# 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


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