Editing Particle velocity (section)

==Progressive sine waves==
The particle displacement of a ''progressive [[sine wave]]'' is given by
:<math>\delta(\mathbf{r},\, t) = \delta_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}),</math>
where
*<math>\delta_\mathrm{m}</math> is the [[amplitude]] of the particle displacement;
*<math>\varphi_{\delta, 0}</math> is the [[phase shift]] of the particle displacement;
*<math>\mathbf{k}</math> is the [[angular wavevector]];
*<math>\omega</math> 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
:<math>v(\mathbf{r},\, t) = \frac{\partial \delta(\mathbf{r},\, t)}{\partial t} = \omega \delta \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = v_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}),</math>
:<math>p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \delta(\mathbf{r},\, t)}{\partial x} = \rho c^2 k_x \delta \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = p_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),</math>
where
*<math>v_\mathrm{m}</math> is the amplitude of the particle velocity;
*<math>\varphi_{v, 0}</math> is the phase shift of the particle velocity;
*<math>p_\mathrm{m}</math> is the amplitude of the acoustic pressure;
*<math>\varphi_{p, 0}</math> is the phase shift of the acoustic pressure.

Taking the Laplace transforms of <math>v</math> and <math>p</math> with respect to time yields
:<math>\hat{v}(\mathbf{r},\, s) = v_\mathrm{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},</math>
:<math>\hat{p}(\mathbf{r},\, s) = p_\mathrm{m} \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}.</math>

Since <math>\varphi_{v,0} = \varphi_{p,0}</math>, the amplitude of the specific acoustic impedance is given by
:<math>z_\mathrm{m}(\mathbf{r},\, s) = |z(\mathbf{r},\, s)| = \left|\frac{\hat{p}(\mathbf{r},\, s)}{\hat{v}(\mathbf{r},\, s)}\right| = \frac{p_\mathrm{m}}{v_\mathrm{m}} = \frac{\rho c^2 k_x}{\omega}.</math>

Consequently, the amplitude of the particle velocity is related to those of the particle displacement and the sound pressure by
:<math>v_\mathrm{m} = \omega \delta_\mathrm{m},</math>
:<math>v_\mathrm{m} = \frac{p_\mathrm{m}}{z_\mathrm{m}(\mathbf{r},\, s)}.</math>