Acoustic waves: should they be propagated forward in time, or forward in space?
P. Kinsler,
Department of Physics,
Imperial College London,
Prince Consort Road,
London SW7 AZ,
United Kingdom.
http://arxiv.org/abs/1202.0714
The evolution of acoustic waves can be evaluated in two ways: either
as a temporal, or a spatial propagation. Propagating in space provides
the considerable advantage of being able to handle dispersion and
propagation across interfaces with remarkable efficiency; but
propagating in time is more physical and gives correctly behaved
reflections and scattering without effort. Which should be chosen in a
given situation, and what compromises might have to be made? Here the
natural behaviors of each choice of propagation are compared and
contrasted for an ordinary second order wave equation, the
time-dependent diffusion wave equation, an elastic rod wave equation,
and the Stokes'/ van Wijngaarden's equations, each case illuminating a
characteristic feature of the technique. Either choice of propagation
axis enables a partitioning the wave equation that gives rise to a
directional factorization based on a natural "reference" dispersion
relation. The resulting exact coupled bidirectional equations then
reduce to a single unidirectional first-order wave equation using a
simple "slow evolution" assumption that minimizes effect of subsequent
approximations, while allowing a direct term-to-term comparison
between exact and approximate theories.