Page 264 - Introduction to Continuum Mechanics

248 The Elastic Solid

5.10 Reflection of Plane Elastic Waves.

In Fig. 5.5, the plane #2 = 0 is the free boundary of an elastic medium, occupying the lower
half-space X2 ^0. We wish to study how an incident plane wave is reflected by the boundary.
Consider an incident transverse wave of wavelength /j, polarized in the plane of incidence with
an incident angle a j, (see Fig. 5.5). Since*2 = 0 is a free boundary, the surface traction on the
plane is zero at all times. Thus, the boundary will generate reflection waves in such a way that
when they are superposed on the incident wave, the stress vector on the boundary vanishes at
all times.
Let us superpose on the incident transverse wave two reflection waves (see Fig. 5.5), one
transverse, the other longitudinal, both oscillating in the plane of incidence. The reason for
superposing not only a reflected transverse wave but also a longitudinal one is that if only one
is superposed, the stress-free condition on the boundary in general cannot be met, as will
become obvious in the following derivation.

Fig. 5.5

Let «,• denote the displacement components of the superposition of the three waves, then
from the results of Example 5.9.4, we have

where