Using IWAVE

Acoustodynamics

The IWAVE acoustics application is based on the pressure-velocity form of acoustodynamics, consisting of two coupled first-order partial differential equations:

$$\rho \frac{\partial {\bf v}}{\partial t} = - \nabla p$$ (1)

$$\frac{1}{\kappa}\frac{\partial p}{\partial t} = -\nabla \cdot {\bf v}+ g$$ (2)

In these equations, $p({\bf x},t)$ is the pressure (excess, relative to an ambient equilibrium pressure), ${\bf v}({\bf x},t)$ is the particle velocity, $\rho({\bf x})$ and $\kappa({\bf x})$ are the density and particle velocity respectively. Bold-faced symbols denote vectors; the above formulation applies in 1, 2, or 3D.

The inhomogeneous term $g$ represents externally supplied energy (a ``source''), via a defect in the acoustic constitutive relation. A typical example is the isotropic point source

$$g({\bf x},t) = w(t) \delta({\bf x}-{\bf x}_s)$$

at source location ${\bf x}_s$.

Virieux (1984) introduced finite difference methods based on this formulation of acoustodynamics to the active source seismic community. Virieux (1986) extended the technique to elastodynamics, and Levander (1988) demonstrated the use of higher (than second) order difference formulas and the consequent improvement in dispersion error. IWAVE's acoustic application uses the principles introduced by these authors to offer a suite of finite difference schemes, all second order in time and of various orders of accuracy in space.

The bulk modulus and buoyancy (reciprocal density) are the natural parameters whose grid samplings appear in the difference formulae. I will display velocity and density instead in the examples below. IWAVE's acoustic application converts velocity and density to bulk modulus and buoyancy as part of the problem setup phase; the user may supply any equivalent combination of parameters.

Using IWAVE