Thermoviscous Attenuation of Plane, Periodic, Finite-Amplitude Sound Waves

The propagation of a plane progressive wave of finite amplitude in a thermoviscous fluid is considered. The wave is taken to be purely sinusoidal in shape at its source. The approach is through Burgers equation, which is a very good approximation of the exact equations of fluid motion when effects of nonlinearity and dissipation are relatively small but definitely not negligible. A complicated but exact solution of Burgers' equation is analyzed. The nature of the solution is found to depend strongly on a parameter Γ, which represents the importance of nonlinearity relative to dissipation. Nonlinear effects prove to be significant when Γ1, a finding in agreement with the criterion proposed by Gol'dberg (Akust. Zh. 2, 325 (1956) English transl.: Soviet Phys.—Acoust. 2, 346(1956)) concerning the appearance of shock waves. When Γ1, simple asymptotic representations of the exact solution in terms of Fourier series may be obtained. One of these corresponds to Fay's solution J. Acoust. Soc. Am. 3, 222 (1931). An equivalent “time-domain” representation shows clearly the sawtooth behavior of the wave. The sawtooth region is found to extend approximately to the point x=O.6/α, where α is the dimensional small-signal attenuation coefficient. Curves of the extra attenuation suffered by the fundamental as a result of nonlinear effects are given for values of Γ in the range 1–100 000. If Γ is large, the extra attenuation in decibels approaches the asymptotic value −12+20 log10Γ as the distance from the source becomes large. A value 1 dB below the asymptote is attained at a distance x=1/α. Application to waves in argon and air is discussed. It is found that nonlinear effects may be employed to increase the efficiency of long-range sound transmission. A possible application to low-frequency sound in the ocean is also discussed.