Case 1: constant mass density

Assume the mass density is constant: $\rho({\bf \underline{x}}) = \rho_o$. Then the general result (13) can be evaluated as

 

$$\rho_o\d_i\ddot{u}_i = \d_i\d_i\lambda\d_k u_k \,.$$ (14)

Recalling the pressure-displacement relation (8), equation (14) can be rewritten as

 

$$\rho_o \partial_{tt}P = \lambda\d_i\d_i P \,,$$ (15)

or,

 

$$\{ \partial_{tt}- v_1^2({\bf \underline{x}})\nabla^2 \} P({\bf \underline{x}},t) = 0\,,$$ (16)

where

 

$$v_1^2 = \lambda({\bf \underline{x}})/\rho_o$$ (17)

is the propagation phase velocity of the pressure waves.

Hence, for a linear isotropic non-viscous fluid medium with constant mass density and spatially variable bulk modulus, the variable velocity acoustic wave equation (1) is rigorously valid for use in wave propagation, imaging and inversion applications.