# Engineering Acoustics/Reflection, transmission and refraction of planar waves

## Two-dimensional planar waves

Two-dimensional planar pressure waves can be described in Cartesian
coordinates by decomposing the wave number into x and  y components,

$\mathbf{p}(x,y,t)=\mathbf{P}e^{j(\omega t -K_x x-K_y y)}.$

Substituting into the general wave equation yields:

$\nabla^2 \mathbf{p}- \frac{1}{c_o^2} \frac{\partial^2 \mathbf{p}}{\partial t^2}=0,$

$\mathbf{P}(-K_x^2-K_y^2)+\frac{\omega^2}{c_o^2}\mathbf{P}=0,$

$K=\frac{\omega}{c_o}=\sqrt{K_x^2+K_y^2}.$

The wave number becomes a vector quantity and may be expressed using the
directional cosines,

$\vec{K}=K_x \boldsymbol{\hat{\imath}} + K_y \boldsymbol{\hat{\jmath}} = K \cos(\alpha) \boldsymbol{\hat{\imath}} + K \cos(\beta) \boldsymbol{\hat{\jmath}}.$

## Obliquely incident planar waves

Consider an obliquely incident planar wave in medium 1 which approaches
the boundary at an angle $\theta_i$ with respect to the normal. Part of
the wave is reflected back into medium 1 at an angle $\theta_r$ and the
remaining part is transmitted to medium 2 at an angle $\theta_t$.

$\mathbf{p_1}=\mathbf{P_i}e^{j(\omega t - \cos\theta_i K_1 x - \sin\theta_i K_1 y)} + \mathbf{P_r}e^{j(\omega t + \cos\theta_r K_1 x - \sin\theta_r K_1 y)}$

$\mathbf{p_2}=\mathbf{P_t}e^{j(\omega t - \cos\theta_t K_2 x - \sin\theta_t K_2 y)}$
!Reflection and transmission of obliquely incident planar
wave.\|440x440px
Notice that the wave frequency does not change across the boundary,
however the specific acoustic impedance does change from medium 1 to
medium 2. The propagation speed  is different in each medium, so the
wave number changes across the boundary. There are two boundary
conditions to be satisfied.

1.  The acoustic pressure must be continuous at the boundary.
2.  The particle velocity component normal to the boundary must be
    continuous at the boundary.

Imposition of the first boundary condition yields

$\mathbf{p_1}(x=0)=\mathbf{p_2}(x=0),$

$\mathbf{P_i}e^{-j \sin\theta_i K_1 y} + \mathbf{P_r} e^{-j \sin\theta_r K_1 y}= \mathbf{P_t} e^{-j \sin\theta_t K_2 y}.$

For continuity to hold, the exponents must be all equal to each other

$K_1 \sin\theta_i =K_1 \sin\theta_r=K_2 \sin\theta_t.$

This has two implications. First, the angle of incident waves is equal
to the angle of reflected waves,

$\sin\theta_i = \sin\theta_r$

and second, Snell\'s law is recovered,

$\frac{\sin\theta_i}{c_1}=\frac{\sin\theta_t}{c_2}.$

The first boundary condition can be expressed using the pressure
reflection and transmission coefficients

$1+\mathbf{R}=\mathbf{T}.$

Imposition of the second boundary condition yields

$\mathbf{u_{1x}}(x=0)=\mathbf{u_{2x}}(x=0),$

$\mathbf{u_i}\cos\theta_i+ \mathbf{u_r}\cos\theta_r= \mathbf{u_t}\cos\theta_t.$

Using the specific acoustic impedance definition yields

$\frac{\mathbf{P_i}}{r_1}\cos\theta_i- \frac{\mathbf{P_r}}{r_1}\cos\theta_r= \frac{\mathbf{P_t}}{r_2}\cos\theta_t.$

Using the reflection coefficient, the transmission coefficient and the
acoustic impedance ratio leads to

$1- \mathbf{R}= \frac{\cos\theta_t}{\cos\theta_i}\frac{\mathbf{T}}{\zeta}.$

Solving for the pressure reflection coefficient yields:

$\mathbf{R}=\mathbf{T}-1=\frac{\frac{\cos\theta_i}{\cos\theta_t}\zeta-1}{\frac{\cos\theta_i}{\cos\theta_t}\zeta+1}=\frac{\frac{r_2}{\cos\theta_t}-\frac{r_1}{\cos\theta_i}}{\frac{r_2}{\cos\theta_t}+\frac{r_1}{\cos\theta_i}}.$

Solving for the pressure transmission coefficient yields:

$\mathbf{T}=\mathbf{R}+1=\frac{2 \frac{\cos\theta_i}{\cos\theta_t}\zeta}{\frac{\cos\theta_i}{\cos\theta_t}\zeta +1}=\frac{2\frac{r_2}{\cos\theta_t}}{\frac{r_2}{\cos\theta_t}+\frac{r_1}{\cos\theta_i}}.$

Solving for the specific acoustic impedance ratio yields

$\zeta = \frac{\cos\theta_t}{\cos\theta_i}\Big(\frac{1+\mathbf{R}}{1-\mathbf{R}}\Big) = \frac{\cos\theta_t}{\cos\theta_i}\Big(\frac{\mathbf{T}}{2-\mathbf{T}}\Big) .$

## Rayleigh reflection coefficient

The Rayleigh reflection coefficient relates the angle of incidence from
Snell\'s law to the angle of transmission in the equations for
$\mathbf{R}$, $\mathbf{T}$ and $\zeta$. From the trigonometric identity,

$\cos^2\theta_t+\sin^2\theta_t=1$

and using Snell\'s law,

$\cos\theta_t=\sqrt{1-\Big( \frac{c_2}{c_1}\sin\theta_i \Big)^2}.$

Notice that for the angle of transmission to be real,

$c_2<\frac{c_1}{\sin\theta_i}$

must be met. Thus, there is a critical angle of incidence such that

$\sin{\theta_c}=\frac{c_1}{c_2}.$

The Rayleigh reflection coefficient are substituted back into the
equations for $\mathbf{R}$, $\mathbf{T}$ and $\zeta$ to obtain
expression only in term of impedance and angle of incidence.

$\mathbf{R}==\frac{\cos\theta_i\zeta-\sqrt{1-\Big( \frac{c_2}{c_1}\sin\theta_i \Big)^2}}{\cos\theta_i\zeta+\sqrt{1-\Big( \frac{c_2}{c_1}\sin\theta_i \Big)^2}}$

$\mathbf{T}=\frac{2 \cos\theta_i\zeta}{\cos\theta_i\zeta +\sqrt{1-\Big( \frac{c_2}{c_1}\sin\theta_i \Big)^2}}$

$\zeta = \frac{\sqrt{1-\Big( \frac{c_2}{c_1}\sin\theta_i \Big)^2}}{\cos\theta_i}\Big(\frac{1+\mathbf{R}}{1-\mathbf{R}}\Big) = \frac{\sqrt{1-\Big( \frac{c_2}{c_1}\sin\theta_i \Big)^2}}{\cos\theta_i}\Big(\frac{\mathbf{T}}{2-\mathbf{T}}\Big) .$


# Engineering Acoustics/Wave Motion in Elastic Solids

## Wave types

In an infinite medium, two different basic wave types, dilatational and
distortional, can propagate in different propagation velocities.
Dilatational waves cause a change in the volume of the medium in which
it is propagating but no rotation; while distortional waves involve
rotation but no volume changes. Having displacement field, strain and
stress fields can be determined as consequences.

`                                                       Figure 1: Dilatational wave`

`                                                       Figure 1: Distortional wave`

## Elasticity equations

Elasticity equations for homogeneous isotropic elastic solids which are
used to derive wave equations in Cartesian tensor notation are

**Conservation of momentum**

$$\tau_{ij,j} + \rho f_i = \rho{ \ddot{u_i}} ,\ (1)$$

**Conservation of moment of momentum**

$$\tau_{ij} = \tau_{ji} ,\ (2)$$

**Constitutive equations (which relate states of deformation with states
of traction)**

$$\tau_{ij,j} = \lambda \epsilon_{kk}\delta_{ij} + 2 \mu \epsilon_{ij} ,\ (3)$$

**Strain-displacement relations**

$$\epsilon_{ij} = {1 \over 2}(u_{i,j}+u_{j,i}) ,\ (4a)$$

$$\omega_{ij} = {1 \over 2}(u_{i,j}-u_{j,i}) ,\ (4b)$$

in which $\scriptstyle\tau$ is the stress tensor, $\scriptstyle\rho$ is
the solid material density, and $\scriptstyle\mathbf{u}$ is the vector
displacement. $\scriptstyle f$ is body force, $\scriptstyle\lambda$ and
$\scriptstyle\mu$ are Lame constants. $\scriptstyle\epsilon$ and
$\scriptstyle\omega$ are strain and rotation tensors.

