Particle displacement

Progressive sine waves

The particle displacement of a progressive sine wave is given by

${\displaystyle \delta (\mathbf{r},t)=\delta \sin (\mathbf{k}\cdot \mathbf{r}-\omega t+{\phi }_{\delta ,0}),}$

where

• ${\displaystyle \delta }$ is the amplitude of the particle displacement;
• ${\displaystyle {\phi }_{\delta ,0}}$ is the phase shift of the particle displacement;
• ${\displaystyle \mathbf{k}}$ is the angular wavevector;
• ${\displaystyle \omega }$ is the angular frequency.

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by

${\displaystyle v(\mathbf{r},t)=\frac{\mathrm{\partial }\delta (\mathbf{r},t)}{\mathrm{\partial }t}=\omega \delta \cos \left(\mathbf{k}\cdot \mathbf{r}-\omega t+{\phi }_{\delta ,0}+\frac{\pi }{2}\right)=v\cos (\mathbf{k}\cdot \mathbf{r}-\omega t+{\phi }_{v,0}),}$

${\displaystyle p(\mathbf{r},t)=-\rho c^2\frac{\mathrm{\partial }\delta (\mathbf{r},t)}{\mathrm{\partial }x}=\rho c^2{k}_x\delta \cos \left(\mathbf{k}\cdot \mathbf{r}-\omega t+{\phi }_{\delta ,0}+\frac{\pi }{2}\right)=p\cos (\mathbf{k}\cdot \mathbf{r}-\omega t+{\phi }_{p,0}),}$

where

• ${\displaystyle v}$ is the amplitude of the particle velocity;
• ${\displaystyle {\phi }_{v,0}}$ is the phase shift of the particle velocity;
• ${\displaystyle p}$ is the amplitude of the acoustic pressure;
• ${\displaystyle {\phi }_{p,0}}$ is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields

${\displaystyle \hat{v}(\mathbf{r},s)=v\frac{s\cos {\phi }_{v,0}-\omega \sin {\phi }_{v,0}}{s^2+{\omega }^2},}$

${\displaystyle \hat{p}(\mathbf{r},s)=p\frac{s\cos {\phi }_{p,0}-\omega \sin {\phi }_{p,0}}{s^2+{\omega }^2}.}$

Since ${\displaystyle {\phi }_{v,0}={\phi }_{p,0}}$, the amplitude of the specific acoustic impedance is given by

${\displaystyle z(\mathbf{r},s)=|z(\mathbf{r},s)|=\left|\frac{\hat{p}(\mathbf{r},s)}{\hat{v}(\mathbf{r},s)}\right|=\frac{p}{v}=\frac{\rho c^2{k}_x}{\omega }.}$

Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by

${\displaystyle \delta =\frac{v}{\omega },}$

${\displaystyle \delta =\frac{p}{\omega z(\mathbf{r},s)}.}$