Simulating an audio beam: A second-generation wavelet collocation method to numerically solve the Khokhlov–Zabalotskaya–Kuznetsov equation
Kelvin C. M. Lee, Yew-Hin Liew, Woon‐Seng Gan · The Journal of the Acoustical Society of America · 2001
The Khokhlov–Zabalotskaya–Kuznetsov (KZK) equation is a nonlinear parabolic wave equation which describes an intense finite-amplitude directional sound beam and accounts for the combined effects of diffraction, absorption, and the nonlinearity behavior of the sound beam in air. A general exact solution has not been found for the KZK equation. Nevertheless, it is usually solved by approximate methods or numerical techniques. A second-generation wavelet collocation method using a lifting scheme is used to numerically solve the KZK equation in a highly accurate and efficient manner and has been found to have significant improvements over the conventional finite-difference scheme. Using a new approach to construct wavelets on the interval in the spatial domain (independent of the Fourier transform), the method uses a computational grid which adapts dynamically in time to allow for solution refinement in local regions with sharp transitions, e.g., development of shocks. It is also found to be particularly effective in the treatment of nonlinear terms and general boundary conditions in the equation. Furthermore with a lifting scheme, the wavelet transform can be computed in place of the original signal without the need for auxiliary memory.