## Wave equations in infinite media

Substituting Eq. (4) in Eq. (3), and the result into Eq. (1) gives
Navier's equation (governing equations in terms of displacement) for the
media

$$( \lambda + 2\mu )u_{j,ji} + \mu u_{i,jj}+ \rho f_i = \rho{ \ddot{u_i}} .\ (5)$$

The displacement equation of motion for a homogeneous isotropic solid in
the absence of body forces may be expressed as

$$( \lambda + 2\mu )\nabla(\nabla \cdot \mathbf{u}) + \mu\nabla^2\mathbf{u} = \rho{ \ddot{\mathbf{u}}} .\ (6)$$

Displacement can advantageously be expressed as sum of the gradient of a
scalar potential and the curl of a vector potential

$$\mathbf{u} = \nabla \phi\ + \nabla \times\psi ,\ (7)$$

with the condition $\nabla \cdot \psi =0$. The above equation is called
Helmholtz (decomposition) theorem in which $\scriptstyle\phi$ and
$\scriptstyle\psi$ are called scalar and vector displacement potentials.
Substituting Eq. (7) in Eq. (6) yields

$$[( \lambda + 2\mu )\nabla^2 \phi\ - \rho\frac{\partial^2 \phi}{\partial t^2}]+\nabla \times\ [\mu\nabla^2\psi - \rho\frac{\partial^2 \psi}{\partial t^2}] =0 .\ (8)$$

Equation (8) is satisfied if

$$c_p^2 \nabla^2 \phi =\frac{\partial^2 \phi}{\partial t^2}$$ where
$c_p^2 =\frac{( \lambda + 2\mu )}{\rho} ,\ (9a)$

$$c_s^2 \nabla^2 \psi =\frac{\partial^2 \psi}{\partial t^2}$$ where
$c_s^2 =\frac{\mu }{\rho} .\ (9b)$

Equation (9a) is a dilatational wave equation with the propagation
velocity of $\scriptstyle c_p$. It means that dilatational disturbance,
or a change in volume propagates at the velocity $\scriptstyle c_p$. And
Eq. (9b) is a distortional wave equation; so distortional waves
propagate with a velocity $\scriptstyle c_s$ in the medium. Distortional
waves are also known as rotational, shear or transverse waves.

It is seen that these wave equations are simpler than the general
equation of motion. Therefore, potentials can be found from Eq. (9) and
the boundary and initial conditions, and then the solution for
displacement will be concluded from Eq. (7).

## References

\[1\] Wave Motion in Elastic Solids; Karl F. Graff, Ohio State
University Press, 1975.

\[2\] The Diffraction of Elastic Waves and Dynamic Stress Concentration;
Chao-chow Mow, Yih-Hsing Pao, 1971.


# 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


# Engineering Acoustics/The Rankine-Hugoniot Jump Equations

## Conservation Equations and Derivation

For the purpose of performing engineering calculations, equations
linking the pre- and post- shock states are required. One of the most
fundamental expressions relating states is the $(P,\rho)$ Hugoniot which
relates pressure and density as:

$\ P = f(\rho)$

This expression can be derived via simplification of the canonical
conservation equations:

Conservation of mass:

$$\rho_1 \nu_1=\rho_o \nu_o\,$$

Conservation of momentum:

$$p_1+\rho_1\nu_1^2=p_o+\rho_o \nu_o^2$$

Conservation of energy:

$$e_1+\frac{p_1}{\rho_1}+\frac{1}{2}\nu_{1}^2=e_o+\frac{p_o}{\rho_o}+\frac{1}{2}\nu_{o}^2$$

The parameters of a shock required to completely solve for the jump
conditions are pressure, particle velocity,specific internal
energy,density and shock speed. With 4 state variables and only 3
equations an additional equation is required to relate some of the
states and make the problem tractable. This equation is referred to as
an equation of state (EOS) - of which many exist for a variety of
applications. The most common EOS is the ideal gas law and can be used
to reduce the system of equations to the familiar Hugoniot expression
for fluids with constant specific heats in steady flow:

$$\frac{p_1}{p_2}=
\frac{(\gamma+1)-(\gamma-1)\frac{\rho_2}{\rho_1}}
{(\gamma+1)\frac{\rho_2}{\rho_1}-(\gamma-1)}$$

For general, non-linear elastic materials there exists no equation of
state that can be derived from first principles. However, a huge
database of experimental data has revealed that virtually all materials
display a linear relationship between particle velocity and shock speed
(the voracity of the linear assumption in the method of characteristics
example is now even more clear!):

$$\ U = C_{o} + s\nu$$

This equation is also known as the shock Hugoniot in the $U-\nu$ plane.

Combination of this linear relation with the momentum and mass equations
yields the desired expression for the Hugoniot in the $P-\rho$ plane for
virtually all solid materials:

$\ P = C_{o}^2\frac{\frac{1}{\rho_{0}}-\frac{1}{\rho}}{\left(\frac{1}{\rho_{0}}-\frac{s}{\rho}\right)^2}$

## Paths and Jump Conditions

The Hugoniot describes the locus of all possible thermodynamic states a
material can exist in behind a shock, projected onto a two dimensional
state-state plane. It is therefore a set of equilibrium states and does
not specifically represent the path through which a material undergoes
transformation.

Consider again our discussion of strong and weak shocks. It was said
that weak shocks are isentropic and that the isentrope represents the
path through which the material is loaded from the initial to final
states by an equivalent wave with converging characteristics (termed a
compression wave). In the case of weak shocks, the Hugoniot will
therefore fall directly on the isentrope and can be used directly as the
equivalent path.

In the case of a strong shock we can no longer make that simplification
directly, howevewer for engineering calculations it is deemed that the
isentrope is close enough to the Hugoniot that the same assumption can
be made.

If the Hugoniot is approximately the loading path between states for an
\"equivalent\" compression wave, then the jump conditions for the shock
loading path can be determined by drawing a straight line between the
initial and final states. This line is called the Rayleigh line and has
the following equation:

$\ P_1 - P_0 = U^2\left(\rho_0 - \frac{\rho_0^2}{\rho_1}\right)$

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

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## References


# Engineering Acoustics/Detonation

## Detonation

A detonation wave is a combustion wave propagating at supersonic speeds.
It is composed of a leading shock which adiabatically compresses the
reactants, followed by a reaction zone which converts the reactants into
products. During this process, a significant amount of heat is released,
increasing the temperature and decreasing the pressure and density. The
products are expanded in the reaction zone, giving the detonation a
forward thrust.

In contrast, a deflagration wave, which can be thought of as a
propagating flame, is a combustion wave which propagates at subsonic
speeds.A deflagration consist of a precursor shock followed by a
reaction zone. A deflagration propagates via heat and mass diffusion
from the reaction zone to ignite the reactants ahead of it. Being a
subsonic wave, information downstream can travel upstream and change the
initial thermodynamic state of the reactants.

A qualitative difference between these two combustion modes are
tabulated below.

                                  Detonation   Deflagration
  ------------------------------- ------------ --------------
  Mach number $u_0/c_0$           5-10         0.00001-0.03
  Velocity ratio $u_1/u_0$        0.4-0.7      4-6
  Pressure ratio $p_1/p_0$        13-55        0.98
  Temperature ratio $T_1/T_0$     8-21         4-16
  Density ratio $\rho_1/\rho_0$   1.7-2.6      0.06-0.25

According to this table, the burned products (with respect to a
stationary combustion wave) experience a deceleration across a
detonation wave, and an acceleration in a deflagration wave. The
pressure and density rise across the detonation which is why a
detonation wave is known as a compression wave. In contrast, the
pressure decreases slightly across a deflagration, hence it is
considered an expansion wave.

## Rayleigh line and Hugoniot curve

Let's first treat the detonation as a black box that brings reactants of
state 0 to products of state 1.

\[\[Image:1d steady flow across combustion
wave.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 1: 1-D steady flow across a combustion wave

```{=html}
<center>
```
\]\]

The basic conservation equations are applied to relate state 0 to
state 1. Considering a one dimension steady flow across the combustion
wave, the basic equations are:

Conservation of mass:

`<big>`{=html}`<big>`{=html}$\rho_0 u_0=\rho_1 u_1$`</big>`{=html}`</big>`{=html}

Conservation of momentum:

$p_0+\rho_0 u_0^2=p_1+\rho_1 u_1^2$

Conservation of energy:

$h_0+q+\frac{u_0^2}{2}=h_1+\frac{u_1^2}{2}$

where ρ, p,u,h and q are the density, pressure, velocity, enthalpy and
the difference between the enthalpies of formation of the reactants and
the products, respectively.

Combining the conservation of mass and momentum, the following equation
is obtained

$(p_1 - p_0)/(\nu_0 - \nu_1)=\rho_0^2u_0^2=\rho_1^2u_1^2= \dot{m^2}$,

where $\nu$ is the specific volume and $\dot{m}=\rho u$ is the mass flux
per unit area. The mass flux can also be written as

$\dot{m} = \sqrt{(p_1 - p_0)/(\nu_0 - \nu_1)}$.

Remember that the mass flux must be a real number. Therefore, if the
numerator is positive so must the denominator and vice versa.

If we define `<big>`{=html}$y= p_1 / p_0$`</big>`{=html} and
`<big>`{=html} $x= \nu_1 / \nu_0$`</big>`{=html} then we can obtain the
following equation

$\dot{m^2} = (y - 1)/(1 - x)$

Substituting expressions for the speed of sound of the reactants
`<small>`{=html}$c_0= \sqrt{\gamma_0p_0/\rho_0}$`</small>`{=html} and
the Mach number of the combustion wave
`<big>`{=html}$M= u_0/c_0$`</big>`{=html}, the above equation can be
re-casted as

$y= (1+\gamma_0M_0^2)- (\gamma_0M_0^2)x$

The above equation defining the thermodynamic path linking state 0 to
state 1is also known as the Rayleigh line. Isolating $u_0^2$ or $u_1^2$
from the following equation

$(p_1 - p_0)/(\nu_0 - \nu_1)=\rho_0^2u_0^2=\rho_1^2u_1^2= \dot{m^2}$

we can eliminate the velocity terms from the energy equation and obtain
the equation of the Hugoniot
curve,

$h_1-(h_0+q)=\frac{1}{2} (p_1-p_0)(\nu_0-\nu_1)$.

