The semi-analytic theory of standing waves

Abstract
Time-dependent particle-like pulses are considered as asymptotic solutions of the classical wave equation. The wave packets are localized in space with gaussian envelopes. The pulse centres propagate along the rays of the wave equation, and the envelope parameters satisfy evolution equations very similar to the ray equations for time-harmonic disturbances. However, the present theory contains an extra degree of freedom not found in the time-harmonic theory. Explicit results are presented for media with constant velocity gradients, and interesting new phenomena are identified. For example, a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction, maintains its initial spatial orientation even as the propagation direction rotates. The reflection and transmission of a pulse incident upon an interface are also discussed. The various theoretical results are illustrated by numerical simulations. This method of solution could be very useful for fast forward modelling in large-scale structures. It is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics.

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