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Peano
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Peano
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The acoustic wave equation describes the propagation of a wave through a material. This equation assumes that all movements are only small perturbations over the background flow field. They can be obtained by linearising the more general Euler equation over this background field. This clearly cannot work for complex flows, e.g., for supersonic “booms”. We describe our material by the parameters K_0, which is the so-called bulk-modulus and by the density \( \rho_0 \). Waves propagate through the material with a velocity of \( c=\sqrt{\rho K_0} \).
We solve here for velocity v and pressure p in one dimension
\( \partial_t \begin{pmatrix} p\\v_x\\v_y \end{pmatrix} + \nabla \cdot F \begin{pmatrix} p\\v_x\\v_y \end{pmatrix} = 0 \)
Where v_x and v_y denote the velocity in x- and y-direction respectively.
Further, the flux is given by:
\( F = \left[ \begin{pmatrix} 0 & K_0 & 0\\ \frac{1}{\rho_0} & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & K_0\\ 0 & 0 & 0\\ \frac{1}{\rho_0} & 0 & 0 \end{pmatrix} \right] \)
The eigenvalues of the flux are -c, 0 and c. These equations are telling us that the change in time of the velocity is proportional to the gradient of pressure.
| t=0.5 s | t=1.0 s |
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The computed pressure of the 2D acoustic wave equation at times t=0.5 s and t=1.0 s.
1.9.1 