The Hugoniot curve represents the locus of all possible downstream
states, given an upstream state. It is also possible to express the
Hugoniot curve with respect to the variables x and y, similar to the
Rayleigh line, as

$y= \frac{\frac{\gamma_0+1}{\gamma_0-1} -x+\frac{2q}{p_0\nu_o}}{\frac{\gamma_0+1}{\gamma_0-1}x -1}$.

Note that the case where q=0 corresponds to a non reacting shock wave
and the Hugoniot curve will pass by the point (1,1) in the x-y plane.
When the Rayleigh line is tangent to the Hugoniot curve, the tangent
points are called Chapman-Jouquet (CJ) points. The upper CJ point
corresponds to the CJ detonation solution whereas the lower CJ point is
referred as the CJ deflagration solution.

\[\[Image:CJ detonation and deflagration
points.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 2: CJ detonation and deflagration solution

```{=html}
<center>
```
\]\]

Note that the CJ theory does not take into account the detailed
structure of the detonation wave. It simply links upstream conditions to
downstream conditions via steady one dimensional conservation laws.

## Detonation wave structure

### ZND model

Assuming a one dimensional steady flow, the Zel'dovich, von Neumann and
Döring (ZND) model is an idealized representation of the detonation
wave. The model essentially describes the detonation wave as a leading
shock followed by chemical reactions. The leading shock adiabatically
compresses the reactants, increasing the temperature, pressure and
density across the shock. An induction zone is followed where the
reactants are dissociated into radicals and free radicals are generated.
The induction zone is thermally neutral in the sense that the
thermodynamic properties remain relatively constant. When enough active
free radicals are produced, a cascade of reactions occurs to convert the
reactants into products. Chemical energy is released resulting in a rise
in temperature and a drop in pressure and density. The decrease in
pressure in the reaction zone is further decreased by expansion waves
and creates a forward thrust that will support the leading shock front.
In other words, the proposed propagation mechanism of a detonation wave
is autoignition by the leading shock which is supported by the thrust
provided by the expansion of the products.

The variation of thermodynamic properties is illustrated in the
following sketch.

\[\[Image:Variation of properties across ZND
structure.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 3: Variation of properties across a ZND detonation wave

```{=html}
<center>
```
\]\]

Although the ZND provides a description of the structure of the
detonation wave, it does not consider any boundary conditions. In
reality, both the initial conditions ( thermodynamic states, mixture
composition) as well as boundary conditions ( geometry, degree of
confinement, nature of walls) affect the detonation velocity. Under
certain conditions, initial and boundary conditions can even make the
propagation of detonations impossible. To-date, no quantitative
theoretical model can accurately predict the limits of detonations.

### Experimental observations

Although the ZND models the detonation in a one-dimensional frame,
experimental observations indicate that the detonation front is actually
three-dimensional and unstable. Instabilities are manifested in both
longitudinal ( pulsating detonation) and transverse directions. The
front consists of many curved shocks composed of Mach stems and incident
shocks. At the intersection between these curved shocks, reflected
shocks, also referred to as transverse waves, extend into the reacting
mixture. The intersection of these three shocks is referred as the
triple point. These transverse waves move back and forth sweeping across
the entire front. The trajectories of these triple points can be
recorded on a sooth covered surface as the detonation passes by. The
wave spacing can be measured from smoked foils and is referred to as the
cell size.

A sketch of a simplified cellular detonation structure is shown below. λ
represents the cell size.

\[\[Image:Det_front_structure.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 4: Detonation front structure

```{=html}
<center>
```
\]\]

## How to initiate a detonation

There are a few ways to initiate a detonation. Here are some examples:

-   Deflagration-to-detonation transition (DDT): assuming that a
    deflagration has already been formed, it needs to accelerate to a
    certain velocity via turbulence. When conditions permit, the
    deflagration abruptly transits into a detonation.Few key processes
    of the development of the detonation wave are summarized as follows:
    -   Generation of compression waves ahead of the propagating flame
    -   Coalescence of the compression wave to form a shock wave.
    -   Generation of turbulence
    -   Creation of blast wave from a local explosion in the reaction
        zone resulting into a detonation "bubble". A detonation bubble
        catches up to the precursor shock and an overdriven detonation
        is formed. Transverse pressure waves are generated.
-   Direct initiation: Method of initiating a detonation bypassing the
    deflagration phase. A detonation may be formed directly through the
    use of a powerful ignition source.
-   Diffraction of a planar detonation into a larger volume to form a
    spherical detonation.

## Limits of detonation

Detonation limits refer to the critical set of conditions outside of
which a self-sustained detonation can no longer propagate. The
detonation velocity is actually affected by initial conditions of the
explosive mixture (thermodynamic states, composition, dilution, etc.)
and the boundary conditions (degree of confinement, geometry and
dimensions of confinement, type of wall surface, etc.). For example,
given a set of thermodynamic states and a particular experimental
apparatus, we change the mixture composition rich or lean. At a
particular fuel concentration the detonation will cease to propagate.
This type of limit approach reveals the composition limit of an
explosive mixture. For given initial conditions, we can also vary the
dimensions of the experimental apparatus. Detonation propagating inside
a tube below a critical tube diameter, no detonation can propagate. This
yields the critical tube diameter. It is worth noting that it becomes
more difficult to initiate a detonation as the detonation limit is
approached. The critical energy required for initiation increases
exponentially. The limit is not a characteristic property of the
explosive mixture since it is affected by both initial and boundary
conditions.

Since the steady propagation velocity of the detonation depends on
initial and boundary conditions, a common observation as the detonation
limit is approached is the presence of velocity deficit. Studies in
round tubes reveal a velocity deficit of about 15% that of CJ velocity
before failure of the detonation wave. Near the limit, longitudinal
fluctuation of the velocity has also been observed. Depending on the
magnitude of these fluctuations and their duration, different unstable
behaviour such as stuttering and galloping can be manifested.

Another characteristic indicating the approach of the limit is the
detonation cell size compared to the tube dimension. Away from the
limit, the detonation cell size is small compared to the dimensions of
the detonation tube. As the limit is approached, there are less
transverse waves and the wave spacing increases until a single
transverse wave propagates around the perimeter of the tube, indicative
of a spinning detonation. The magnitude of the transverse pressure wave
oscillations become larger and larger as the limit is approached.

## Prevention

Since it takes less energy to initiate a deflagration, this is the mode
of combustion most likely to occur in industrial accidents. Although the
key factors required for the formation of a detonation or a deflagration
( explosive mixture and ignition source) may not be eliminated in a
chemical plant, some prevention mechanisms have been developed to stop
the propagation of a deflagration and prevent a detonation from forming.
Here are a few examples:

-   Inhibition of flames : once a flame is detected, flame suppressant
    is injected. The suppressant will combine with active radicals. By
    taking away the radical necessary for the chemical reactions to take
    place, the flame will cease to spread.
-   Venting : to avoid formation of a detonation, any pressure build up
    is released. However, by actually releasing the pressure, the
    turbulence created can accelerate the flame.
-   Quenching : it is possible to quench (or suppress) the flame, even
    near detonation velocities.

## References

-   Lee, J.H.S.,*The Detonation Phenomenon*, Cambridge University Press,
    2008
-   Kuo, K.K.,*Principles of Combustion*, John Wiley and Sons, Inc.
    2005, 2nd ed.
-   Fickett, W. and Davis, W.C.,*Detonation*, University of California
    Press, 1979

## Other links

A Detonation
database containing
a collection of experimental data related to detonations is available
online.


# Engineering Acoustics/The Acoustic Parameter of Nonlinearity

## The Acoustic Parameter of Nonlinearity

For many applications of nonlinear acoustic phenomena the magnitude of
an acoustic medium\'s nonlinearity can be quantified using a single
value known as the parameter of nonlinearity. This convention is often
attributed to Robert T. Beyer, stemming from his influential 1960 paper
titled *Parameter of Nonlinearity in Fluids* [^1] and his subsequent
texts on the subject Nonlinear Acoustics.[^2] It is worthwhile to note
that in Hamilton's text on Nonlinear Acoustics,[^3] Beyer attributes the
concept to an earlier work by Fox and Wallace (1954).[^4]

The mathematical grounds for the parameter of nonlinearity stem from the
Taylor series relating the perturbed pressure, *p* \', to the perturbed
density, *ρ* \'. Physically, these small perturbation values are
referenced to an ambient state defined by a density value, *ρ~o~*, and a
constant entropy, *s* = *s*~o~.

$$p' =  A\left(\frac{\rho'}{\rho_o}\right)
+ \frac{B}{2!}\left(\frac{\rho'}{\rho_o}\right)^2
+ \frac{C}{3!}\left(\frac{\rho'}{\rho_o}\right)^3
+ ...$$

where the coefficients *A*, *B*, *C*, give the magnitude for each term
in the Taylor expansion. As the coefficients *B*, *C*, apply to squared
and cubic terms they represent a nonlinearity in the relation between
pressure, *p* \', and density *ρ* \'. Values for *A*, *B*, and *C* can
be determined experimentally using a several techniques.[^5][^6][^7] It
is also possible to use the Taylor series definition of *A*, *B*, and
*C* to calculate values when a constitutive relation between pressure
and density is known. Using the ideal gas or Tait equation of state for
this purpose is discussed in a subsequent section. The Taylor series
coefficient definitions are:

