Causality and mathematical models in vibration and acoustics: A realistic perspective.

Abstract

Acoustic and vibrations applications require stricter causality conceptions than primitive causality (effect never precedes cause). Relativistic causality (nothing travels faster than light) is irrelevant; there is no practical reason that a satisfactory continuum‐mechanical model, holding for low to moderate frequencies, be relativistically invariant. Modifying the requirement so that the speed of light is replaced by some acoustic speed is not satisfactory, as the insertion of dissipative mechanisms into any continuum‐mechanical model invariably results in small precursors which may precede sonic‐velocity wavefronts. These are small, but they are formally nonzero at arbitrarily large distances in advance of the front. The present paper follows Ginzberg (1955) and advocates acoustic causality: the requirements that (i) the precursors die out rapidly with distance, (ii) up to some frequency of interest acoustic disturbances genuinely propagate, with the attenuation per wavelength being substantially less than unity, and (iii) vibrations and propagation are governed by coupled partial differential equations. With these principles as a guide, approximate relations involving extrapolations into the complex plane are derived, and it is shown how error bounds can be placed on applications of various members of a derived family of Kramers–Kronig relations. Very low‐frequency relaxation processes account for proportional damping in vibrations.