Equation 8–12 contains the fluid pressure p and the structural displacement components u_x,F, u_y,F, and u_z,F as the dependent variables to solve. The finite element approximating shape functions for the spatial variation of the pressure and displacement components are given by:

(8–31)

(8–32)

where:

From Equation 8–31 and Equation 8–32, the second time derivative of the variables and the virtual change in the pressure can be expressed as follows:

(8–33)

(8–34)

(8–35)

After substituting Equation 8–31 and Equation 8–32 into Equation 8–12, the finite element statement of the wave Equation 8–5 is expressed as:

(8–36)

where:

{n} = outward normal vector at the fluid boundary

{q} = nodal mass source vector

Other terms are defined in Acoustic Fundamentals. Equation 8–36 can be written in matrix notation to create the following discretized wave equation:

(8–37)

where: