Page 267 - Introduction to Continuum Mechanics

Vibration of an Infinite Plate 251

Thus, the problem of the reflection of a transverse wave polarized in the plane of incidence is
solved. We mention that if the incident transverse wave is polarized normal to the plane of
incidence, no longitudinal component occurs. Also, when an incident longitudinal wave is
reflected, in addition to the regularly reflected longitudinal wave, there is also a transverse
wave polarized in the plane of incidence.
Equation (xvii) is analogous to Snell's law in optics, except here we have reflection instead
of refraction. If sinaj >n, then sin a 3> 1 and there is no longitudinal reflected wave but rather,
—i
waves of a more complicated nature will be generated. The angle aj = sin n is called the
critical angle.

5.11 Vibration of an Infinite Plate

Consider an infinite plate bounded by the planes x\ = 0 and x\ — I. These plane faces may
have either a prescribed motion or a prescribed surface traction.

The presence of these two boundaries indicates the possibility of a vibration ( a standing
wave). We begin by assuming the vibration to be of the form

and, just as for longitudinal waves, the displacement must satisfy the equation

A steady-state vibration solution to this equation is of the form

where the constant A,B,C, D, and A are determined by the boundary conditions. This
vibration mode is sometimes termed a thickness stretch vibration because the plate is being
stretched through its thickness. It is analogous to acoustic vibration of organ pipes and to the
longitudinal vibration of slender rods.
Another vibration mode can be obtained by assuming the displacement field

In this case, the displacement field must satisfy the equation

and the solution is of the same form as in the previous case. This vibration is termed
thickness-shear and it is analogous to the vibrating string.