Page 255 - Introduction to Continuum Mechanics

Plane Irrotational Wave 239

Consider the motion

representing an infinite train of sinusoidal plane waves. In this motion, every particle executes
simple harmonic oscillations of small amplitude £ around its natural state, the motion being
always parallel to the QI direction. All particles on a plane perpendicular to ej are at the same
phase of the harmonic motion at any one time [i.e., the same value of (2ji/l)(x\ - c^ t)\, A
particle which at time/ isatjti + ^acquires at t + dt the same phase of motion of the particle
whichisatJCiattimeHf(^i 4- dx\ )—c^(t + dt) = x\ — c^t,i.Q. tdxi/dt = c/,. Thus c^ is known
as the phase velocity (the velocity with which the sinusoidal disturbance of wavelength / is
moving in the ej direction). Since the motions of the particles are parallel to the direction of
the propagation of wave, it is a longitudinal wave.
We shall now consider if this wave is a possible motion in an elastic medium.
The strain components corresponding to the «,- given in Eq. (5.8.1) are

The stress components are (note e - EH +0 + 0 = EH )

Substituting Ty and w/ into the equations of motion in the absence of body forces, i.e.,

we easily see that the second and third equations of motion are automatically satisfied (0 = 0)
and the first equation demands that

or