The particle displacement of a
progressive sine wave is given by :\delta(\mathbf{r},\, t) = \delta \sin(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}), where • \delta is the
amplitude of the particle displacement; • \varphi_{\delta, 0} is the
phase shift of the particle displacement; • \mathbf{k} is the
angular wavevector; • \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 :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 \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}), :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 \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}), where • v is the amplitude of the particle velocity; • \varphi_{v, 0} is the phase shift of the particle velocity; • p is the amplitude of the acoustic pressure; • \varphi_{p, 0} is the phase shift of the acoustic pressure. Taking the Laplace transforms of
v and
p with respect to time yields :\hat{v}(\mathbf{r},\, s) = v \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2}, :\hat{p}(\mathbf{r},\, s) = p \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}. Since \varphi_{v,0} = \varphi_{p,0}, the amplitude of the specific acoustic impedance is given by :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 :\delta = \frac{v}{\omega}, :\delta = \frac{p}{\omega z(\mathbf{r},\, s)}. ==See also==