# Engineering Acoustics/Qualitative Description of Shocks

## Defining a Shock-Wave

In the general case of mechanical wave propagation it is assumed that
the intensive properties of the medium can be described by continuous
functions of space and time. In the limit where pressure amplitudes
become very large, the wave propagation can evolve such that the wave
front becomes discontinuous and must be described in terms of a jump
from the undisturbed thermodynamic states in front of the wave to the
final thermodynamic states behind the wave.

A propagating disturbance of this type, which generates a discontinuous
change in pressure, temperature, enthalpy, internal energy, density and
particle velocity is referred to as a shock wave. A shock-wave is
depicted schematically in the following figure:

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![](Basic_Shock_Sketch.png "Basic_Shock_Sketch.png"){width="350"}

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A shock-wave can ultimately be interpreted as a transverse mechanical
wave with an undefinable pulse wavelength that discontinuously changes
the state of the medium at a propagation velocity greatly exceeding the
sound speed of the medium.

## Shock Formation and Attenuation

### \"Shocking Up\"

It is most intuitive to consider how a shock wave is formed by
considering the process in an elastic solid as the behaviour can then be
extended in principle to fluids. For a linear-elastic material,
behaviour under compressive loading can be broadly described by two
regimes. In the elastic regime the deformation (strain) is directly
proportional to the stress applied to it. Above a certain critical
stress level (termed the yield stress) the strain is no longer directly
proportional to the stress and the material begins to behave
nonlinearly - this is the plastic regime.

If we define the sound speed in the material as:

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$C^2=\frac{\partial P}{\partial \rho}$

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It is clear that in the elastic regime pressure and density are linearly
related and thus the speed of propagation of a wave is constant if their
pressure amplitudes are below the yield stress of the material. However,
consider a wave whose amplitude is in the regime of pressure beyond the
yield strength as depicted in the following figure:

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![](stressstrainVSwave.png "stressstrainVSwave.png"){width="550"}

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Since the pressure amplitudes are past the regime of linear
proportionality between stress and strain the wave speed is no longer
constant. By consulting the stress-strain curve it is apparent that wave
velocity increases with increasing pressure beyond the elastic limit.
Consequently, point C of the waveform will have the lowest local wave
speed while points B and A will have consecutively increasing wave
speeds. As a result the highest pressure parts of the waveform travel
faster than the lower parts and must eventually overtake them. A time
lapse of this process is depicted as follows:

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![](ShockingUp.png "ShockingUp.png"){width="550"}

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As the smooth wave pulse propagates through the material the
instantaneously faster parts of the wave overtake the slower ones and
the pulse itself becomes increasingly steep until it adopts the
familiar, discontinuous profile associated with a shock-wave.
Consequently any wave with pressure amplitudes greater than the yield
strength of the material will eventually \"shock-up\" and become
discontinuous due to the non-linear increase in wave speed with
increasing pressure.

It is tempting to assume that if we were to play the evolution of the
shock-wave even further in time, the top of the vertical line would
continue to outpace the bottom and the shock wave front would become
sloped. This does not happen in reality due to a competing wave process
that serves to attenuate the shock.

### Shock Rarefaction

Once a shock-wave has established itself in a material it cannot
propagate indefinitely unless it is either driven mechanically via a
piston or self supported via coupled chemical reaction in a detonation
wave. It will be shown that the attenuation and eventual dissipation of
a shock wave is also the natural result of the non-linear relation
between pressure, density and wave speed above the elastic limit.

Consider the following square pulse shock wave:

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![](ShockPulse.png "ShockPulse.png"){width="350"}

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Examine point A\': It is moving into un-shocked material with wave speed
$C_{0}$ and associated particle speed $\nu_{0}$. In contrast, point A is
moving into already shocked material at significantly higher pressure
and density and thus with higher particle velocity, $\nu_{A'}$ and wave
speeds $C_{A'}$. Consequently, point A will be moving significantly
faster than point A\' and will soon overtake the front. Now examine
point C. It has been relieved down to ambient conditions and thus has a
low associated wave speed. It will thus lag progressively further and
further behind point A. As the shock-wave wave propagates, the line A-C
will stretch out and thus angle down. This can be see as an averaging
out of the shock pulse amplitude over a larger and large front
thickness. This averaging serves to attenuate the pulse until the
pressure decays below the elastic limit and the shock-wave devolves into
an acoustic wave. The line A-C is in fact a wave process with a
propagation velocity faster than that of the shock front. Such a wave is
known as a rarefaction and it is a fundamental characteristic of
shock-wave processes. The rarefaction attenuation process is depicted in
the following figure:

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!350

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## Shock Description via the Method of Characteristics

While the previous discussion is extremely intuitive in understanding
shock behaviour, all of these results can be obtained directly via
mathematical solution of the non-linear wave equations through the
method of characteristics and
Riemann Invariants. The method of characteristics is a technique for
solving partial differential
equations by reducing the
PDE to a set of ordinary differential equations through the
parametrization of the existing coordinate system into a new system
where properties of the PDE remain constant over curves in the new
system. The contours revealed during this method are called
characteristics.

