**4. Plane Waves**

Here we consider the properties of a single plane wave in free space. A spectrum of plane waves can be used to represent a general solution of the sound equations; see wave spectra.

The solution for a plane wave propagating in the ±x direction is:

The function f is arbitrary. Substituting into equation (26) or (27) shows that (28) and (29) are valid solutions iff:

For N_{2} at 20 ^{0}C the value
computed earlier is σ_{o}=295 m-s^{-1}; then from equation
(30) the indicated speed of sound is c=349 m-s^{-1}. The correct value
for air at 20 ^{0}C is 344 m-s^{-1}, which is pretty close.
For a monatomic molecule γ=5/3.

The relative values of
u_{o} and ρ_{o} are obtained by substituting equations
(28) and (29) into either equation (24) or (25)

For air at 20 ^{0}C, ρ_{o}=1.205
kg-m^{-3}, and with σ_{o}=295 m-s^{-1}as above,
Z_{o}= 420.6 kg-m^{-2}-s^{-1}.
Making an analogy to electromagnetic waves, sound pressure and velocity are
analogous to electric and magnetic fields, and Z_{o} is analogous to
the impedance of space.

By substituting equation (28) into equation (11), and neglecting **u**⋅
∇, the variable part of the density is found to be

so the density variation is in phase with the velocity variation. Using equations (23) and (31) the variance variation is equal to

I recently (September 2005) discovered that due to the approximate nature of the linearized equations this value is correct to the first order, but leads to an error for the wave momentum. There is further discussion of this issue below. Equations (31)-(33) are valid for monatomic gases as well, when the appropriate value of c is used.

In the books I have seen, including physics references [1,2], the wave equation solution is derived from a potential function ɸ which in the case of a harmonic plane wave can be written

where λ is the wavelength. This solution leads to a paradox, the resolution of which seems quite obvious in hindsight, but which caused me quite a few hours of head-scratching. Equation (34) is indeed a valid solution of the wave equation - no problem there. The paradox arises if one attempts to apply equation (34) to the problem of a harmonically vibrating piston in an infinitely long tube - as done in [Vincenti and Kruger]. Conceptually, the solution should be identical to a plane wave. If one then evaluates the RMS values of the fluxes in Table 3 of Section 1, it is found that there is a net RMS flow of mass. The RMS mass flow is proportional to the square of the magnitude of the sound, so it is quite small for most sound levels. However a 117 dB SPL sound would still move close to 1/2 kg per square meter in 24 hours. Obviously this is impossible for the piston problem, where it would soon produce a perfect vacuum at the piston face!

The energy flux exceeds the work done by the piston
by a factor of 7/2. This is even more serious than the mass flux problem, since
it is independent of the magnitude of the sound wave. I finally realized that
there is a missing term in equation (34), and that the correct solution for
**u**_{x} is:

The heretofore missing constant term b_{o}
is determined such that the RMS mass flux is zero. (It is very hard for me to
believe that I am the first person to notice this, but if this result is in
any of the texts I have read, I have missed it). When b_{o} is evaluated,
the result is

Note added August 2007: I finally found a reference
to this term, called *Stokes drift*, in an article by Stone.
I am still surprised that it isn't mentioned in many textbooks.

The RMS energy flux is then found to be

At the face of a piston, **u** must equal the piston velocity, and then
equation (37) also equals the RMS value of the product of the piston velocity
and the pressure on the piston face, which is the work done by the piston on
the gas. So now all is well, and equation (35) is the solution to the wave generated
by a piston harmonically vibrating with a peak velocity u_{o}.

This result has been confirmed by a numerical solution of the problem of a piston vibrating in a tube, where the piston excursion is also taken into account; this section also includes a heuristic description of the mass transport.

For reasonable sound levels the constant term in equation (35) is negligible compared to the variable term. So it is unimportant in that regard. It also does not effect the energy coupled out of a sound wave, as shown in the next section. But in terms of an overall mathematically consistent solution, it is quite important, specifically with regard to molecular mass and energy transport. This is a second order effect, and one might ask if the approximations used in developing the linear wave equations might be the cause. However it turns out that when the mass flux is computed for a solution to the nonlinear equations, the result is the same. The nonlinear solution is discussed below.

A peak power level of 1 Watt per square meter (1/2 Watt per square meter RMS) corresponds sound level of 117 dB SPL, which is a very loud sound level, and as large as we need to deal with. The corresponding peak values of velocity, pressure, density, and velocity variance for this level are:

The neglected dot product term in equation (11) is
then found to be .00014 as large as the retained term. The ratio of the variable
parts to the constant parts of equations (20) and (21) are found to be .00014
and .000055 respectively. This peak density change corresponds to a surplus
or deficit of 140 molecules per million, and the peak variance change corresponds
to a temperature fluctuation of 0.015^{0}C. The neglected drift velocity
terms in equations (17) and (19) are .00014 as large as the retained terms.
So all approximations are validated to within a small fraction of a percent.

A piston coupled
into a closed cavity can generate a large peak pressure, but obviously the minimum
pressure is zero. If the quiescent pressure is 1 atmosphere, and the piston
excursion is symmetrical with respect to its starting position, the maximum
un-clipped sinusoidal wave amplitude p_{0} is 1 atmosphere. This would
be a SPL level of 191 dB. Incredibly, competition automotive sound systems ("Extreme
Class") reached a level of 177.6 dB in 2002! Needless to say, the linearized
equations are not accurate in this case.

