6.2 Going Further: Nonlinear Propagation
291
propagation distance. At a particular point, called the shock formation point (or
after a distance called the shock formation distance) , the derivative of the pressure
becomes infinite, and the wave has a sawtooth profile.
The nonlinear travelling wave equation obtained can be solved accurately by
the method of Riemann invariants, also called the method of characteristics. In
Sect. 6.1.3 it was noted that for a periodic but not necessarily sinusoidal input
pressure p m , the shock formation distance is
L s
2γp 0 c
[(γ + 1)(∂p m /∂t) max ]
.
(6.3)
For times later than the onset time of the shock, the plot of p versus x derived from
the method of Riemann invariants becomes multivalued, which cannot happen in
reality. In this model without friction or heat conduction, the dilemma is resolved by
using the so-called equal-area rule (see, e.g. Pierce (1989)). The wave profile retains
a sawtooth profile as the propagation increases, its amplitude decreasing towards 0.
6.2.2 The Burgers Equations
The model outlined in Sect. 6.2.1, which ignores friction and heat transfer, predicts
the gradual distortion of the wave and the formation of the sawtooth profile
described as an N-wave. After the shock formation point, the model predicts a
continuous decrease in the amplitude of the propagating shock, despite the fact
that no dissipative mechanism is taken into account (Pierce 1989). Including
viscothermal losses helps in obtaining a model which is closer to reality. The
goal here is to estimate a priori the order of magnitude of competing phenomena:
nonlinear effects and viscothermal losses (Menguy and Gilbert 2000). If losses
dominate they may damp the signal before it has time to distort significantly. In this
case, the context of linear acoustics is sufficient to model the phenomena. Strong
nonlinear phenomena can lead to the formation of a shock wave, but beyond the
shock point, the amplitude of the N-wave may decrease until dissipative effects
dominate over nonlinear distortion. Since the viscothermal losses at walls increase
with frequency, the sawtooth profile is damped and deformed over time, tending
ultimately to a sinusoidal signal of very low amplitude. During the formation of the
shock wave, both the volume viscothermal losses (usually neglected in the pipe) and
viscothermal losses in the boundary layer describe correctly the shock wave shape.
In practice, the angles of the N-wave are rounded by the dissipation. It is the aim of
the following paragraphs to explore this using the Burgers equations (Hamilton and
Blackstock 1998).
The exact analytical solution of the equations of nonlinear acoustics including
losses is not possible. An approximate way to tackle the problem is to use a
perturbation method: the ‘method of multiple scales’ (see, e.g. Menguy and Gilbert
(2000)). This method is based on the presence in the equation of a parameter small
291
propagation distance. At a particular point, called the shock formation point (or
after a distance called the shock formation distance) , the derivative of the pressure
becomes infinite, and the wave has a sawtooth profile.
The nonlinear travelling wave equation obtained can be solved accurately by
the method of Riemann invariants, also called the method of characteristics. In
Sect. 6.1.3 it was noted that for a periodic but not necessarily sinusoidal input
pressure p m , the shock formation distance is
L s
2γp 0 c
[(γ + 1)(∂p m /∂t) max ]
.
(6.3)
For times later than the onset time of the shock, the plot of p versus x derived from
the method of Riemann invariants becomes multivalued, which cannot happen in
reality. In this model without friction or heat conduction, the dilemma is resolved by
using the so-called equal-area rule (see, e.g. Pierce (1989)). The wave profile retains
a sawtooth profile as the propagation increases, its amplitude decreasing towards 0.
6.2.2 The Burgers Equations
The model outlined in Sect. 6.2.1, which ignores friction and heat transfer, predicts
the gradual distortion of the wave and the formation of the sawtooth profile
described as an N-wave. After the shock formation point, the model predicts a
continuous decrease in the amplitude of the propagating shock, despite the fact
that no dissipative mechanism is taken into account (Pierce 1989). Including
viscothermal losses helps in obtaining a model which is closer to reality. The
goal here is to estimate a priori the order of magnitude of competing phenomena:
nonlinear effects and viscothermal losses (Menguy and Gilbert 2000). If losses
dominate they may damp the signal before it has time to distort significantly. In this
case, the context of linear acoustics is sufficient to model the phenomena. Strong
nonlinear phenomena can lead to the formation of a shock wave, but beyond the
shock point, the amplitude of the N-wave may decrease until dissipative effects
dominate over nonlinear distortion. Since the viscothermal losses at walls increase
with frequency, the sawtooth profile is damped and deformed over time, tending
ultimately to a sinusoidal signal of very low amplitude. During the formation of the
shock wave, both the volume viscothermal losses (usually neglected in the pipe) and
viscothermal losses in the boundary layer describe correctly the shock wave shape.
In practice, the angles of the N-wave are rounded by the dissipation. It is the aim of
the following paragraphs to explore this using the Burgers equations (Hamilton and
Blackstock 1998).
The exact analytical solution of the equations of nonlinear acoustics including
losses is not possible. An approximate way to tackle the problem is to use a
perturbation method: the ‘method of multiple scales’ (see, e.g. Menguy and Gilbert
(2000)). This method is based on the presence in the equation of a parameter small