\
$$A = \rho_o\left(\frac{\partial P}{\partial \rho}\right)_{\rho_o,s_o} = \rho_o c_o^2$$

$$B = \rho_o^2\left(\frac{\partial^2 P}{\partial \rho^2}\right)_{\rho_o,s_o}$$

$$C = \rho_o^3\left(\frac{\partial^3 P}{\partial \rho^3}\right)_{\rho_o,s_o}$$

where a definition for ambient sound speed, (*∂p*/*∂ρ*)*~ρ~o~,s~o~~* =
*c~o~*^2^, has been applied to shown: *A* = *ρ*~o~ *c*~o~^2^. For a
majority of problems in nonlinear acoustics, utilizing the first two
terms of this expansion is sufficient to represent the range of density
perturbations encountered. In this case the series reduces to:

\
$$p' = A\left(\frac{\rho'}{\rho_o}\right) + \frac{B}{2}\left(\frac{\rho'}{\rho_o}\right)^2$$

This truncation leads to what is commonly referred to as the parameter
of nonlinearity in a fluid, or *B/A*. By factoring *A* = *ρ*~o~
*c*~o~^2^ from both terms in the truncated series, the physical
relevance of *B/A* becomes more apparent:

\
$$p' = \rho' c_o^2\left[1+\frac{1}{2}\frac{B}{A}\left(\frac{\rho'}{\rho_o}\right)\right]$$

This expression shows the ratio *B/A* quantifies the influence of
nonlinearity on the local pressure perturbation for a given state, *ρ*\'
/ *ρ*~o~. Similarly, it can also be shown the parameter *B/A* quantifies
the variation of local sound speed as a function of perturbed density
according to:[^8]

\
$$c = c_o\left[1+\frac{1}{2}\frac{B}{A}\left(\frac{\rho'}{\rho_o}\right)\right]$$

## B/A In Relation to Power Law Equations of State

For power law equations of state (EOS), such as the Tait-Kirkwood EOS
for liquids,[^9] or the isentropic compression of an ideal gas, the
*B/A* parameter can be related to know power law coefficients. To
demonstrate this relation, the partial derivatives of pressure with
respect to density, *∂p*/*∂ρ*, are calculated for the Tait EOS and are
then applied in the Taylor series for perturbed pressure, *p\'*. The end
result when using the ideal gas EOS instead of Tait is identical to that
shown.

\
$$\frac{P+D}{P_o+D} = \left(\frac{\rho}{\rho_o} \right )^{\gamma}$$

$$\frac{\partial P}{\partial \rho} = \left(P_o+D \right )\frac{\gamma}{\rho_o}\left(\frac{\rho}{\rho_o} \right )^{\gamma-1}$$

Evaluating the first derivative at the ambient state (*ρ* = *ρ~o~*)
gives a useful equation for the linear sound speed: *c~o~^2^* =
*γ*(*P~o~*+*D*) / *ρ~o~*. Including this expression to simplify
*∂p*/*∂ρ*, and continuing on to calculate *∂^2^p* / *∂ρ^2^* gives:

\
$$\frac{\partial P}{\partial \rho} = c_o^2\left(\frac{\rho}{\rho_o} \right )^{\gamma-1}$$

$$\frac{\partial^2P}{\partial \rho^2} = c_o^2  \frac{\left(\gamma-1 \right )}{\rho_o}  \left(\frac{\rho}{\rho_o} \right )^{\gamma-2}$$

Incorporating these first and second derivatives into a Taylor series of
*p\'*on *ρ\'*gives equations derived from the power law EOS that can be
compared to the previous series containing the B/A parameter.

\
$$p'=c_o^2 \rho' + \frac{c_o^2\left(\gamma-1 \right )}{2\rho_o}\rho'^2+...$$

$$p'=c_o^2 \rho'\left[1 +  \frac{\left(\gamma-1 \right )}{2}\left(\frac{\rho'}{\rho_o} \right ) \right ] + ...$$

This final equation showns the two Taylor series for *p\'*are identical,
with (*γ*-1) in place of *B/A*. Thus, for liquids obeying the
Tait-Kirkwood EOS and ideal gasses under isentropic conditions with
known adiabatic index:

\
$$\frac{B}{A} = \gamma - 1$$

## Sample Values for B/A

Table 1 provides a sample of values for the B/A parameter in various
gases, liquids, and biological materials. Reference temperatures are
included along with each sample as the value of *B/A* for a particular
material will vary with temperature. Several organic materials are
included in Table 2 as nonlinear acoustic effects are particularly
prominent in biomedical ultrasound applications.[^10]

\
:{\| class=\"wikitable\" style=\"text-align: center;\" \|+ Table 2:
Sample *B/A* values for fluids. \|- ! Material !! *B/A* !! Ref. Temp.
^o^C !! Reference \|- \| style=\"width: 160px;\" \| Diatomic Gases (Air)
\|\| style=\"width: 60px;\" \| 0.4 \|\| 20 \|\| [^11] \|- \| Distilled
Water \|\| 5.0 \|\| 20 \|\| [^12] \|- \| Distilled Water \|\| 5.4 \|\|
40 \|\| [^13] \|- \| Salt Water \|\| 5.3 \|\| 20 \|\| [^14] \|- \|
Ethanol \|\| 10.5\|\| 20 \|\| [^15] \|}\
:{\| class=\"wikitable\" style=\"text-align: center;\" \|+ Table 2:
Sample *B/A* values for organic materials. \|- ! Material !! *B/A* !!
Ref. Temp. ^o^C !! Reference \|- \| style=\"width: 160px;\" \| Glycerol
\|\| style=\"width: 60px;\" \| 9.1 \|\| 30 \|\| [^16] \|- \| Hemoglobin
(50%) \|\| 7.6 \|\| 30 \|\| [^17] \|- \| Liver \|\| 6.5 \|\| 30 \|\|
[^18] \|- \| Fat \|\| 9.9 \|\| 30 \|\| [^19] \|- \| Collagen \|\| 4.3
\|\| 25 \|\| [^20] \|}

## References

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]:

[^7]:

[^8]:

[^9]:

[^10]:

[^11]:

[^12]:

[^13]:

[^14]:

[^15]:

[^16]:

[^17]:

[^18]:

[^19]:

[^20]:


# Engineering Acoustics/Harmonic Generation

## Nonlinear Generation of Harmonics

As described in the entry for the qualitative description of
shocks,
finite amplitude waves in any medium will undergo a steepening phenomena
cumulating in the formation of shock wave. For strong flows and wave
conditions where fluid velocity is similar in magnitude to sound speed,
*u*/*c~o~* ≥ *O*(1), where *u* is the particle velocity and *c~o~* is
the ambient sound speed, the transition to a shockwave occurs rapidly
and can be termed a local effect. For weaker wave conditions where
*u*/*c~o~* \<\< 1, but nonlinear effects are still observable, wave
steepening occurs over many wavelengths and can be termed a cumulative
effect. In this regime of wave strengths an important result of wave
deformation is the accumulation of harmonic content in the propagating
waveform.

\
== Progressive Wave Deformation ==

To describe the harmonic content of the deformed wave profile consider
some analysis can be carried out for the case of a plane wave
propagating in the x^+^ direction driven by a boundary piston with
velocity *u~o~* = *sin*(*ωt*), where *ω* is the driving frequency, and
*t* is the time variable. In this case the resulting sound field depends
only on the *x*^+^ wave, thus falls under the simple wave
assumption
and can be defined using a reduced equation for progressive waves in an
inviscid fluid:[^1][^2]

\
$$\frac{\partial u}{\partial t}+\left(c + u \right )\frac{\partial u}{\partial x} = 0$$

$$\frac{\partial u}{\partial t}+\left(c_o + \beta u \right )\frac{\partial u}{\partial x} = 0$$

$$\beta = 1 + \frac{1}{2}\frac{B}{A}$$

The relation between *β* and *B/A* is given to highlight the relation to
the acoustic parameter of nonlinearity. The equations given describe a
propagating planar wave, in which the propagation velocity for any
particular point is given by local value of (*c~o~* + *βu*), as opposed
to simply *c~o~* in the case of an assumed linear wave. For an initially
sinusoidal wave profile, the wave apexes propagate with the greatest
velocity, while equilibrium points propagate only at the ambient sound
speed. This progression is qualitatively depicted in Figure 1, where the
trajectory (wave velocity) of the wave maximum, minimum, and equilibrium
points are plotted. As the trajectories of different points on the wave
are non-parallel, the wave will deform as it propagates. The rate of
deformation depends on the magnitude of difference between the various
trajectories in the wave profile, and those depend on both the induced
particle velocity and the fluids value of *B/A*. As a result of this
dependence, a fluid with a higher *B/A* value will exhibit more rapid
wave deformation than a fluid with a lower *B/A* value if the same
boundary velocity is applied to both.

