Approximate solution methods for nonlinear acoustic propagation over long ranges

This paper gives asymptotic solutions to generalized Burgers equations governing the propagation of weakly nonlinear acoustic waves under the influence of geometrical spreading and thermoviscous diffusion. Geometrical effects are included through a general ray tube area function, $\mathscr{A}$(r), and solutions are obtained up to arbitrarily large ranges for initially sinusoidal waves by the use of rational asymptotic techniques in the two cases of weak diffusivity and of high initial wave amplitude. These solutions use results obtained earlier by Nimmo & Crighton. Simpler approximate techniques to obtain similar solutions are then discussed. The two approximate methods considered, proposed by Shooter et al. and Rudnick, are based on physical considerations, rather than asymptotic theory. The validity of such methods is demonstrated for a broad, though restricted, range of physical situations.