A numerical tour of wave propagation

Basic wave equation

Define $\textbf{x}=(x,y,z)$ , time $t$ , the $s$ -th energy source function $\tilde{S}(\textbf{x},t;\textbf{x}_s)$ , pressure $p(\textbf{x},t; \textbf{x}_s)$ , particle velocity $\textbf{v}(\textbf{x},t)$ , material density $\rho(\textbf{x})$ , the bulk modulus $\kappa(\textbf{x})$ . Now we have

$\bullet$ Newton's law

$$\rho(\textbf{x})\frac{\partial\textbf{v}(\textbf{x},t;\textbf{x}_s)}{\partial t}=\nabla p(\textbf{x},t;\textbf{x}_s).$$ (1)

$\bullet$ Constitutive law

$$\frac{1}{\kappa(\textbf{x})}\frac{\partial p(\textbf{x},t;\textbf... ...dot \textbf{v}(\textbf{x},t;\textbf{x}_s)+\tilde{S}(\textbf{x},t;\textbf{x}_s).$$ (2)