\
! Figure 1: Progressive deformation of an initially sinusoidal wave
profile. Local trajectories are plotted on the x-t axis for the wave
maximum (*c* + *v*), the local equilibrium (*c~o~*), and the wave
minimum (*c* +
*v*)., the local equilibrium (co), and the wave minimum (c + v)."){width="420"}

According to the progressive wave equation given for an inviscid fluid -
and the process depicted in Figure 1 - the wave maximum will eventually
catch up and overtake the wave front to form a discontinuous shock. In
real fluids this is not necessarily the only possible outcome, as all
sound waves in real fluids will attenuate as they propagate to some
extent. For many dissipative processes, the effect on the wave is
proportional to *ω*^2^,[^3] thus the generated higher harmonics are
dissipated more severely than the fundamental frequency. In this regard
the effects of dissipation hamper wave steepening, and for some wave
amplitudes a quasi-steady wave-form can be reached where the nonlinear
steeping effects are perfectly balanced by dissipative effects.

Provided sufficient amplitude, or a nearly inviscid fluid, shock waves
are formed in a progressive wave. It is for this reason that cumulative
deformation is more likely to result in shocks in water than air, as air
can be highly dissipative at the required wave amplitudes. Although not
discussed in the sections to follow, the generation of new harmonic
content continues after shock formation. The analytical description of
this process is not fundamentally different from the analysis given, as
the weak shock assumption is employed. For more details on the post
shock regime refer to the seminal paper by Blackstock [^4] or in a
variety of reference texts on the nonlinear acoustics.[^5][^6][^7][^8]

## Frequency Analysis of Solutions Obtained Using the Method of Characteristics

The most intuitive analytical solution to the deformed wave profile,
*u*(*x*,*t*), is obtained when using an approach method based on the
method of
characteristics.
When applied to planar progressive waves this approach is described in
the entry for the qualitative description of shock
waves.
Discussion focusing on the intermediate wave profile in addition to the
shock formation properties are given in many reference texts on
nonlinear acoustics, including those by Hamilton and Blackstock,[^9]
Elfno,[^10] Beyer,[^11] or Pierce.[^12]

As depicted to in Figure 1, the essence of this approach is to project
know values or ordinary differential eqations along characteristic paths
defined by *dx*/*dx* = (*c* + *u*). For the case of a single propagating
wave driven from a sinusoidal boundary, *u~o~* = *sin*(*ωt*), the
solution is particularly simple as each location in the domain
corresponds to only one characteristic path carrying constant values of
*u*, *c*, etc. At any position throughout the domain the characteristic
trajectories can be identified and the corresponding value of *u* is
used to construct the solution. In the following example, the implicit
equations described in Elfno [^13] were used to calculate the wave
profile:

\
$$u = u_o \sin\left(\tau + \frac{\beta u}{c_o^2}x \right ),$$

$$\tau = t - \frac{x}{c_o}$$

For this particular example the wave amplitude was set to achieve shock
formation after five wavelengths. For generality all magnitudes of the
solution are given in non-dimensional form. For the fluid, the parameter
of nonlinearity was set to B/A = 5.0, which corresponds to fresh water
at 20^o^C.[^14] The upper portion of Figure 2 gives the spatial wave
profile in which the wave steepening is readily apparent. In the lower
three panels of Figure 2, the frequency spectrum of the velocity wave is
shown as while passing through the regions indicated by red bands. The
frequency spectrum plots were obtained using Discrete Fourier transform
(DFT) of the calculated time domain signal at each indicated location.

\
! Figure 2: Harmonic contnet of a deforming progressive wave. *β* =
3.5, corresponding to *B/A* = 5.0 for distilled water at
20^o^C.{width="420"}

In the frequency spectra, where *f~o~* = 2*πω*, the progressive wave
contains only the fundamental frequency at the driven boundary, x = 0.
Importantly, it can also be seen the propagated and deformed wave
profile contains only integer harmonics of the initial frequency, and
the magnitude of these harmonics increases with propagation distance.

## Direct Analytical Solution to Harmonic Profile

While the method of characteristics provides spatial and time domain
wave profiles, from which frequency content can be analyzed, a more
direct approach is available to directly solve for the harmonic content
of the deformed progressive wave. Again consider the example of a planar
wave driven by a sinusoidal boundary condition, *u~o~* = *sin*(*ωt*).
One common approach for obtaining a general solution when system input
is periodic, is to assume a periodic system response expressed as a
Fourier Series. As shown
by Pierce,[^15] applying this approach to the progressive wave problem
gives the series solution as:

\
$$u\left(\omega \tau, \sigma \right ) = \sum_{n=1}^{\infty} B_n\left(\sigma \right ) \sin\left(n \omega \tau \right )$$

$$\sigma = \frac{x}{x_\text{shock}}$$

$$\tau = t - \frac{x}{c_o}$$

$$B_n\left(\sigma \right ) = \frac{2}{\pi} \int_{0}^{\pi}u\left(\omega \tau, \sigma \right ) \sin \left(n \omega \tau \right ) d\left(\omega \tau \right )$$

From this form of the solution the coefficient magnitudes directly yield
the harmonic amplitudes, while the complete Fourier series yields the
solution in the time and space domains. The challenge in this approach
is the evaluation of the Fourier coefficients. In the context of
nonlinear acoustics this solution is first attributed to Fubini in
1935,[^16] where manipulation of the integral terms was used to achieve
the integral form of Bessel's function, thus the harmonic components can
be directly defined according to:

\
$$B_n(\sigma) = \frac{2u_o}{n \sigma} J_n(n\sigma)$$

Where *J~n~* is the Bessel
function of the first
kind. Historically, the subsequent work by Blackstock in 1966 [^17]
clarified the physical implications of this solution somewhat and also
brought this solution to a wider audience in nonlinear acoustics. For
additional details on this derivation refer to the texts by Pierce,[^18]
Elfno,[^19] or Hamilton.[^20] For completeness, Figure 3 plots the
continuous harmonic profile as a function of normalized propagation
distance. The wave conditions are the same as those used in calculations
for Figure 2. In this particular example the shock formation occurs at
five wavelengths; however, the plotted harmonic profile is general to
any value of x~shock~.

! Figure 3: Continuous harmonic profile of a deforming progressive
wave. *β* = 3.5, corresponding to *B/A* = 5.0 for distilled water at
20^o^C.{width="480"}

## References

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]:

[^7]:

[^8]:

[^9]:

[^10]:

[^11]:

[^12]:

[^13]:

[^14]:

[^15]:

[^16]:

[^17]:

[^18]:

[^19]:

[^20]:


# Engineering Acoustics/Thunder acoustics

**Thunder** is defined as the sound signature associated to the shock
wave produced after a lightning discharge. Thunder has intrigued and
frightened humans for millennia. Early explanations of this phenomenon
included fights between Zeus and his subordinates in Greek mythology, or
the collision of clouds in the sky by Aristotle. It was only in the late
1800s that the true physical causes were discovered by the scientific
community i.e. the heating of a narrow channel to \~24000K. The air
molecules within ionize to generate a highly powerful shock wave that
can be heard over distances up to 25 km, depending on the wind.

## Lightning fundamentals

!CGtypes\|400px\|Figure 2. The 4 groups of cloud-to-ground (CG)
discharges (adapted from *Lightning: Physics and Effects*
(2003)). discharges (adapted from Lightning: Physics and Effects (2003)).")

There are two main types of lightning discharges: cloud-to-ground (CG)
and intra-cloud (IC), the latter accounting for about 3/4 of all
discharges (there are however other types of discharges that are less
commonly encountered such as ball
lightning).

The CGs can be categorized into 4 groups:[^1]

:   \(a\) downward negative (90% of all CGs are of this group),
:   \(b\) upward positive,
:   \(c\) downward positive, and
:   \(d\) upward negative.

The charges have three main modes in which they can be sent to the
ground from the cloud (see diagram):

1.  Dart-leader-return stroke sequences.
2.  Continuing currents (that last for hundreds of milliseconds) which
    are long quasi-stationary arcs.
3.  M-components (named after D.F. Malan who first studied these
    processes in the 1930s) which are transient processes during a
    continuing current (2nd mode).

!modesoftransfer\|400px\|Figure 3. The three main modes of charge
transfer to ground (adapted from *Lightning: Physics and Effects*
(2003)).).")

## Thunder generation

\"Thunder can be defined as the acoustic emission associated with a
lightning discharge\".[^2]

### Types

All processes in CG and IC discharges produce thunder which can be
divided into 2 categories:

-   audible (frequencies greater than 20 Hz) which come from a series of
    decaying shockwaves produced by the expansion of various portions of
    the nearly instantaneous heated lightning channel which is filled
    with ionized air molecules
    (plasma)). Of these
    there are: a) rumbling thunder which is a long low series of
    growling-like sounds and b) clapping thunder that is usually loud
    and quick.
-   non-audible or infrasonic thunder (frequencies lower than 20 Hz)
    which is thought to originate from ICs where large volumes of air
    are displaced by the rapid removal of electrons or protons from the
    cloud itself. This category of thunder has only recently received
    the attention of the scientific community.

### Maximum amplitude frequency

!Figure 4. The frequency spectrum of thunder (and unfiltered rain
sounds) obtained using Audacity 1.3
(Beta). obtained using Audacity 1.3 (Beta)."){width="300"}
It has been empirically found that the loudest frequency in thunder is:

$$f_{\rm max} = c_0 \left( \frac{p_0}{E_0} \right)^\frac{1}{2}$$ where
$c_0$ is the speed of sound, $p_0$ is the ambient pressure, and $E_0$ is
the energy per unit length of lightning channel which is defined as:

$E_0 = \frac{1}{\pi {R_0}^2} \int\limits_{0}^{t_{\rm disch}}\rho I^2\,dt$

where $R_0$ is the initial channel radius, $\rho$ is the resistivity of
the plasma and $t_{\rm disch}$ is the discharge duration. The values of
$E_0$ have been found to vary around 50 kJ/m.