Consider the basic set of nonlinear-elastic wave equations:

$$\ \epsilon = \frac{\rho_0}{\rho}$$ strain-density relation

$$\ \sigma = \sigma (\epsilon)$$ constitutive equation

$$\ \frac{\partial \rho}{\partial t} + \nu \frac{\partial \rho}{\partial x} +\rho \frac{\partial \nu}{\partial t} = 0$$
conservation of mass

$$\ \rho \left(\frac{\partial \nu}{\partial t} + \nu \frac{\partial \nu}{\partial x}\right) = \frac{\partial \sigma}{\partial x}$$
conservation of momentum

And characteristic coordinates:

$$\ \xi = \xi(x,t)$$

$$\ \zeta = \zeta(x,t)$$

Subject to the constraints where $\ x = x(X,t)$:

$$\ \frac{\partial \zeta}{\partial t} = C\frac{\partial \zeta}{\partial X}$$

$$\ \frac{\partial \xi}{\partial t} = -C\frac{\partial \xi}{\partial X}$$

Where we define:

$$\ C = \left( \frac{1}{\rho_{0}} \frac {d \sigma}{d \epsilon} \right)^{ \frac{1}{2}}$$

Note that here we have effectively employed the expression for wave
speed as a function of pressure and density!

Yielding the constraint equations:

$$\ \frac{\partial \zeta}{\partial t} = (C \epsilon - \nu)\frac{\partial \zeta}{\partial x}$$

$$\ \frac{\partial \xi}{\partial t} = -(C\epsilon + \nu)\frac{\partial \xi}{\partial x}$$

Combining the constraint equations with the differentials
$d \zeta = \frac{\partial \zeta}{\partial x}dx + \frac{\partial \zeta} {\partial t}d t$
and
$d \xi = \frac{\partial \xi}{\partial x}dx + \frac{\partial \xi}{\partial t}  d t$
yields :

$$d \zeta = \frac{\partial \zeta}{\partial x} [dx - (\nu - C \epsilon)dt]$$

$$d \xi = \frac{\partial \xi}{\partial x} [dx - (\nu + C \epsilon)dt]$$

These equations then yield the slopes of the contours of the new
coordinate system:

$$\ \frac{d x}{d t} = \nu + C\epsilon$$ when $\ d \xi = 0$

$$\ \frac{d x}{d t} = \nu - C\epsilon$$ when $\ d \zeta = 0$

We must now apply these relations to transform the expression of the
wave equations from $\ (x , t)$ space into $\ (\zeta , \xi)$ space. Take
the time derivative of constitutive equation and substitute the slope of
the characteristic lines to obtain:

$$\ \frac{\partial \sigma}{\partial x} = - (C \epsilon)^{2}\frac{\partial \rho}{\partial x}$$

$$\ \frac{\partial \sigma}{\partial t} = - (C \epsilon)^{2}\frac{\partial \rho}{\partial y}$$

For simplicity make the substitution $\ c = C \epsilon$

Substitution into the conservation of mass and momentum equations
yields:

$$\ \frac{\partial \sigma}{\partial t} +\nu \frac{\partial \sigma}{\partial x} = \rho c^{2} \frac{\partial \nu}{\partial x}$$

$$\frac{\partial \nu}{\partial t} + \nu \frac{\partial \nu}{\partial x} = \frac{1}{\rho} \frac{\partial \sigma}{\partial x}$$

These can be combined and solved via partial derivative chain rule
expansion to yield:

$$\ \frac{\partial \nu '}{\partial \zeta} = \frac{1}{\rho c}\frac{\partial \sigma '}{\partial \zeta '}$$

$$\ \frac{\partial \nu '}{\partial \xi} = -\frac{1}{\rho c}\frac{\partial \sigma '}{\partial \xi '}$$

These are the characteristic equations in $\zeta$ and $\xi$ space.

