Definition of a Linear Medium for One-Dimensional Longitudinal Motion

Abstract

If the longitudinal stress is linear in the extension, then the exact equation of one-dimensional motion, in a material (“Lagrangian”) description, is the linear wave equation. This result provides a rational basis for defining a hypothetical linear solid or fluid medium. Such a medium has nonzero higher-order elastic constants. The expected consequences of linearity of the equation of motion are noted. These include distortionless propagation of finite-amplitude waves, the absence of interaction of waves with each other, stress independence of the travel time or resonance frequency, and the absence of acoustic radiation pressure. The idea of distortionless propagation of traveling waves is treated in detail in both the material and spatial descriptions, and it is emphasized that distortion of longitudinal mechanical waves is not a necessary consequence of kinematics. In a spatial (“Eulerian”) description, the linear medium has a nonlinear equation of motion, but this nonlinear equation has distortionless traveling wave solutions of the form f(t±x/V). Some examples are given of three dimensional constitutive equations that yield the described linear behavior in one dimension. The equation of state of the hypothetical linear medium is highly unrealistic, but is of interest because nonlinear phenomena can be ascribed to departures from it. [An appendix discusses the momentum in a plane progressive wave. Consistent with the absence of radiation pressure, the instantaneous momentum in a plane progressive wave in a semi-infinite linear medium that was initially at rest is simply proportional to the displacement of the face. The still persistent idea that every wave train of intensity J propagated with speed W has momentum J/W2 per unit volume is shown to be an unwarranted generalization. Such a wave train can be set up in the semi-infinite linear medium if the prescribed velocity at the face is the sum of a sinusoidal part and a properly chosen constant term. A special equation of state is found for which power and momentum are related as above in the initially formed simple (purely progressive) wave resulting from a sinusoidal velocity of the face that starts at time zero.]