Energy conservation for inhomogeneous plane waves

Abstract

In studying the reflection and transmission of inhomogeneous plane waves at liquid–solid interfaces, it is found that the theory predicts a minimum in the reflection coefficient at the Rayleigh angle. This phenomenon has not been predicted by previous treatments of homogeneous plane waves. A second surprising point is that the theory predicts that the modulus of the reflection coefficient becomes greater than unity for angles larger than the Rayleigh angle. A physical interpretation of this result is given, and it is shown that it is consistent with the conservation of energy.

Use of evanescent plane waves for low-frequency energy transmission across material interfaces

“…In contrast, if an evanescent plane wave is incident on the solid, bulk evanescent waves (both longitudinal and shear) are transmitted, and the amplitudes are greatest at the Rayleigh angle. A minimum in the reflection coefficient is observed at this angle, 41,45 owing to the resonance phenomenon (and increased transmission) that occurs when the excitation is coincident with the free wave solution. Thus, the use of an incident evanescent wave, in generating the transmitted bulk waves, provides a mechanism for energy to propagate below the interface, which is maximized at the Rayleigh angle.…”

“…Examples include surface waves, such as Rayleigh, Lamb, and Stoneley waves, as well as bulk evanescent waves. 9,41,45 In general, such inhomogeneous waves may simultaneously decay and propagate in arbitrary directions. Each of the wavevector components is represented as a complex quantity, where the real part represents propagation and the imaginary part represents exponential decay of the wave, in the respective dimensions:k x ¼ a x À jb x ;k y ¼ a y À jb y , andk z ¼ a z À jb z .…”

“…It can be shown that, in the absence of material dissipation, there is no net energy flux through any closed control surface S, which may be constructed in either medium or which may stretch across the interface, since the energy flux is continuous through the interface plane. 44,45 (The uppercase S used to denote the control surface should not be confused with the lowercase s used in subscripts to denote quantities associated with shear waves.) The net energy flux through the closed surface is thus given by the surface integral ðð…”

“…The energy flux in the presence of such waves, as well as the phenomena occurring at material interfaces, have been studied both theoretically and experimentally. 41,[44][45][46] Notably, a minimum of the reflection coefficient is observed near the Rayleigh angle for incident inhomogeneous waves in both lossless and dissipative media. The representation of such waves by complex wavevectors is analogous to the representation of waves in dissipative or heterogeneous media, 10,11,47,48 where the imaginary part of each wavevector component corresponds to decay in that direction.…”

On the use of evanescent plane waves for low-frequency energy transmission across material interfaces

“…The intensity is represented as a vectorĨ, where the components correspond to the acoustic intensities in the respective directions. For stress tensorr mn and velocity vectorũ m , the components of the instantaneous energy flux vector (per unit area) in lossless media are expressed as 44,45,56 …”