New Asymptotics for Old Wave Equations

Science  12 Feb 1988:
Vol. 239, Issue 4841, pp. 760-762
DOI: 10.1126/science.239.4841.760


Wave equations govern the propagation of acoustical or optical fields in diverse physical settings of interest to oceanographers, geologists, atmospheric scientists, among others. For wavelengths much smaller than all other length scales in the system the wave equation solution is generally expressed as a superposition of waveforms, each of which is determined by properties of the rays of geometrical acoustics or optics. Typically such solutions are accurate except in the vicinity of one or another caustic surface such as those defined as a surface across which a jump in the number of rays tracing through each point occurs. When numerous caustic surfaces exist, which is the generic situation, standard asymptotic solutions prove unsuitable. In this report a new asymptotic expression that overcomes deficiencies in previous approximations is introduced and characterized in an elementary way.