Particle displacement

Particle displacement, denoted δ, is given by

{{cite book

| author = John Eargle

| author-link = John M. Eargle

| title = The Microphone Book: From mono to stereo to surround – a guide to microphone design and application | publisher = Focal Press | date = January 2005

| location = Burlington, Ma

| pages = 27

| url =https://books.google.com/books?id=w8kXMVKOsY0C&q=instantaneous+particle+displacement&pg=PA27

| isbn =978-0-240-51961-6

}}

:$\backslash mathbf\; \backslash delta\; =\; \backslash int\_\{t\}\; \backslash mathbf\; v\backslash ,\; \backslash mathrm\{d\}t$

where v is the particle velocity.

The particle displacement of a progressive sine wave is given by

:$\backslash delta(\backslash mathbf\{r\},\backslash ,\; t)\; =\; \backslash delta\; \backslash sin(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\_\{\backslash delta,\; 0\}),$

where

• $\backslash delta$ is the amplitude of the particle displacement;
• $\backslash varphi\_\{\backslash delta,\; 0\}$ is the phase shift of the particle displacement;
• $\backslash mathbf\{k\}$ is the angular wavevector;
• $\backslash omega$ 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

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

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

where

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

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

:$\backslash hat\{v\}(\backslash mathbf\{r\},\backslash ,\; s)\; =\; v\; \backslash frac\{s\; \backslash cos\; \backslash varphi\_\{v,0\}\; -\; \backslash omega\; \backslash sin\; \backslash varphi\_\{v,0\}\}\{s^2\; +\; \backslash omega^2\},$

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

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

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

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

:$\backslash delta\; =\; \backslash frac\{v\}\{\backslash omega\},$

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

• Sound
• Sound particle
• Particle velocity
• Particle acceleration

{{Reflist}}

Related Reading:

• {{cite book | author=Wood, Robert Williams

| title=Physical optics | publisher=New York: The Macmillan Company | year=1914}}

• {{cite book |author1=Strong, John Donovan |author2=Hayward, Roger |name-list-style=amp | title=Concepts of Classical Optics | publisher=Dover Publications | date=January 2004

| isbn= 978-0-486-43262-5 }}

• {{cite book | last = Barron | first = Randall F.

| title = Industrial noise control and acoustics

| publisher = CRC Press | date = January 2003

| location = NYC, New York | pages = 79, 82, 83, 87

| url =https://books.google.com/books?id=k1tXPl2hC-cC&q=instantaneous+particle+displacement&pg=PA82

| isbn =978-0-8247-0701-9}}

• [http://acoustics.open.ac.uk/802574C70048F266/%28httpAssets%29/EC0466002A53AFDD802574E200381B0C/$file/ali_tonddast_navaei_thesis.pdf Acoustic Particle-Image Velocimetry. Development and Applications]
• [http://www.sengpielaudio.com/calculator-ak-ohm.htm Ohm's Law as Acoustic Equivalent. Calculations]
• [http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave]

Category:Acoustics

Category:Sound

Category:Sound measurements

Category:Physical quantities