Acoustic wave equation#Equation

The wave equation describing a standing wave field in one dimension (position $x$) is

$$p\_\{xx\}\; -\; \backslash frac\{1\}\{c^2\}\; p\_\{tt\}\; =0,$$

where $p$ is the acoustic pressure (the local deviation from the ambient pressure) and $c$ the speed of sound, using subscript notation for the partial derivatives.Richard Feynman, Lectures in Physics, Volume 1, Chapter 47: [https://feynmanlectures.caltech.edu/I_47.html Sound. The wave equation], Caltech 1963, 2006, 2013

{{see also|Euler equations (fluid dynamics)#Euler equations}}

Start with the ideal gas law

:$P\; =\; \backslash rho\; R\_\backslash text\{specific\}T,$

where $T$ the absolute temperature of the gas and specific gas constant $R\_\backslash text\{specific\}$.

Then, assuming the process is adiabatic, pressure $P(\backslash rho)$ can be considered a function of density $\backslash rho$.

File:Derivation of acoustic wave equation.png

The conservation of mass and conservation of momentum can be written as a closed system of two equations{{sfn | LeVeque | 2002 | p=26}}

$$\backslash begin\{align\}$$

\rho_{t} + (\rho u)_{x} &= 0,\\

(\rho u)_{t} + (\rho u^2 + P(\rho))_{x} &=0.

\end{align}

This coupled system of two nonlinear conservation laws can be written in vector form as:

$$q\_t\; +\; f(q)\_x\; =\; 0,$$

with

$$q\; =\; \backslash begin\{bmatrix\}\backslash rho\; \backslash \backslash \; \backslash rho\; u\backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}q\_\{(1)\}\; \backslash \backslash \; q\_\{(2)\}\backslash end\{bmatrix\},\; \backslash quad\; f(q)\; =\; \backslash begin\{bmatrix\}\backslash rho\; u\; \backslash \backslash \; \backslash rho\; u^2\; +\; P(\backslash rho)\backslash end\{bmatrix\}=\backslash begin\{bmatrix\}\; q\_\{(2)\}\; \backslash \backslash \; q\_\{(2)\}^2/q\_\{(1)\}\; +\; P(q\_\{(1)\})\backslash end\{bmatrix\}.$$

To linearize this equation, let{{sfn | LeVeque | 2002 | pp=27-28}}

$$q(x,t)\; =\; q\_0\; +\; \backslash tilde\{q\}(x,t),$$

where $q\_0\; =\; (\; \backslash rho\_0\; ,\; \backslash rho\_0\; u\_0)$ is the (constant) background state and $\backslash tilde\{q\}$ is a sufficiently small perturbation, i.e., any powers or products of $\backslash tilde\{q\}$ can be discarded. Hence, the taylor expansion of $f(q)$ gives:

$$f(q\_0\; +\; \backslash tilde\{q\})\; \backslash approx\; f(q\_0)\; +\; f\text{'}(q\_0)\backslash tilde\{q\}$$

where

This results in the linearized equation

$$\backslash tilde\{q\}\_t\; +\; f\text{'}(q\_0)\backslash tilde\{q\}\_x\; =\; 0\; \backslash quad\; \backslash Leftrightarrow\; \backslash quad\; \backslash begin\{align\}$$

\tilde{\rho}_{t} + (\widetilde{\rho u})_{x} &= 0\\

(\widetilde{\rho u})_{t} + (-u_{0}^2 + P'(\rho_{0}))\tilde{\rho }_{x} + 2u_{0}(\widetilde{\rho u})_x &=0

\end{align}

Likewise, small perturbations of the components of $q$ can be rewritten as:

$$\backslash rho\; u\; =\; (\backslash rho\_0\; +\backslash tilde\{\backslash rho\})(u\_0\; +\backslash tilde\{u\})\; =\; \backslash rho\_0\; u\_0\; +\; \backslash tilde\{\backslash rho\}u\_0\; +\; \backslash rho\_0\; \backslash tilde\{u\}\; +\; \backslash tilde\{\backslash rho\}\backslash tilde\{u\}$$

such that

$$\backslash widetilde\{\backslash rho\; u\}\; \backslash approx\; \backslash tilde\{\backslash rho\}u\_0\; +\; \backslash rho\_0\; \backslash tilde\{u\},$$

and pressure perturbations relate to density perturbations as:

$$p\; =\; p\_\{0\}\; +\; \backslash tilde\{p\}=\; P(\backslash rho\_0\; +\; \backslash tilde\{\backslash rho\})\; =\; P(\backslash rho\_\{0\})\; +\; P\text{'}(\backslash rho\_\{0\})\backslash tilde\{\backslash rho\}\; +\; \backslash dots$$

such that:

$$p\_0\; =\; P(\backslash rho\_0),\; \backslash quad\; \backslash tilde\{p\}\backslash approx\; P\text{'}(\backslash rho\_0)\backslash tilde\{\backslash rho\},$$

where $P\text{'}(\backslash rho\_0)$ is a constant, resulting in the alternative form of the linear acoustics equations:

$$\backslash begin\{align\}$$

\tilde{p}_{t} + u_0 \tilde{p}_x + K_0 \tilde{u}_x &= 0,\\

\rho_0\tilde{u}_t + \tilde{p}_x + \rho_0 u_0 \tilde{u}_x &=0.