In order to solve for the characteristic contours we must integrate from
some reference to a final state with reference to the invariants, thus:

$$\ J_{+}(\xi) = \nu - \nu^{0} - \int_1^\epsilon{C(\epsilon)\,d\epsilon}$$

$$\ J_{-}(\zeta)= \nu - \nu^{0} + \int_1^\epsilon{C(\epsilon)\,d\epsilon}$$

These two equations represent the Riemann invariants for the wave
system. They can be combined to yield the characteristic equations for
which the combination of stress and particle velocity does not change as
follows:

$$\ \nu - \nu^{0} = \frac{1}{2}(J_{+}(\xi) +J_{-}(\zeta))$$

$$\ \int_1^\epsilon{C(\epsilon)\,d\epsilon} = \frac{1}{2}(-J_{+}(\xi) + J_{-}(\zeta))$$

### Simple Wave Solution

A simple wave is a solution to the wave equations in characteristic
space for which one of the invariants is constant. Consider a non-linear
wave where we set:

$$\ J_{-}(\zeta) = 0$$

This yields stress, strain, particle velocity, and sound speeds that are
solely a function of $\xi$. From the slopes of the coordinate contours
derived previously we obtain:

$$\ \frac{d X}{dt} = \nu(\xi^{i})+c(\xi^{i})$$

Integration of these contours directly yields :

$$\ \xi = t - \frac{x}{\nu(\xi)+c(\xi)}$$

Consequently the simple wave solution of the non linear equations in
characteristic space can described as a transformation of a specified
transversely moving pulse into a collection of straight line
characteristics with differing slopes. Each line can be interpreted as
the x-t history of one specific point on the pulse.

If we compute the Riemann integrals:

$$\ \nu - \nu^{0} = \frac{1}{2}J_{+}\left[ t-\frac{x}{\nu+c}\right]$$

$$\ \int_1^\epsilon{C(\epsilon)\,d\epsilon} = -\frac{1}{2}J_{+}\left[ t-\frac{x}{\nu+c}\right]$$

We obtain the important result:

$$\ \frac{d(v+c)}{d\epsilon} \le 0$$

This inequality mathematically substantiates our previous statement that
wave speed increases with increasing pressure in the non-linear
(plastic) regime and illustrates an important concept: the wave speed is
equal to the sum of the particle velocity and the sound speed.

The relation between simple wave solutions in characteristic space can
be linked to the formation of shock waves through the concept that
characteristics with varying slopes must eventually intersect at some
point in time.

### Shocks From Waves

It can be shown mathematically that the point of intersection of the
contours of a non-linear wave is a mathematical discontinuity, thereby
recovering our concept that shock-waves are state discontinuities

For simplicity sake consider a wave that is a ramp of particle velocity
with respect to position - this is analogous our previous qualitative
example from the first section but with particle velocity plotted
instead of pressure (where the two are related). The equation describing
this ramp is:

$\ \nu = -mx$

Applying our solution for the characteristics of a simple wave we
obtain:

$\ \nu = m[(\nu+c)t -x]$

In order to make this solution tractable we must employ an equation of
state to mathematically link the wave velocity and particle speeds.
Since we have not specified the material in question it is expedient to
simply assume that the relationship is linear, thus:

$$\ c = \lambda u + c_{0}$$

Substituting this into our relationship we obtain for velocity:

$$\nu = \frac{m(c_{0}t-x)}{1-\lambda m t}$$

Clearly when $t = \frac{1}{\lambda m}$, the value for the particle
velocity is undefined and the simple wave solution breaks down and we
have a shock-wave formed. However, when $m$ is negative, the value of
$\nu$ never tends to infinity and we have a rarefaction wave. If we plot
the position history of one point at the bottom of the ramp
(corresponding to the point C in our previous discussion) and one point
at the top of our ramp (corresponding to point A) on an x-t diagram we
can visualize how these characteristics behave:

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![](XTShockvsRare.png "XTShockvsRare.png"){width="700"}

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It is important to note that our mathematical and conceptual discussions
have ultimately yielded analogous descriptions of shock-wave formation
and behaviour. In the mathematical discussion we can see that each
characteristic corresponds to trajectory of a specific point on the
waveform depicted in either $(P,x)$ or $(P,t)$ space. The history of
these characteristics tracks how specific portions of the waveform
overtake or lag behind the others. In the case of a set of converging
characteristics the point of intersection corresponds to a mathematical
singularity and the formation of a shockwave. In the case of diverging
characteristics we can see that the waveform points begin smearing out
in space - this is clearly analogous to the description of rarefaction
waves.

### Strong and Weak Shocks

In the context of the method of characteristics a shock-wave is any
discontinuity produced by the convergence of characteristic lines. A
distinction is made between two types of shock solutions depending on
how they affect the locus of thermodynamic states. A weak shock is
define as the case when the change between final and reference states is
nearly identical to that of the equivalent simple, characteristically
convergent wave. In this case the process by which the change in states
is effected is isentropic and the path through which the material is
loaded from the reference state to the final state is described by the
isentrope.

Conversely a strong shock is the discontinuous solution for which the
locus of all possible states does not coincide with the isentrope but
rather a different loading path.

This discussion of state loci will become more clear with the
introduction of the Hugoniot.

## References

1.  Introduction to Wave Propagation in Nonlinear Fluids and Solids;
    D.S. Drumheller; 1998
2.  Explosives Engineering; Paul W. Cooper; 1996
3.  Shock Waves and Explosions; P.L. Sachdev; 2004