The energy density of a plane wave consists of kinetic
energy arising from u_{x}, and potential energy arising from p. Neglecting
second order terms the two energy components are equal and the total energy
density times c equals the RMS energy flow W.

**Plane wave solution of the non-linear equations**

For reasonable sound levels the above plane wave solution is a very accurate
approximation. The more accurate equations (11), (17), and (19) are non-linear,
and much more difficult to solve than the wave equations. A perfect sine wave
is __not__ a valid solution to these equations, and the solution inherently
includes distortion. One of the remarkable characteristics of the plane wave
solution of the non-linear equations is that the harmonic distortion grows as
the wave propagates. For the 2nd harmonic the increase is linear with propagation
distance (until the harmonic gets large), and higher order harmonics increase
with higher powers of distance. One case where this distortion becomes significant
is in the design of horn speakers, where the sound level in the narrow throat
can reach high levels (see Beranek).
A separate subsection on the solution of the non-linear equations can
be found here.

**Sound Wave Momentum **(see
the note at the end of this section)

Like light waves, sound waves in air carry a momentum flux. I recently had the realization that the clearest exhibition of wave momentum is the increase in pressure due to wave reflection by a rigid surface. This is directly analogous to "light radiation pressure." The pressure at the surface, given by equation (8), is equal to the product of the density and the variance. The density and variance, given by equations (20) and (21), are each equal to a large constant plus a small sinusoidal term; the sinusoidal terms are given by equations (32) and (33). The product of density and variance then yields three terms:

- the product of the two large constants equals the static atmospheric pressure
P
_{0}, independent of the wave amplitude and time; - the cross products of each constant and a sinusoidal term corresponds to twice (due to the reflection) the sound wave pressure p, given by equation (23). This term is proportional to the wave amplitude, and varies sinusoidally at the wave frequency;
- Third is the product of the two sinusoids; this produces a component that varies at twice the wave frequency, plus a constant component. These terms are proportional to the square of the sound wave amplitude. The constant component is the wave momentum.

For a peak velocity of u_{o}=0.0488 the magnitudes of the three terms
are P_{0}=1.05x10^{5}, 2p=40.1, and .0017 for the wave momentum.
The term b_{0} of equation (33), that eliminates any time-average mass
transfer, has an insignificant effect on the wave momentum, equal to 4x10^{-11}
for this case. The wave momentum is a second-order effect, and it is tricky
to evaluate. It is neglected in the derivation of the linearized wave equations.

The general equation for momentum flux is given in Table 3 of section 1. When I originally computed this flux for a traveling wave, using the variable variance value given in equation (33), I got the strange result that the ratio of energy flux to momentum flux was 7/9 the speed of sound. When I recently computed the flux for a standing wave created by an incident and perfectly reflected wave, I got the result that the time-average momentum flux was not independent of x. This would mean that the momentum in certain regions steadily increases without limit, which is clearly impossible. An obvious correction is to determine the variable variance value such that momentum is conserved; that produces a value given by equation (33) with γ equal to infinity. With this value, the ratio of energy flux to momentum flux for a traveling wave is 1/2 the speed of sound, exactly corresponding to the classical relationship between energy and momentum.

However we have an inconsistency: to satisfy
the differential equations we need the value given by equation (33), and to
conserve momentum we need a different value. The problem is that the linearized
equations provide a first order solution for a sound wave, and the momentum
term is a second order effect. A solution to the non-linear equations is potentially
more accurate. Unfortunately a traveling wave solution to the non-linear equations
contains an ambiguity, involving the mass transfer term b_{0}, that
also obscures the small wave momentum term. But a solution for a standing wave
in a cavity does not require this term, and it (finally!) provides the answer.

For a light wave the ratio of energy carried by the wave divided by the wave momentum is equal to the speed of light. The standard quantum physics relation is that energy comes in lumps of Planck's constant times frequency, and momentum in lumps of Planck's constant divided by the wavelength. It would seem that this implies that for any non-dispersive wave the ratio of energy to momentum should be the speed of propagation. But the rather interesting result for a sound wave is that the ratio is equal to two times the number of degrees of freedom divided by the number of degrees of freedom plus one. So only if there were one degree of freedom would the result equal the speed of propagation. For a diatomic molecule the result is 5/3 the speed of propagation. The momentum flux for the non-linear solution initially varies with x just like the first order solution, but the problematic term decays with time, so it does not lead to infinite values of momentum.

Phonons, conventionally defined as energy quanta in lattice vibration modes, are another of my recent interests. Goodstein states that phonons have momentum equal to the energy divided by the speed of sound. But the result I get when I bounce a lattice wave off a perfect reflecting boundary is that the momentum is zero (see lattice waves). Unlike sound in air, lattice vibration phonons do not carry real momentum.

This subsection on momentum was revised January 2, 2006. Issues like this are one reason I continue to be fascinated by this subject. As always, I welcome feedback, and I will be happy to post any interesting comments I receive, along with an acknowledgement.

**Onward **

Having determined how to put energy into a gas to create a sound wave, it is also interesting to consider how one gets energy back out of the sound wave - and that is the topic of the next section.

_________________________________________

Note:
I was embarrassed to learn in July 2007 that the subject of sound wave momentum
has a significant history (e. g. see the 1978 paper by Beyer),
which I was unaware of when I wrote the above section. My embarrassment is relieved
a bit by the fact that the answer I obtained using a numerical method agrees
with the analytical result derived by Beyer in his book Nonlinear
Acoustics. Also see the discussion and derivation given by Faber.
Thanks
to Dr. Krzysztof Szymanski for steering me towards this literature, and for
several interesting interchanges on the topic.