Mondaic - Full Waveform Solutions

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Due to the variety of wave-equations and source-time functions one can use in Salvus Compute, the question of "which units are my results out put in?" can be non-trivial. If you are interested in precicely determining the units of input and output, please read on.

Within Salvus we consider a formulation of the acoustic wave equation based on a scalar displacement potential. The strong form of this equation is:

\rho ^{-1} c ^{-2} \partial _t^2 \phi = \nabla \cdot (\rho ^{-1} \nabla \phi) + f.

Here $\rho$ represents density, $c$ is the speed of sound, and by definition the units of the state variable $\phi$ are $\text{m}^2$ . Depending on the units of our forcing term $f$ , physically useful units can be extracted from the solution as follows.

Desired output	Symbol	Field to save	Further operations
Pressure ( $\text{N}\cdot \text{m}^{-2}$ )	$\phi _t$	`["phi_t"]`	None
Particle velocity ( $m \cdot s^{-1}$ )	$\nabla \phi \cdot \rho ^{-1}$	`["gradient-of-phi"]`	Multiply by the inverse density

Desired output	Symbol	Field to save	Further operations
Pressure ( $\text{N}\cdot \text{m}^{-2}$ )	$\phi \cdot \rho^{-1}$	`["phi"]`	Multiply by the inverse density
Particle velocity ( $\text{m}\cdot s^{-1}$ )	$\int \nabla \phi\;dt \cdot \rho ^{-2}$	`["gradient-of-phi"]`	Multiply by the inverse density squared

For more information on the physical equations solved in Salvus, please check out our paper here. Additionally, a complete list of fields which can be output can be found in the documentation.

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