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

Abstract

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 Γ>1, 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 Γ>>1, 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.