In this chapter we will consider the symmetry of the acoustic equation for nonhomogeneous locally-isotropic media. Let us write 42(t, r) as the displacement vector of a point of an elastic medium, r = (x, y, z), where t is the time. +Z obeys the linear differential equation (Brehovskih and Godin 1989)
p 5 = (,? + ,u) grad div 4+P + PA % + grad II * div %? + grad p x rot 4P
+ 2(grad p * grad)%, (1.1)
where p(r) is the density of the medium, p(r) is the shift modulus, and l(r) is the elastic modulus. This equation may be re-expressed in a more compact form using tensor symbols
where we write % = (U,, U,, U,), r = (x,, x2, x3) and repeated indexes denote summation. Moreover, sometimes (2.1) will be written in the form
which may be considered as the definition of the acoustic operator 8,, . In order to clarify the physical meaning, some comments connected with (1.1)
seem to be in order. This equation is based on two fundamental mechanical laws: Newton’s second law and Hooke’s law for an elastic isotropic medium. Equation (1.1) shows that an elementary volume of the medium is accelerated by a resultant force, which arises from many forces, acting on the boundaries of the volume.