Page 354 - Introduction to Continuum Mechanics

338 The Elastic Solid

5.26. (a) Write a displacement field for an infinite train of longitudinal waves propagating in
the direction 3 ej + 4e 2.
(b) Write a displacement field for an infinite train of transverse waves propagating in the
direction 3 ej + 4e 2 and polarized in thejfj^ plane.

5.27. Consider a material with Poisson's ratio equal 1/3 and a transverse elastic wave (as in
Section 5.10) of amplitude ej and incident on a plane boundary at an angle a\. Determine the
amplitudes and angles of reflection of the reflected waves if
(a)« 1 = 0

(b)«! = 15°.
5.28. Consider an incident transverse wave on a free boundary as in Section 5.10. For what
particular angles of incidence will the only reflected wave be transverse? (Take v - 1/3 ).
5.29. Consider a transverse elastic wave incident on a traction-free plane surface and polarized
normal to the plane of incidence. Show that the boundary condition can be satisfied with only
a reflected transverse wave that is similarly polarized, what is the relation of the amplitudes,
wavelengths, and direction of propagation of the incident and reflected wave?
5.30. Consider the problem of Section 5.10 and determine the characteristics of the reflected
waves if the boundary *2 = 0 is fixed (no motion). How are the results different from the case
of a free boundary.

5.31. A longitudinal elastic wave is incident on a fixed boundary
(a) Show that in general there are two reflected waves, one longitudinal and the other
transverse (polarized in plane normal to incident plane).
(b) Find, as in Section 5.10, the amplitude ratio of reflected to incident elastic waves.

5.32. Do the previous problem for a free boundary.
533. Verify that the thickness stretch vibration given by Eq. (5.11.3) does satisfy the lon-
gitudinal wave equation.
5.34. Do Example 5.11.1 if the right face x\ = / is free.

5.35. (a) Find the thickness stretch vibration if the x\ - 0 face is being forced by a traction
t = (/3 cos a) t )ei and the right-hand face*i = / is fixed.

(b) Find the resonant frequencies.
536. (a) Find the thickness-shear vibration if the left-hand face jcj = 0 has a forced displace-
ment u = (a cos CD t )e 3 and the right-hand facejq = / is fixed.

(b) Find the resonant frequencies.
5.37. Do the previous problem if the forced displacement is given by
u = a (cos co t e 2 + sin CD t e 3 ). Describe the particle motion throughout the plate.