Page 265 - Introduction to Continuum Mechanics

Reflection of Plane Elastic Waves. 249

On the free boundary fa = 0), where n = -62, the condition t = 0 leads to

i.e.,

Using Hooke's law, and noting that u$ = 0 and HI does not depend on x$, we easily see that
the condition 732 = 0 is automatically satisfied. The other two conditions, in terms of displace-
ment components, are

Performing the required differentiation, we obtain from Eqs. (v) and (vi)

Since these equations are to be satisfied onj^ = 0 for whatever values of xi and t, we must
have

so that they drop out from Eq. (vii) and (viii). Thus, at

where pand^ are integers, i.e.,

where ?? 2 ' = r] 2-(±p / 2) and ^ 3 ' = r\^-(±p / 3)

Equation (x) can be satisfied for whatever values of x\ and t only if