The linear wave equations are approximate. My interest in a more accurate solution of the nonlinear equations was originally motivated by a paradox involving the energy and momentum of a sound wave solution of the linear equations. I am happy that this led me to the interesting subject of non-linear acoustics. After a bit of a struggle, I finally have an answer for the momentum issue as well.
This section is limited to a plane wave solution of the equations. Other cases can be found in the book by Enflo and Hedberg. A numerical solution for a standing wave in a cavity, which provides the momentum result, in given in another section.
Plane Wave Solution of the Nonlinear Equations
Three basic equations of fluid dynamics are derived elsewhere on this site. In the case of variation in one dimension only they are: the equation of continuity,
Euler's equation, with the velocity variance denoted as s≡σ2
and the energy equation (for a diatomic gas)
For the plane wave solution to the linear wave equations, the variable part of the density ρ and velocity variance s are equal to a constant times the velocity u. For the nonlinear case we will look for a solution with additional quadratic terms
In the equations below, only terms up to the first power of u are retained. Using equation (4), equation (1) becomes
Equation (2) becomes
and equation (3) becomes
If the following substitutions are made
Equations (5) and (6) become identical, yielding the same equation for u
Equation (7) becomes nearly the same as equation (9), except that the factor of two is replaced by 12/5. The first order terms ρ1 and s1 are the same values that appear in the solution of the linear equations. For u<<c equation (9) is very nearly equal to the Riemann wave equation, which is Burgers' equation for the case of zero dissipation (see Enflo and Hedberg)
Somewhat surprisingly, for the case of γ=1, equation (10) is exactly the same, and the values of ρ1 and ρ2 are exactly the same. The only difference is that s1 and s2 are both exactly zero.
A solution to equation (10) is the Bessel-Fubini solution
Where Jn is the Bessel function of the 1st kind, and σBF (which is not related to the velocity variance σ2) is defined by
The solution is not valid if σBF >1. I checked the solution given by equations (4), (8) and (11) directly against equations (1), (2), and (3). Each error was the square root of the integral over one cycle of the square of the difference between the right and left hand sides of each equation, normalized by the integral of the left hand side squared. For u0=1 and x=5.5 wavelengths, σBF=0.1 and the maximum RMS error for all three equations is 8x10-6. By contrast, the linear solution for the same case has a maximum error of 3x10-3, more than two orders of magnitude higher.
Characteristics of the Nonlinear Solution
At x=0 the solution for u is a sine wave at the single frequency ω. The density and velocity variance contain harmonics due to the second order terms in equation (4). The magnitudes of these harmonics are small, but they do greatly reduce the error in equations (1)-(3) compared to the linear solution. As the wave propagates, as long as σBF<<1, the magnitude of the 2nd harmonic grows linearly with the propagation distance x, a third harmonic grows with the square of the distance, and so on. As σBF → 1 the wave develops a vertical front face - it turns into a shock wave. As an example, for the sound level of 117 dB, where u0=.0488 meters per second, this point occurs at a distance of a little over 1000 wavelengths. It is important to note that this result only applies for plane wave behavior, which in reality would only occur in a constrained space, like a tube. In contrast, for a wave created by a piston in an infinite baffle, the amplitude decays (approximately) like 1/x, and σBF is approximately constant. Since equation (11) is only valid for a plane wave, this is not a precise description for this case, but is roughly correct.
Enflo and Hedberg also give a generalized Bessel-Fubini solution for a wave velocity that consists of sine waves at two frequencies at x=0. The nonlinearity in this case introduces intermodulation products. As an example, consider the case of frequencies of 100 and 1000 Hz, with peak velocities of .1 and .01 meters per second respectively. At x=1 meter, the harmonics at 200 and 2000 Hz are down more than 80 dB relative to the respective fundamentals, and the intermodulation products at 900 and 1100 Hz are down 70 dB.
Momentum and Energy Flux Calculations
The energy flux is relatively easy to calculate. As long as the wave solution is corrected such that there is no time-average mass flow, the energy flow equals the expected result given by equation (37) in the section on the plane wave solution to the linear equations. The energy flux is quite stable, and is not sensitive to differences between the linear and non-linear solutions. The mass flow correction for both the linear and non-linear solutions is obtained by adding the same constant term to the velocity u, as defined by equations (35) and (36) in the plane wave section (there is actually a tiny difference in the constant, but it is insignificant).
For the linear solution, if a constant is added to the velocity it is still a solution to the wave equations, and the solution for the variable part of the density and velocity variance are still given by equations (32) and (33) in that section. The momentum is not significantly changed by this constant velocity factor. For a solution to the nonlinear equation (10) above, adding a constant to a solution means it is no longer a solution. It is a bit ambiguous if the variable part of the density and velocity variance, as defined by equation (4) above, should be based on the velocity before or after adding this constant. In a formal sense it should be the final value of the velocity, but the solution is not a perfect solution to begin with, and the constant is small enough that it is still a very good approximate solution regardless of how these details are handled. However there is a very significant effect on the momentum flux: if equation (4) uses the velocity with, or without, the added constant, the ratio of energy to momentum flux is 2, or 2/3, of the speed of sound, respectively. This brackets the result I originally expected, but disagrees with the linear case.
I finally obtained an unambiguous momentum result by solving equations (1)-(3) numerically for the case of a standing wave in a cavity. The above solution method does not work for this case, so a perturbation technique was used, writing the solution as the first order solution plus correction terms. The solution is described in a separate section. The result is that the ratio of energy flux to momentum flux is 5/3 of the speed of sound for a diatomic gas. In general it is equal to two times the number of degrees of freedom divided by the number of degrees of freedom plus one.
To the Physics Contents
Note: This is a completely revised version of a previous section "Plane Wave solution of the Exact Equations," and also replaces a section "Wave Created by a Piston Moving at a Constant Velocity." The derivation in this section, up to equation (9), is original as far as I know. Originally Posted October 3, 2004. Correction regarding the momentum term posted October 11, 2004, and additional revisions posted October 16, 2004, September 20, 2005, and January 2, 2006.