Engineering Rock Mechanics

Stress waves 209

an infinitesimal cube of material is accelerating, and applying Newton’s
Second Law of Motion, these equations become the differential equations
of motion:

30, I ar,, ar,, - a2 u,

ax ay aZ at2

+-- -P-

Although these equations may appear daunting, they are quite simple to
understand. The three components on the left-hand side of the equations
are the increments of stress in each Cartesian direction-note that in each
equation the last subscript in the numerators is the same, indicating that the
stress increments are all in the same direction. p is the density, the u variable
is for displacement and t is for time. In static equilibrium the right-hand
side of the equations is zero, because the infinitesimal cube is static: in the
equations above, the right-hand side is the equivalent of the mass x
acceleration term associated with dynamics.
If we consider a compressive stress wave travelling in the x-direction,
independent of its position in the y-z plane, then the equations of motion
reduce to

ao,- a2 u,

ax at2

• -p-

xy ar = -p- a2

ax at2

az,- a2 u,

ax at2

--p-.

It is possible, through the differential forms of the constitutive relations,

to modify the left-hand sides of these equations to give

a2u a2 u,

-PL ax2 = -p7 at

where d and ,u are the Lam6 elastic constants, which are related to the