### A.A. Few\'s Model of thunder generation

It is widely accepted that audible thunder is generated by the lightning
channel and the subsequent shock wave that travels extremely rapidly
(\~3000 m/s).[^3] A.A. Few provides a experimentally-proved thunder
generation mechanism.

#### Assuming perfectly cylindrical/spherical expansion

The shock wave time history can be cut into three intervals: the first
consists of a strong shock with an extremely high pressure ratio across
the boundary. The second section is a weak shock that travels at a
relatively slower pace. And finally the third section of the shockwave
is the acoustic wave that propagates at 343 m/s i.e. the speed of sound
at 293K.

The distance traveled by the strong shock wave before it turns into a
weak shock can be found by performing a work-energy balance on the fluid
that has been compressed by the strong shock (i.e. work is done on the
fluid by volume and pressure changes). A so-called relaxation radius,
$R_s$ (for spherical shock waves or $R_c$ for cylindrical shock waves)
can thus be defined to account for the distance traveled by the strong
shock:

Spherical relaxation radius:

$$R_s=\left(\frac{3E_t}{4 \pi p_0}\right)^\frac{1}{3}$$

Cylindrical relaxation radius:

$$R_c=\left(\frac{E_0}{\pi p_0}\right)^\frac{1}{2}$$

where $E_t$ is the total energy released by the spherical shock wave. It
is in this last section of the shock wave that the thunder is heard.

Most studies that were done on thunder could not analyze from close
range a naturally produced sound. This is because of the impossibility
to predict exactly where lightning will strike and thus to place a
microphone near it. The most common way of artificially generating
lightning is with a rocket, that is connected to a steel wire and fired
into a thundercloud to create a short circuit near ground. This
\"forces\" an electrical current from the electrically-charged
thundercloud to the ground via the attached electrical wire.

This type of discharge is commonly called artificially triggered
lightning, and was used, among others, by Depasse in 1986, 1990 and 1991
at Saint-Privat D\'Allier, France, where the pressure profile behind a
lighting-generated shockwave was matched to the theoretical profile
obtained from cylindrical shockwave theory developped by Few in
1969.[^4]

!Figure 5. The different sizes of tortuous segments in a lightning
rod.-3.png "Figure 5. The different sizes of tortuous segments in a lightning rod."){width="400"}

#### Effect of tortuosity on rumbling or clapping thunder

Lightning channels are not straight channels with perfectly circular gas
dynamic expansion and hence the tortuosity of these must be accounted
for. It was for this reason that A. A. Few derived three different
levels of tortuosity in lightning rods. Macro, meso and micro-tortuous
segments can be qualitatively observed in any CG discharge. It was found
that macro- and meso-tortuous segments are very important in organizing
the pulses and acoustic behaviour of a CG discharge.[^5] Through
computational studies, it was found that 80% of the acoustic energy is
released at a 30 degree angle of the plane perpendicular to the main axe
of a macro tortuous discharge. It is in this region that an observer
will hear a loud clap sort of thunder whereas one placed outside this
region will hear a rumbling sort of thunder.

## Thunder propagation

!Thunder propagation
schematics.{width="400"}

### Attenuation

There are two types of attenuation effects. The first is due to the
finite amplitude of the propagating acoustic waves which cause a
non-negligeable amount of stretching to the wave. There is a so-called
\"eroding\" effect which tries to break down the sudden pressure jump at
the wave front into a more rounded profile. This is a form of
dissipation due to visco-thermal losses that affects the higher
frequencies and so explains why only lower frequencies are heard when
lightning strikes 1 or more kilometres away. The second form of
attenuation is due to the scattering and aerosol effect left by the rain
drops and thunderclouds (filled with water vapour) commonly found in
most lightning conditions. These micro particles also attenuate the
higher frequencies of a thunder clap or rumble. See
shockwave and
detonation wikis for more
information about the decay of a strong shock.

### Environment

Recalling that the speed of sound, c, is dependent on the density of the
medium it is thus likely that, depending on the conditions surrounding
the lightning rod such as the air composition, atmospheric pressure, the
thunder will travel at a unique velocity, pitch, frequency band and
duration depending on the characteristics of the lightning rod. Indeed,
as shown in the study by Blanco et al. (2009) [^6] the geometry plays a
vital role in the perceived resulting sound. Furthermore, there is a
level of attenuation that must be accounted for as the sound travels
through the atmosphere and ground obstacles (such as trees, buildings,
bridges, land).

Back to Main page

## References

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]:


# Engineering Acoustics/Basic Concepts of Underwater Acoustics

The study of Underwater Acoustics was always of great importance, when
it comes to navigation instruments, for those who depend on the sea.
Methods to find the position on earth according to the stars were
available since long ago, but the first apparatus that track what is
underwater are relatively recent. One of these instruments, which
improved the safety of navigation is the fathometer. It has the simple
concept of measuring how much time a sound wave generated at the ship
takes to reach the bottom and a reflected wave returns. If one knows the
speed of sound in the medium, the depth could be easily determined.
Another mechanism consists of underwater bells on lightships or
lighthouses and hydrophones on ships that are to find the distance
between them. These could be considered the forerunners of the SONAR
(SOund Navigation And Ranging). There are a lot of animals that also
take advantage of underwater sound propagation to communicate.

# Speed of Sound

In 1841, Jean-Daniel Colladon
1 was able to
measure the speed of sound underwater for the first time. He conducted
experiments in Lake Geneva, where he was able to transmit sound waves
from Nyon to Montreux (50 km). The idea of the experiment was to
propagate a sound wave making use of a hammer and an anvil to generate
the wave and a parabolic antenna to capture the sound at distance. A
flash of light is emitted at the same time that the hammer hit the anvil
and the delay between the light and sound is used to determine the speed
of sound.

The difference in speed of sound with depth is much more significant
than its difference along the surface.

The equation for speed of sound(m/s) in water developed by Del
Grosso,[^1] applicable in Neptunian[^2] waters, depends on the
Temperature(T) in Celsius, Salinity(S) in ppt(part per thousand) and
gauge Pressure(P) in atmospheres.:

$c(T,S,P) = 1449.08 + 4.57Te^{-(T/86.9+(T/360)^2)} + 1.33(S-35)e^{-T/120} + 0.1522Pe^{T/1200+(S-35)/400} + 1.4610^{-5}P^{2}e^{-T/20+(S-35)/10}$

where the pressure is a function of depth\[Km\] and the Latitude, given
by:

$P=99.5(1-0.00263cos2\phi)Z + 0.239Z^{2}$

!Figure 1: Speed of sound profile at low latitudes. Salinity gradient
was not taken into
account.

The Axis passing by the region where the speed of sound is minimum is
known as Deep Sound Channel Axis.

The speed of sound is very sensible to the temperature, which changes
considerably on the thermocline
2. Beyond 1000 m meters in
depth, the pressure governs the equation, increasing slowly the speed
with depth. The salinity has very low effect on the equation unless in
very specific situations, such as heavy rain or the encounter between a
river and the sea. The shape of the curve may change drastically from
one place to another, e.g., the curve would be more like a linear
function of depth in very cold places.

# Refraction

The gradient of speed of sound within the water causes a phenomena
similar to a mirage, in which rays of light are bent. If we divide the
water in multiple layers parallel to surface, we should get various
medium with different speed of sound i.e., different specific
characteristic impedance. Considering a source of sound pressure
underwater and making use of Snell\'s Law
3 we can see the path the
wave will follow. Snell\'s Law tells us that the sound bends towards the
lower sound speed layer. If the sound wave angle with the horizontal is
too high (higher than $\theta_{max}$), the wave will eventually hit the
bottom or the surface, otherwise it will bend continuously towards the
horizontal until it passes the critical angle($\theta_c$) and then will
be completely reflected back.

$\theta_{max}=(2\Delta c/c_{max}))^{1/2}$

($c_{max}$ is the maximum speed found in the SOFAR channel.)

$sin\theta_c = c_1/c_2$

This process happens over and over again causing the sound to be trapped
in certain depth range known as SOFAR (Sound Fixing and Ranging) channel
4. As the sound cannot
reach the bottom nor the surface, the losses are small and no sound is
transmitted to the air nor the seabed, helping sound propagate through
large distances. Signals have been detected in ranges that exceed
3000 km.

This channel can be used for communication successfully by some species
of *cetacea*.

We can see that the sound concentrates at some depths and is much less
present in others, causing some regions to be more noisy than others.

!Sound trapping in the SOFAR
channel{width="800"}

Note that if the surface temperature is very low this phenomena may no
longer occur. The wave would be bouncing on the surface and being
reflected back just like we can see on the graph for 15.19° angle. The
same effect occurs on the *mixed layer* which is the layer affected by
the agitation of waves, causing the speed of sound to be only dependent
on pressure. This effect may cause shadow zones.

If you have a source that is between the deep sound channel axis and the
surface, only the rays making an angle less than $\theta_o$ with the
horizontal would be trapped.

$\theta_o=\theta_{max}(z_s/D_s)^{1/2}$

where $z_s$ is the depth of the source and $D_s$ is the depth of the
sound axis

# Reflection

Reflection also occurs when the sound wave hits a another body, such as
the seabed, the surface, animals, ships and submarines.

