principles and applications of microearthquake networks

may be rewritten in vector form as
4.1. Elastic WnLre Propagatiorl 77
(4.2) p(a2u/at‘) = ( A + p) v(v u) + /l.v*u
If we differentiate both sides of Eq. (4.1) with respect to xi, sum over
the three components, and bring p to the right-hand side, we obtain
(4.3) m/at2 = [(A + 2p)/plv20
If we apply the curl operator (VX) to both sides of Eq. (4.2), bring p to the
right-hand side, and note that V (V x u) = 0, we have
(4.4) a*(v x u)/arz = (p/p)v*(v x u)
Now Eqs. (4.3)-(4.4) are in the form of the classical wave equation
(4 * 5) azq,/atz t’’ v‘q,
where II, is the wave potential, and c is the velocity of propagation. Thus a
dilatational disturbance 8 (or a compressional wave) may be transmitted
through a homogeneous elastic body with a velocity V, where
(4.6)
v, = [(A + ZjL)/p]”*
according to Eq. (4.3), and a rotational disturbance V X u (or a shear
wave) may be transmitted with a velocity V, where
(4.7) v, = (p/p)*’*
according to Eq. (4.4). In seismology, these two types of waves are called
the primary (P) and the secondary (S) waves, respectively.
For a heterogeneous, isotropic, and elastic medium, the equation of
motion is more complex than Eq. (4.2). and is given by (Karal and Keller,
1959, p. 699)
(4.8) p(dZu/dt2) = ( A + p) V(V *u) + pV2u + VA(V *u)
+ vp x (V x u) + 2(Vp*V)u
where u is the displacement vector. Furthermore, the compressional wave
motion is no longer purely longitudinal, and the shear wave motion is no
longer purely transverse.
A significant portion of seismological research is based on the solution
of the elastic wave equations with the appropriate initial and boundary
conditions. However, explicit and unique solutions are rare, except for a
few simple problems. One approach is to develop through specific boundary
conditions a solution in terms of normal modes (e.g., Officer, 1958,
Chapter 2). Another approach is to transform the wave equation to the
eikonal equation and seek solutions in terms of wave fronts and rays that
are valid at high frequencies (e.g., Cerveny et al., 1977).