Wave Equation and Its General Solution in the Time Domain

Abstract

Classical themes concerning the general solutions of the wave equations in the time domain are briefly summarized in this chapter. The solution of a one-dimensional wave equation or plane waves may be formulated from point of view of the linear system theory using the convolution of the impulse responses and virtual sources converted from the initial excitation. These plane wave solutions may be extended to spherically symmetric waves following the three-dimensional wave equation. The propagation of transient waves rendered by releasing an initial condensation such as a balloon can be formulated using the general solutions for the spherically symmetric waves or the three-dimensional wave equations. Interestingly, after releasing the initial disturbance, the propagating positive condensation wave is followed by a negative condensation wave. Negative condensation waves without followers may arise from an initial condition in, for example, the limit case of a light tube. The difference in the sound perception of an isolated positive pulse and a following negative pulse would be an intriguing topic in the field of acoustics. Power spectral differences in the low-frequency components might partly explain the difference in the perception of the transient waves rendered by a balloon and a light tube.