$R = (r_2/r_1 - cos\theta_t/cos\theta_i)/(r_2/r_1 + cos\theta_t/cos\theta_i)$

where $r_1$ is the characteristic acoustic impedance of water and $r_2$
is the characteristic acoustic impedance of the other body, $\theta_i$
is the incident angle and $\theta_t$ is the angle of the transmitted
wave, which can be obtained via Snell\'s Law. The formula is for 2D
case, but we can easily recall the 1D case by setting
$\theta_i=\theta_t=0$

If we can measure the reflected wave, we can determine the reflection
factor and with it we are able to determine the characteristic acoustic
impedance of the body that the wave hit to then have an idea of what the
body might be.

# Transmission Loss

The transmission loss is defined as

$TL=10log[I(1)/I(r)]$

where $I(r)$ is the intensity of sound measured at a distance $r$
Sometimes it is useful to separate $TL$ in loss due to geometrical
spreading and loss due to absorption

$TL=TL(geom)+TL(losses)$

If the sound is trapped between two perfect reflecting surfaces

$TL=10log r + ar$

where a is the absorption coefficient in dB/m.

# Sonar Equations

The passive sonar measures the incoming waves and is able to determine
the position of the target if there is more than one device, by
triangulation. Its equation determines that the *Sound Level* coming
from the source reduced by the *Transmission Loss* has to be higher than
the background noise (generated by waves, wind, animals, ships and
others) in order to get any measurement.

Passive Sonar Equation

$SL - TL >= NL - DI + DT_N$

where *SL* is the sound emitted by the target, *NL* is the noise level,
*DI* is the directivity index and $DT_N$ is the detection threshold for
noise-limited performace and *TL* is the transmission loss.

The active sonar emits a wave and measures the reflected sound waves.
Since the wave will propagate the double distance. The transmission loss
term is multiplied by two. The equation determines the conditions to get
valid measures (higher than background noise).

Active Sonar Equation

$SL - 2TL + TS >= NL - DI + DT_N$

where *SL* is the sound emitted by the source, *NL* is the noise level,
*DI* is the directivity index and $DT_N$ is the detection threshold for
noise-limited performace and *TL* is the transmission loss and*TS* is
the target strength, a measure of how good acoustic reflector the target
is.

# References

`3. Lawrence E. Kinsler, Austin R. Frey, Alan B. Coppens, James V. Sanders(2000) ,`*`Fundamentals of Acoustics 4th ed`*`, Wiley`

[^1]:

[^2]:


# Engineering Acoustics/Analogies in aeroacoustics

## Acoustic Analogies

!Test facility and testing of supersonic jet engine to assess noise
emissions at NASA Langley Research
Center
A direct way to predict Aerodynamic noise would be to solve the
Navier-Stokes Equations in a general three-dimensional unsteady
compressible numerical simulations. Unfortunately, it is hardly
achievable except for very simple academic configurations within a
delimited region. The idea of an acoustic analogy is to restate the full
equations of gas dynamics providing an equivalent wave equation in
homogeneous medium in uniform motion from the point of view of a distant
observer. This condition leads to simplifications of usual linear
acoustics problems\[1\]. The most systematically used formalisms are
Lighthill\'s analogy and the extensions made by Curle and Ffowcs
Williams & Hawkings, because they offer of wide range of applicability.

It must be started clearly that the aim of an analogy is not essentially
to deduce exacts results or numerical coefficients, but to infer general
laws from the standard procedures associated with the classical wave
equation. A preliminary knowledge of the main flow features coming from
either experiments, CFD or analytical methods is needed to apply these
analogies. Moreover, the degree of accuracy in the flow variables to
extract acoustic results is crucial to ensure the relevance of the
prediction method.

## Governing gas dynamics equations to Lighthill\'s equation

References \[2\] \[3\], development below originates from Aeroacoustics
wikipage.

$$\frac{\partial \rho}{\partial t} + \nabla\cdot\left(\rho\mathbf{v}\right)=\frac{D\rho}{D t} + \rho\nabla\cdot\mathbf{v}= 0,$$\...\...\...\...\...\...\...\.....*conservation
of mass equation (E1)*

where $\rho$ and $\mathbf{v}$ represent the density and velocity of the
fluid, which depend on space and time, and $D/Dt$ is the substantial
derivative.

Next is the conservation of momentum equation, which is given by

$${\rho}\frac{\partial \mathbf{v}}{\partial t}+{\rho(\mathbf{v}\cdot\nabla)\mathbf{v}} = -\nabla p+\nabla\cdot\sigma,$$\...\...\...\...\...\...\...\.....*conservation
of momentum equation (E2)*

where $p$ is the thermodynamic pressure, and $\sigma$ is the viscous (or
traceless) part of the Cauchy stress tensor from the Navier--Stokes
equations.

_Step 1:_ Multiplying (E1) by $\mathbf{v}$ and
adding it to (E2) yields

$$\frac{\partial}{\partial t}\left(\rho\mathbf{v}\right) + \nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v}) = -\nabla p + \nabla\cdot\sigma.$$

_Step 2:_ Differentiating (E1) with respect to
time, taking the divergence of (E2) and subtracting the latter from the
former, we get

$$\frac{\partial^2\rho}{\partial t^2} - \nabla^2 p + \nabla\cdot\nabla\cdot\sigma = \nabla\cdot\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v}).$$

_Step 3:_ Subtracting $c_0^2\nabla^2\rho$, where
$c_0$ is the speed of sound in the medium in its equilibrium (or
quiescent) state, from both sides of the last equation and rearranging
it results in:

$$\frac{\partial^2\rho}{\partial t^2}-c^2_0\nabla^2\rho = \nabla\cdot\left[\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v})-\nabla\cdot\sigma +\nabla p-c^2_0\nabla\rho\right],$$

which is equivalent to

$$\frac{\partial^2\rho}{\partial t^2}-c^2_0\nabla^2\rho=(\nabla\otimes\nabla) :\left[\rho\mathbf{v}\otimes\mathbf{v} - \sigma + (p-c^2_0\rho)\mathbb{I}\right],$$
where $\mathbb{I}$ is the identity tensor, and $:$ denotes the (double)
tensor contraction operator.

Using *Einstein notation*, Lighthill's equation can be written as

$$\frac{\partial^2\rho}{\partial t^2}-c^2_0\nabla^2\rho=\frac{\partial^2\sigma_{ij}}{\partial x_i \partial x_j},\quad$$

where $\sigma_{ij}$ is the so-called Lighthill\'s stress tensor
$\sigma_{ij} = \rho V_i V_j + (P-c_0^2 \rho)\delta_{ij} - \tau_{ij}$,
further details are provided in the next section. with: $\rho V_i V_j$
is the
inertial
stress tensor of Reynolds $\tau_{ij}$ is the viscous stress tensor of
Reynolds $(P-c_0^2 \rho)$ represents all effects due to entropy
non-homogeneities (important for hot jets with high temperature
gradients)

## Lighthill\'s Acoustic Analogy

Aeroacoustic engineers need to predict noise arising with turboengines
id est localized unsteady flow in a propagating medium, at a first step
of approximation. The basic idea of Lighthill (1952) is to reformulate
the general equations of gas dynamics, in order to derive a wave
equation. The featured variable is the density fluctuation, naturally
preferred because acoustic waves in a gas are due to compressibility. No
special assumption is made nor linearisation introduced.

$$\ \frac{\partial^2 \rho}{\partial t^2} - \frac{\partial^2 (\rho V_i V_j)}{\partial x_i \partial x_j} = - \frac{\partial^2 \sigma_{ij}}{\partial x_i \partial x_j}$$\...\...\...\...\...\...\...\...\...\....*Lighthill\'s
equation*

with
$$\ \sigma_{ij} = \rho V_i V_j + (P-c_0^2 \rho)\delta_{ij} -\tau_{ij}$$
called *Lighthill\'s tensor*

This equation remains true if the same quantity is added on both sides.
Add the $$\ c_0^2 \frac{\partial^2 \rho}{\partial^2 x_j^2}$$ with
$\ c_0$ being the characteristic speed of sound of the undisturbed gas
(precisely the speed of sound in the medium surrounding the flow region,
in the applications; this is different from the local speed of sound
$\ c^2$ in the flow). Then, forming a wave operator on the left-hand
side and removing all other terms on the right-hand side leads to:

$$\ \frac{\partial^2 \rho}{\partial t^2} - c_0^2 \frac{\partial^2 \rho}{\partial x_j^2} = \frac{\partial^2}{\partial x_i \partial x_j} [\rho V_i V_j + (P-c_0^2 \rho)\delta_{ij} -\tau_{ij} ]$$

This result is the well-known Lighthill\'s equation. When applied to a
true problem of acoustics, it reduces to the homogeneous wave equation
at large distances from the flow, since all terms in the right-hand side
can be considered negligible (according to the reasonable assumptions
related to the propagation of acoustic waves, as a small-amplitude,
isentropic motion\[4\]). An alternative form of Lighthill\'s equation
can be written with the pressure instead of the density, as follows:

$$\ \frac{1}{c_0^2} \frac{\partial^2 P}{\partial t^2} - \frac{\partial^2 P}{\partial x_j^2} = \frac{\partial^2}{\partial x_i \partial x_j} (\rho V_i V_j  + \frac{1}{c_0^2} \frac{\partial^2}{\partial t^2}(P-c_0^2 \rho-\tau_{ij}))$$\...\...\...\...\...\...\...*LightHill
equation on the fluctuating pressure*

Maybe this one is less common because density fluctuations are directly
related to compressible effects, whereas pressure fluctuations can exist
to compensate inertial accelerations in the fluid. When temperature
non-homogeneities are involved, however, the fluctuating pressure is
well suited.

