσ
,
(6.28)
where J n is the Bessel function of order n. The Fubini approximation is able
to describe the harmonic cascade phenomenon which occurs before shock wave
formation (see Sect. 6.1.3). After the shock formation, for a distance more than
three times the shock length distance, the behaviour of the harmonic components
corresponding to a N-wave is well represented by the Fay-Blackstock approximation
(Hamilton and Blackstock 1998). The analytic Fourier coefficients Q n are written
as follows:
Q n (σ ) = 2A
1
sinh(n(1 + σ )A)
.
(6.29)
These two asymptotic behaviours are recovered by the solutions of the generalised
Burgers equations discussed below (see Fig. 6.15).
The original ‘Burgers equation’ was developed by the Dutch physicist J.M. Burgers in his work on turbulence. Burgers included the effect of volume viscothermal
losses on plane wave propagation by adding a term to the right-hand side of Eq. 6.26.
Generalised Burgers equations are extensions of the basic Burgers equation which
take additional physical phenomena into account. As explained in Sect. 4.1, for air
in tubes, wall losses are more important than volumic losses. Wall losses display
a square root frequency dependence, which is equivalent to a fractional derivative
6 Shocks and Surprises: Refining the Elementary Model
compared to unity. A suitable parameter in the present case is a Mach number
defined in the vicinity of the acoustic source of the system. This number serves as
a basis for the definition of ‘fast’ and ‘slow’ scales. The fast scale, which is a short
spatial scale, is a dimensionless delayed time θ that describes the wave propagation
locally. The slow scale is a long spatial scale, represented by the variable σ = x/L s ,
that represents the cumulative effects of nonlinear distortion and losses.
After some mathematical operations which will not be detailed here, the weakly
nonlinear lossless wave equation, applied to a function q which is one of the
dimensionless acoustic parameters, takes the form
∂q
∂σ
− q
∂q
∂θ
= 0.
(6.26)
For a sine wave input, the above equation has an exact solution for σ < 1 given by
Fubini (see, e.g. Pierce (1989)):
q(σ, θ) =
∞
n=1
Q n (σ ) sin(nθ ),
(6.27)
with
Q n (σ ) = 2
J n (nσ )
nσ
,
(6.28)
where J n is the Bessel function of order n. The Fubini approximation is able
to describe the harmonic cascade phenomenon which occurs before shock wave
formation (see Sect. 6.1.3). After the shock formation, for a distance more than
three times the shock length distance, the behaviour of the harmonic components
corresponding to a N-wave is well represented by the Fay-Blackstock approximation
(Hamilton and Blackstock 1998). The analytic Fourier coefficients Q n are written
as follows:
Q n (σ ) = 2A
1
sinh(n(1 + σ )A)
.
(6.29)
These two asymptotic behaviours are recovered by the solutions of the generalised
Burgers equations discussed below (see Fig. 6.15).
The original ‘Burgers equation’ was developed by the Dutch physicist J.M. Burgers in his work on turbulence. Burgers included the effect of volume viscothermal
losses on plane wave propagation by adding a term to the right-hand side of Eq. 6.26.
Generalised Burgers equations are extensions of the basic Burgers equation which
take additional physical phenomena into account. As explained in Sect. 4.1, for air
in tubes, wall losses are more important than volumic losses. Wall losses display
a square root frequency dependence, which is equivalent to a fractional derivative