Particle velocity

Progressive sine waves

The particle displacement of a progressive sine wave is given by

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

where

• ${\displaystyle {\delta }_{\mathrm{m}}}$ 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

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

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

where

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

Taking the Laplace transforms of ${\displaystyle v}$ and ${\displaystyle p}$ with respect to time yields

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

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

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

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

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

${\displaystyle v_{\mathrm{m}}=\omega {\delta }_{\mathrm{m}},}$

${\displaystyle v_{\mathrm{m}}=\frac{p_{\mathrm{m}}}{z_{\mathrm{m}}(\mathbf{r},s)}.}$

Particle velocity level

Sound velocity level (SVL) or acoustic velocity level or particle velocity level is a logarithmic measure of the effective particle velocity of a sound relative to a reference value.
Sound velocity level, denoted L_v and measured in dB, is defined by

${\displaystyle L_v=\ln \left(\frac{v}{v_0}\right)\mathrm{N}\mathrm{p}=2{\log }_{10}\left(\frac{v}{v_0}\right)\mathrm{B}=20{\log }_{10}\left(\frac{v}{v_0}\right)\mathrm{d}\mathrm{B},}$

where

• v is the root mean square particle velocity;
• v₀ is the reference particle velocity;
• 1 Np = 1 is the neper;
• 1 B = ⁠1/2⁠ ln 10 is the bel;
• 1 dB = ⁠1/20⁠ ln 10 is the decibel.

The commonly used reference particle velocity in air is

${\displaystyle v_0=5\times {10}^{-8}\mathrm{m}/\mathrm{s}.}$

The proper notations for sound velocity level using this reference are L_{v/(5 × 10⁻⁸ m/s)} or L_v (re 5 × 10⁻⁸ m/s), but the notations dB SVL, dB(SVL), dBSVL, or dB_SVL are very common, even though they are not accepted by the SI.