Isotropic wave equation

The constant-density acoustic wave equation in isotropic media is

$$\frac{\partial^2 u(\vecx ,t)}{\partial t^2} = L ^2 u(\vecx ,t),$$ (41)

where $u(\vecx ,t)$ is the pressure wavefield at spatial location $\vecx =(x,y,z)$ and time $t$ ; $L^2=v^2(\vecx ) \nabla ^2$ , $v(\vecx )$ is the P-wave velocity in the medium, and $\nabla^2$ is the Laplacian defined as $\nabla^2 = \partial_x^2 + \partial_y^2 + \partial_z^2$ . Efficient numerical solutions of the wave equation on a discrete grid is my main interest. To solve the discretized version of equation 4.1, I approximate the temporal (left) and spatial (right) derivatives in the equation, where the time derivative can be approximated by a second-order finite-difference approximation

$$u(\vecx ,t+\dt ) = 2u(\vecx ,t) - u(\vecx ,t-\dt ) - \dt ^2 \bigg[ -L^2 u(\vecx ,t) \bigg],$$ (42)

where $\dt$ denotes the length of a discrete time step.

The pseudospectral method is known as a highly accurate scheme for approximating the Laplacian operator. In doing so, the numerical errors in the solution of the wave equation are only dominated by the temporal discretization. For the isotropic case, the $-L^2$ operator in equation 4.2 can be expressed in the wavenumber domain

$$-L^2 = v_v^2 \bigg(k_x^2 + k_y^2 + k_z^2 \bigg)=v_v^2 (k_r^2+k_z^2)=v_v^2 k_{\rho}^2,$$ (43)

where $v_v$ is the velocity of a wave traveling vertically along the axis of symmetry; $k_x$ , $k_y$ and $k_z$ are spatial wavenumbers, $k_r^2=k_x^2 + k_y^2$ , and $k_{\rho}^2=k_r^2 + k_z^2$ .