Particle velocity

Particle velocity, denoted ${\displaystyle \mathbf {v} }$ , is defined by

${\displaystyle \mathbf {v} ={\frac {\partial \mathbf {\delta } }{\partial t}}}$ 

where ${\displaystyle \delta }$  is the particle displacement.

The particle displacement of a progressive sine wave is given by

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

where

• ${\displaystyle \delta _{\mathrm {m} }}$  is the amplitude of the particle displacement;
• ${\displaystyle \varphi _{\delta ,0}}$  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

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

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

where

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

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

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

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

Since ${\displaystyle \varphi _{v,0}=\varphi _{p,0}}$ , 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

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

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

Sound velocity level (SVL) or acoustic velocity levelorparticle 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^[1]

${\displaystyle L_{v}=\ln \!\left({\frac {v}{v_{0}}}\right)\!~\mathrm {Np} =2\log _{10}\!\left({\frac {v}{v_{0}}}\right)\!~\mathrm {B} =20\log _{10}\!\left({\frac {v}{v_{0}}}\right)\!~\mathrm {dB} ,}$ 

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^[2]

${\displaystyle v_{0}=5\times 10^{-8}~\mathrm {m/s} .}$ 

The proper notations for sound velocity level using this reference are L_{v/(5 × 10⁻⁸ m/s)}orL_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.^[3]

• Sound
• Sound particle
• Particle displacement
• Particle acceleration

• Ohm's Law as Acoustic Equivalent. Calculations
• Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave
• The particle Velocity Can Be Directly Measured with a Microflown
• Particle velocity measured with Weles Acoustics sensor - working principle
• Acoustic Particle-Image Velocimetry. Development and Applications