288
6 Shocks and Surprises: Refining the Elementary Model
the small-amplitude approximation used to derive the linear acoustic wave equation
is not valid, and it is necessary to return to the fundamental fluid dynamic equations.
6.2.1 From the Fundamental Fluid Dynamic Equations to the
Nonlinear Wave Propagation Equation
A criterion for judging the relevance of the linear approximation is that the
dimensionless acoustic Mach number M = v /c 0 is much smaller than unity. Here
v is the acoustic velocity amplitude (the medium being assumed to be quiescent),
and c 0 is the speed of propagation in the linear approximation. For M 1, we
are in the strongly nonlinear regime, and the approach through linearisation of the
fundamental equations fails (Hamilton and Blackstock 1998). Even for M 1,
however, it is possible to observe very pronounced nonlinear distortion effects.
These effects are locally negligible: over distances small compared to a wavelength,
it is still possible, for example, to make the assumption that in the case of a travelling
plane wave, p = ρcv . However, the nonlinear effects are cumulative over distances
of the order of several wavelengths and can generate strongly distorted waveforms
and even the pressure discontinuities called shock waves. This is the weakly
nonlinear regime which we consider here.
We start our analysis by considering plane wave propagation in a tube with
uniform diameter. The mass conservation and force equations in one dimension,
assuming an ideal fluid without friction or heat conduction, have quadratic nonlinear
terms:
∂ρ
∂t
+
∂(ρv)
∂x
=
∂ρ
∂t
+ v
∂ρ
∂x
+ ρ
∂v
∂x
= 0,
(6.9)
ρ
∂v
∂t
+ v
∂v
∂x
= −
∂p
∂x
.
(6.10)
In the above equations and the derivations which follow, p = p 0 + p , ρ =
ρ 0 + ρ , and v = v 0 + v , the variables including both mean values (subscript 0) and
fluctuating values (primed).
The two equations above yield the system of equations:
dρ
dp
∂p
∂t
+
d(ρv)
dp
∂p
∂x
= 0,
(6.11)
ρ
dv
dp
∂p
∂t
+
ρv
dv
dp
+ 1
∂p
∂x
= 0.
(6.12)
These equations have non-trivial solutions for ∂p/∂t and ∂p/∂x if the determinant of the system of equations is zero:
6 Shocks and Surprises: Refining the Elementary Model
the small-amplitude approximation used to derive the linear acoustic wave equation
is not valid, and it is necessary to return to the fundamental fluid dynamic equations.
6.2.1 From the Fundamental Fluid Dynamic Equations to the
Nonlinear Wave Propagation Equation
A criterion for judging the relevance of the linear approximation is that the
dimensionless acoustic Mach number M = v /c 0 is much smaller than unity. Here
v is the acoustic velocity amplitude (the medium being assumed to be quiescent),
and c 0 is the speed of propagation in the linear approximation. For M 1, we
are in the strongly nonlinear regime, and the approach through linearisation of the
fundamental equations fails (Hamilton and Blackstock 1998). Even for M 1,
however, it is possible to observe very pronounced nonlinear distortion effects.
These effects are locally negligible: over distances small compared to a wavelength,
it is still possible, for example, to make the assumption that in the case of a travelling
plane wave, p = ρcv . However, the nonlinear effects are cumulative over distances
of the order of several wavelengths and can generate strongly distorted waveforms
and even the pressure discontinuities called shock waves. This is the weakly
nonlinear regime which we consider here.
We start our analysis by considering plane wave propagation in a tube with
uniform diameter. The mass conservation and force equations in one dimension,
assuming an ideal fluid without friction or heat conduction, have quadratic nonlinear
terms:
∂ρ
∂t
+
∂(ρv)
∂x
=
∂ρ
∂t
+ v
∂ρ
∂x
+ ρ
∂v
∂x
= 0,
(6.9)
ρ
∂v
∂t
+ v
∂v
∂x
= −
∂p
∂x
.
(6.10)
In the above equations and the derivations which follow, p = p 0 + p , ρ =
ρ 0 + ρ , and v = v 0 + v , the variables including both mean values (subscript 0) and
fluctuating values (primed).
The two equations above yield the system of equations:
dρ
dp
∂p
∂t
+
d(ρv)
dp
∂p
∂x
= 0,
(6.11)
ρ
dv
dp
∂p
∂t
+
ρv
dv
dp
+ 1
∂p
∂x
= 0.
(6.12)
These equations have non-trivial solutions for ∂p/∂t and ∂p/∂x if the determinant of the system of equations is zero: