6.2 Going Further: Nonlinear Propagation
289
dρ
dp
ρv
dv
dp
+ 1
−
d(ρv)
dp
ρ
dv
dp
= 0 −→
dρ
dp
− ρ
2
dv
dp
2
= 0.
(6.13)
Two solutions emerge, for waves travelling in the ±x direction, with
dv
dp
= ±
1
ρ
dρ
dp
1/2
.
(6.14)
We can use the conservation of entropy, since in the present discussion, friction
and heat transfer have been neglected. At constant entropy,
dp
dρ
= c
2 ,
(6.15)
(see, e.g. (Pierce 1989, Sect. 1–5). Equation 6.14 can thus be rewritten as
dv
dp
= ±
1
ρc
.
(6.16)
For the case of a forward travelling wave, described in the nonlinear acoustics
literature as a simple wave, some further algebraic operations lead to the nonlinear
partial differential equation (Earnshaw 1860):
∂p
∂t
+ (c + v)
∂p
∂x
= 0.
(6.17)
This equation demonstrates the effect of convection (fluid entrainment). To a
fixed observer, a given point on the wave (such as a crest) travels not at the local
speed of sound c, but at a speed
dx
dt
= c + v
;
(6.18)
the wave crests are therefore speeded up and the troughs slowed down.
The rate of nonlinear distortion is increased by variations in the local speed of
sound c due to the variations in acoustic pressure. In a region of compression, the
temperature rises, and c increases, while in a region of expansion, the temperature
falls, and c decreases. An expression for c can be derived from the equation relating
pressure and density in a perfect adiabatic fluid (Kinsler et al. 1999), which is itself
nonlinear:
c
2
=
dp
dρ
=
γp
ρ
= γ
p 0
ρ
γ
0
ρ
γ −1 ,
(6.19)
where γ is the ratio of heat capacities. For a calorically perfect gas, γ is constant
(equal to 1.4 for air at room conditions), and
289
dρ
dp
ρv
dv
dp
+ 1
−
d(ρv)
dp
ρ
dv
dp
= 0 −→
dρ
dp
− ρ
2
dv
dp
2
= 0.
(6.13)
Two solutions emerge, for waves travelling in the ±x direction, with
dv
dp
= ±
1
ρ
dρ
dp
1/2
.
(6.14)
We can use the conservation of entropy, since in the present discussion, friction
and heat transfer have been neglected. At constant entropy,
dp
dρ
= c
2 ,
(6.15)
(see, e.g. (Pierce 1989, Sect. 1–5). Equation 6.14 can thus be rewritten as
dv
dp
= ±
1
ρc
.
(6.16)
For the case of a forward travelling wave, described in the nonlinear acoustics
literature as a simple wave, some further algebraic operations lead to the nonlinear
partial differential equation (Earnshaw 1860):
∂p
∂t
+ (c + v)
∂p
∂x
= 0.
(6.17)
This equation demonstrates the effect of convection (fluid entrainment). To a
fixed observer, a given point on the wave (such as a crest) travels not at the local
speed of sound c, but at a speed
dx
dt
= c + v
;
(6.18)
the wave crests are therefore speeded up and the troughs slowed down.
The rate of nonlinear distortion is increased by variations in the local speed of
sound c due to the variations in acoustic pressure. In a region of compression, the
temperature rises, and c increases, while in a region of expansion, the temperature
falls, and c decreases. An expression for c can be derived from the equation relating
pressure and density in a perfect adiabatic fluid (Kinsler et al. 1999), which is itself
nonlinear:
c
2
=
dp
dρ
=
γp
ρ
= γ
p 0
ρ
γ
0
ρ
γ −1 ,
(6.19)
where γ is the ratio of heat capacities. For a calorically perfect gas, γ is constant
(equal to 1.4 for air at room conditions), and