Blackbody Sound Radiation

This section replaces an earlier 1-dimensional analysis based on the Nyquist noise theorem for electric circuits. The 3-dimensional approach presented here parallels the analysis of blackbody electromagnetic radiation (see Bohm, Chapter 1, for example). The objective is similar: compute the background noise of air due to random molecular motion. For sound there is an additional constraint, not present for the electromagnetic case, which resolves the infinite power paradox without invoking quantum theory.

I am now convinced that the analysis presented in another section, based directly on the molecular model of sound, is correct, and that the blackbody approach is flawed. However the answers (under certain conditions) are pretty close. I find the analysis, and the comparison with the electromagnetic case, interesting enough to keep this section.

Problem Formulation

Imagine an infinite planar surface with air on one side and a perfect vacuum on the other side. Equilibrium conditions are assumed. The first observation is that the pressure will be statistically uniform everywhere on the surface. Thus we can place a piston of any shape anywhere on the surface, as long as it is very large compared to the spacing between molecules; for convenience, a rectangular shape is selected. The second observation is that a rectangular cavity with one open end can be placed over the piston without changing the pressure. The analysis is made within such a cavity. The cavity can have a cross section identical to the piston's shape, the first case analyzed, or extend beyond the piston perimeter, which is the second case analyzed. The piston is assumed to be immovable, and all surfaces perfectly rigid.

Analysis

Inside the cavity, sound pressure can be written in terms of a series of orthogonal modes (see the section on sound waveguide and cavity modes)

The mode frequencies are given by

where dx, dy, dz are the cavity dimensions, and each coordinate varies between 0 and its associated dimension. When all indices are non-zero the RMS kinetic energy per mode is

where ρo is the density of the gas, and c is the speed of sound in the gas. There is an equal amount of potential energy. The energy increases by a factor of 2 if one index is zero, or 4 if two indices are zero. If all three indices are zero the energy is zero. The classical Rayleigh-Jeans approach is to invoke the equipartition of energy, and assume that at equilibrium the average total energy per mode is kT, where k is Boltzman's constant, and T is the temperature in degrees Kelvin. Then, when all indices are non-zero, the average value for the square of the coefficients is

The phase of pnmq is random. There is a zero-frequency mode where all three indices are zero, but since the mode energy is also zero the coefficient cannot be evaluated using equation (3). Thus one aspect of this approach is that the DC term (i.e. atmospheric pressure) is indeterminate.

The total power is the sum of the individual mode powers. Since the number of modes is infinite, this implies that there is an infinite amount of energy in the cavity. Obviously this is a serious flaw in the classical theory. The same flaw in the electromagnetic case led Planck to postulate quantum theory.

Pressure Inside a Large Cubical Cavity

Consider the case of a cubical cavity, dx=dy=dz=d. As d → ∞, the number of modes within a narrow frequency band fo< f < fo+Δf is given by

Since all modes have equal average power, which simply add for the total power, this indicates that the "blackbody" pressure noise within the gas, in a narrow frequency band, is proportional to the square of the frequency.

The total number of modes from DC to a frequency fmax is

We now turn to the problem of infinite power in the cavity.

Quantum theory

As noted above, for the electromagnetic case the infinite-power paradox was one of the roots of quantum theory (see for example Bohm). Planck provided the solution. His derivation is based on two equations: (1) a result of statistical mechanics that states that the probability that a mode has an energy level E is proportional to exp(-E/kT), and (2) Planck's hypothesis that only quantized energy levels are possible, E=nhf, where n is 0 or a positive integer, f is the mode frequency, and h is Planck's constant. It is then found that the average energy of a mode is equal to

For low frequencies, E = kT in agreement with the classical equation. For frequencies such that hf>kT the energy drops rapidly, and the total energy is finite. At a temperature of 200C the peak of the energy spectrum is at 1.7x1013 Hz. For light this is in the middle of the infrared region, and this agrees with experimental data. For sound it would represent a wavelength smaller than an air molecule! But for sound there is an additional constraint.