Explanation of each term of Lighthill\'s tensor:

Aeoracoustic sources can be separate in three distinct categories,

!Monopole acoustic
source{width="100"}

-   Monopole sources: $\frac{\partial^2}{\partial t^2}(P)$ Spherical
    sources or discrete source delivering flow rate $Q(t)$ fluctuating
    over the time. Can appear only if solid surfaces are encountered.

!Dipole acoustic
source{width="100"}

-   Dipole sources: $-\frac{\partial^2}{\partial t^2}(c_0^2 \rho)$
    Similar to two monopoles placed side by side with opposition phase
    $+Q(t)$ and $-Q(t)$. Dipoles are associated to the force $F(t)$
    according to the axe created by our joined monopoles. Like Monopoles
    they only appear when solid surfaces are involved in the domain.

!Quadrupole acoustic
source{width="100"}

-   Quadrupole sources:
    $\frac{\partial^2}{\partial x_i \partial x_j} (\rho V_i V_j)$
    Constituted of two dipoles side by side in phase opposition, these
    sources cames from turbulent vortices and are usually neglected at
    low velocity flow. They originate from shear-terms in Navier-Stokes
    equations.

As a consequence of the general equations of gas dynamics, Lighthill\'s
equation is exact. All aeroacoustic processes, including generation of
sound by the flow non-homogeneities, sound propagation through the flow
and sound dissipation by viscosity or heat conduction, are accounted
for. Hence, this equation is not tractable as a pure wave equation using
linear acoustics since the right-hand side contains the acoustic field
to be determined and cannot be considered as a true source term. Thus we
need to approximate this term independently of the acoustic variables
which corresponds to neglect some of the mechanisms. To remove this
fundamental difficulty, Lighthill proposed some simplifications
motivated by the thinking that **sound generation by the mixing of fluid
is the dominant mechanism** , especially at the high Reynolds numbers of
interest in aeronautics. This is equivalent to privilege the mechanical
effects related to fluid inertia and discard thermodynamic effects as
secondary ones.

## Lighthill\'s Approximation

Lighthill\'s equation is well posed because it assumes that sources are
contained within the flow and not outside of it. I reduces to the
homogeneous wave equation in the propagation region. But in order to
solve near the sources approximations using comparison between phenomena
has to be done.

Lighthill\'s analogy is often use to calculate jet noise, is this type
of application we have specific conditions leading to simplifications
\[5\]:

!Explain classical approximation made using Lighthill\'s
approximation{width="600"}

Practical approximation used in industrial context:

!Explain classical approximation made using Lighthill\'s
approximation{width="500"}

The approximation makes the equation explicit in the sense of the wave
equation in linear acoustics, to be solved formally by the Green\'s
function technique. When numerical means are used to described the flow,
some assumptions can be removed for a more accurate evaluation and the
equation used to post-process the flow data.

!Description of Pseudo-sound phenomenon and influence on near-field and
far-field assumptions to calculate sound
pressure{width="600"}

If entropic non-homogeneities dominate in a disturbed flow, sources
appears as equivalent monopoles:

Using Ribner\'s splitting and uncompressible fluid laws:

:   ```{=html}
    <center>
    ```
    $\ \Delta p'_t = -\rho_0 \frac{\partial^2(U_i U_j)} {\partial x_i \partial x_j} = -\rho_0 \frac{\partial U_i} {\partial x_j} \frac{\partial U_j} {\partial x_i}$

    ```{=html}
    </center>
    ```

Leading to the final approximate Lighthill equation:

:   ```{=html}
    <center>
    ```
    $\ \Delta p'_a - \frac{1}{c_0^2} \frac{\partial^2p'_a}{\partial t^2} = \frac{1}{c_0^2} \frac{\partial^2p'_t}{\partial t^2}$
    \...\...\...\...\...\...\...*approximate Lighthill equation*

    ```{=html}
    </center>
    ```

## Ffowcs Williams & Hawkings formulation

### History

In 1969, Ffowcs Williams & Hawkings, are the first scientists to express
a fundamental equation to predict noise generate by blades in a flow
\[6\].

### Ffowcs Williams & Hawkings general equation

(They are several formulations of FWH analogy, here is one adapted to
blade noise description phenomena)

Deriving from Lighthill\'s equation, it reveal that fluctuating pressure
generating acoustic sources in a rotor is solution of a specific
inhomogeneous wave equation:

$\frac{\partial^2 p(z,t_z)}{\partial z_i^2} - \frac{1}{c_0^2} \frac{\partial^2 p(z,t_z)}{\partial t_z^2} = S(z,t_z)$

where, $z$ the vector coordinates from a source point and $t_z$ the time
in sources domain.

The source term $S(z,t_z)$ can be written as the following sum:

$S(z,t_z) = Q(z,t_z) + \frac{\partial F_i(z,t_z)}{\partial z_i} + \frac{\partial^2 T_{ij}(z,t_z)}{\partial z_i \partial z_j}$

-   First term $Q(z,t_z)$ represents *thickness noise* generated by
    volume displacement of fluid. A fan blade has its thickness and
    volume. As the rotor rotates, the volume of each blade displaces
    fluid volume, then they consequently fluctuate pressure of near
    field, and noise is generated. This noise is tonal at the running
    frequency and generally very weak for cooling fans, because their
    RPM is relatively low. Therefore, thickness of fan blades hardly
    affects to electronic cooling fan noise. (This kind of noise can
    become severe for high speed turbomachines like helicopter rotors.)
-   Second term is called *loading noise*, comes from fluctuation of the
    force-field $F_i(z,t_z)$ on moving surfaces. In a rotor, It
    originates from nonstationary aerodynamic forces between the fluid
    and the blades. In computational models, this term is represented by
    surface-distributed dipoles. (Dominant for fans)
-   Final term is *shear noise*, which is composed of quadrupoles on the
    surface of the blades \[7\].

The Ffowcs Williams & Hawkings theory allows to solve this equation
knowing the source terms in Green\'s functions.

### Ffowcs Williams & Hawkings extended analogy using permeable control surface

In all applications where the quadrupole term is significant and must be
calculated, which preferentially occurs at high speeds, the computations
can be made cumbersome because the sources are distributed inside a
volume, the boundaries of which are not precisely defined. In contrast,
the surface source terms are much simpler to compute and clearly
delimited. If CFD must be used in a limited domain surrounding the
surfaces, and if the computations are able to reproduce the acoustic
near-field, a more convenient way of solving the acoustic problem can be
proposed by taking the information not on the physical surfaces but on a
delocalised control surface that can be user-defined \[8\]. Double layer
potential description can be apply, to solve the Helmotz equation using
Bessel functions assumption. This generalised form of Ffowcs Williams &
Hawkings' analogy is widely used in recent Computational Aero-Acoustics
(CAA).

!Aeroacoustic computational simulations using FWH analogy and permeable
surface around aerodynamic
profile{width="650"}

N.B: The CFD Domain inside the control surface must be implemented until
turbulence systems generated are fully developed (using for example
k-$\epsilon$criterion).

The formal advantage of an analogy is to state a problem of
aeroacoustics as a usual problem of linear acoustics, by defining
equivalent sources that would produce in a uniform medium the same sound
as what is heard at observer's location from the flow-and-surface
configuration. The difficulty of the initial gas-dynamics equations is
transposed to the description of the source terms. The formal solution
is derived using the theoretical background of linear acoustics but it
may be useless if the equivalent source terms are not determined
elsewhere. Using Lighthill and FWH analogies allow engineers to
calculate aircraft engines noise for example at a lower time-calculation
cost.

Intensive research at NASA and CERFACS facilities try to develop more
efficient calculation schemes to provide improved design tools to
motorists engineer to gain confidence in noise prediction and develop
aero-mechanical-acoustic design for future generation products.

Making approximations means discarding phenomena that are expected
negligible and retaining the dominant features. This is just proposing
an interpretation.

## References

\[1\]Goldstein, M. E. (1976). Aeroacoustics. New York, McGraw-Hill
International Book Co., 1976. 305 p., 1.

\[2\]Tam, C. K. (1995). Computational aeroacoustics-Issues and methods.
AIAA journal, 33(10), 1788-1796.

\[3\]Wang, M., Freund, J. B., & Lele, S. K. (2006). Computational
prediction of flow-generated sound. Annu. Rev. Fluid Mech., 38, 483-512.

\[4\]Colonius, T., Lele, S. K., & Moin, P. (1993). Boundary conditions
for direct computation of aerodynamic sound generation. AIAA journal,
31(9), 1574-1582.

\[5\] Williams, J. F. (1969). Hydrodynamic noise. Annual Review of Fluid
Mechanics, 1(1), 197-222.

\[6\] Williams, J. F., & Hawkings, D. L. (1969). Sound generation by
turbulence and surfaces in arbitrary motion. Philosophical Transactions
of the Royal Society of London. Series A, Mathematical and Physical
Sciences, 264(1151), 321-342.

\[7\] Ianniello, S. (1999). Quadrupole noise predictions through the
Ffowcs Williams-Hawkings equation. AIAA journal, 37(9), 1048-1054.

\[8\]Di Francescantonio, P. (1997). A new boundary integral formulation
for the prediction of sound radiation. Journal of Sound and Vibration,
202(4), 491-509.