Sound Blackbody Spectrum

According to classical theory the total energy in the cavity is equal to kT/2 times the number of degrees of freedom of the molecules in the cavity. Assuming for the moment that this is correct, the total mode energy then must equal this value as well. The probability relation favors modes of lower frequency. If we assume an abrupt cutoff at a frequency fmax, we can use equation (6) to calculate this upper limit. For air at room temperature the molecular density is about 2.7x1025 molecules per cubic meter, and with the 5 degrees of freedom for a diatomic molecule the result is fmax=8.6x1010 Hz. At this frequency hf<<kT, and the classical energy value is correct to within 1%, so the energy in the cavity is very close to kT/2 times the number of degrees of freedom, as assumed. So the final result is that under these conditions the "blackbody" spectrum of sound is proportional to the square of the frequency, up to fmax. In reality the spectrum will not truncate abruptly at fmax, since the higher frequencies are only less probable, not impossible. This is still a very high frequency, but not totally beyond reason.

It is interesting that for the case of sound Planck's hypothesis is not needed to resolve the infinite power paradox.

Force on a Piston at the End of a Tube

We now consider the first case, the force on a rectangular piston located at x=0 with dimensions dz, dy, inside a cavity with a cross-section matching the piston dimensions. Pressure is a rather abstract concept within a gas, and becomes manifest only by exerting force on a surface. The first thing we note is that the spatial average of pressure over the piston face is zero for all modes where m and q are not zero. So the only modes which contribute to the net force on the piston are the pnoo modes. The force is then

where A=dydz is the piston area, and the m and q subscripts have been suppressed. Since the phase of the coefficients pn is random, the expected value of the force is zero (recall that the DC atmospheric pressure term is not included). The absolute value of force-squared will contain cross products involving the phase of one coefficient minus the phase of another. It will also involve the product of two cosines of different frequency. The expected values of these cross products are therefore zero. At a random time the expected value of the square of the cosine is 1/2, the same as its mean-square value over time. The expected value of force-squared is then

For a large value of dx, the number of these modes within a frequency band fo<f<fo+Δf is given by

So now, instead of increasing with the square of the frequency, the energy spectrum of the force noise is flat as a function of frequency. The maximum frequency is the same. It seems odd that the blackbody spectrum of pressure within the gas is completely different than the spectrum of the force exerted by the pressure, but that's the result.

To evaluate the mean-square force on the piston within the bandwidth Δf, we use equation (4), reduced by a factor of 4 since two of the indices are zero, multiplied by the number of modes. The result is

This is exactly the same result obtained in the previous "Nyquist" derivation. If there is air on both sides of the piston, the result is doubled.

Force on a Small Piston in the Corner of a Large Cube

We now consider the second case, where we again place the piston at x=0, but in the corner of a cube of width d. The piston extends from z=0 to z=a, and y=0 to y=a. Integrating the pressure over this area yields the force on the piston

Again the expected value of the force is zero, and the expected values of all cross-products are zero. When m or q equals zero, the corresponding sinc function is replaced by 1.

To evaluate the mean square force we consider the case where the bandwidth extends from DC to a frequency fo; in other words, Δf=fo. With λo=c/fo, the indices must satisfy

There are two cases of interest: for a fixed value of a, select two frequencies f1 and f2, with corresponding wavelengths λ1 and λ2, such that a>>λ1 and a<<λ2. In the first case the arguments of the sinc function range from 0 to values much greater than one, and the series can be approximated by using the relationship

Then, accounting for the terms with zero indices,

As d → ∞, and using equation (4), the result is the same as equation (11). The spectrum is flat with respect to frequency.

In the second case, the arguments of the sinc functions are all much less than one, and

Equation (16) differs from equation (11) by the factor in brackets. In this case the spectrum is proportional to the square of the frequency.

Discussion

The argument that one can place a cavity of any shape over the piston without changing the force is in direct conflict with the results above. Regardless of the size of the piston with respect to a wavelength, in the derivation leading to equation (11), where the piston occupies the entire cavity cross section, we can set dx=dy=a. Then equation (16) should be the same as equation (11). But it is totally different.

Another problem with the modal approach is that the equipartition argument should apply regardless of the number of modes. For a small cavity the spacing between mode frequencies becomes large, and the modal analysis implies that the noise occurs only at these discrete, widely spaced, frequencies. The frequencies change as the cavity size changes. This clearly is incorrect.

Just to make things interesting, the result of the molecular analysis is different from both equation (16) and equation (11), but it is closer to the latter.

To the impulse sequence part of the molecular analysis

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