\end{align}

where $K\_0\; =\; \backslash rho\_0\; P\text{'}(\backslash rho\_0)$ is the bulk modulus of compressibility. After dropping the tilde for convenience, the linear first order system can be written as:

$$\backslash begin\{bmatrix\}$$

p\\

u

\end{bmatrix}_{t} + \begin{bmatrix}

u_{0} & K_0\\

1/\rho_0 & u_0

\end{bmatrix}\begin{bmatrix}

p\\

u

\end{bmatrix}_{x}=0.

While, in general, a non-zero background velocity is possible (e.g. when studying the sound propagation in a constant-strength wind), it will be assumed that $u\_\{0\}=0$. Then the linear system reduces to the second-order wave equation:

$$p\_\{tt\}\; =\; -K\_0\; u\_\{xt\}\; =\; -K\_0\; u\_\{tx\}\; =\; K\_0\backslash left(\backslash frac\{1\}\{\backslash rho\_0\}p\_x\backslash right)\_x\; =\; c\_\{0\}^2\; p\_\{xx\},$$

with $c\_0\; =\; \backslash sqrt\{K\_0/\backslash rho\_0\}$ the speed of sound.

Hence, the acoustic equation can be derived from a system of first-order

advection equations that follow directly from physics, i.e., the first integrals:

$$q\_t\; +\; Aq\_x\; =\; 0,$$

with

q = \begin{bmatrix}p\\ u\end{bmatrix}, \quad A = \begin{bmatrix}

0 & K_0\\

1/\rho_0 & 0

\end{bmatrix}.

Conversely, given the second-order equation $p\_\{tt\}\; =c\_\{0\}^2\; p\_\{xx\}$ a first-order system can be derived:

$$q\_t\; +\; \backslash hat\{A\}q\_x\; =\; 0,$$

with

q = \begin{bmatrix}p_t\\ -p_x\end{bmatrix}, \quad \hat{A} = \begin{bmatrix}

0 & c_{0}^2\\

1 & 0

\end{bmatrix},

where matrix $A$ and $\backslash hat\{A\}$ are similar.{{sfn | LeVeque | 2002 | p=33}}

Provided that the speed $c$ is a constant, not dependent on frequency (the dispersionless case), then the most general solution is

:$p\; =\; f(c\; t\; -\; x)\; +\; g(c\; t\; +\; x)$

where $f$ and $g$ are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one ($f$) traveling up the x-axis and the other ($g$) down the x-axis at the speed $c$. The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either $f$ or $g$ to be a sinusoid, and the other to be zero, giving

:$p=p\_0\; \backslash sin(\backslash omega\; t\; \backslash mp\; kx)$.

where $\backslash omega$ is the angular frequency of the wave and $k$ is its wave number.

FeynmanRichard Feynman, Lectures in Physics, Volume 1, 1969, Addison Publishing Company, Addison provides a derivation of the wave equation for sound in three dimensions as

:$\backslash nabla\; ^2\; p\; -\; \{1\; \backslash over\; c^2\}\; \{\; \backslash partial^2\; p\; \backslash over\; \backslash partial\; t\; ^2\; \}\; =\; 0,$

where $\backslash nabla\; ^2$ is the Laplace operator, $p$ is the acoustic pressure (the local deviation from the ambient pressure), and $c$ is the speed of sound.

A similar looking wave equation but for the vector field particle velocity is given by

:$\backslash nabla\; ^2\; \backslash mathbf\{u\}\backslash ;\; -\; \{1\; \backslash over\; c^2\}\; \{\; \backslash partial^2\; \backslash mathbf\{u\}\backslash ;\; \backslash over\; \backslash partial\; t\; ^2\; \}\; =\; 0$.

In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form

:$\backslash nabla\; ^2\; \backslash Phi\; -\; \{1\; \backslash over\; c^2\}\; \{\; \backslash partial^2\; \backslash Phi\; \backslash over\; \backslash partial\; t\; ^2\; \}\; =\; 0$

and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):

: $\backslash mathbf\{u\}\; =\; \backslash nabla\; \backslash Phi\backslash ;$,

: $p\; =\; -\backslash rho\; \{\backslash partial\; \backslash over\; \backslash partial\; t\}\backslash Phi$.

The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time-dependence factor of $e^\{i\backslash omega\; t\}$ where $\backslash omega\; =\; 2\; \backslash pi\; f$ is the angular frequency. The explicit time dependence is given by

:$p(r,t,k)\; =\; \backslash operatorname\{Real\}\backslash left[p(r,k)\; e^\{i\backslash omega\; t\}\backslash right]$

Here $k\; =\; \backslash omega/c\; \backslash$ is the wave number.

: $p(r,k)=Ae^\{\backslash pm\; ikr\}$.

: $p(r,k)=AH\_0^\{(1)\}(kr)\; +\; \backslash \; BH\_0^\{(2)\}(kr)$.

where the asymptotic approximations to the Hankel functions, when $kr\backslash rightarrow\; \backslash infty$, are

:$H\_0^\{(1)\}(kr)\; \backslash simeq\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; kr\}\}e^\{i(kr-\backslash pi/4)\}$

:$H\_0^\{(2)\}(kr)\; \backslash simeq\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; kr\}\}e^\{-i(kr-\backslash pi/4)\}$.

: $p(r,k)=\backslash frac\{A\}\{r\}e^\{\backslash pm\; ikr\}$.

Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist.

• Acoustics
• Acoustic attenuation
• Acoustic theory
• Differential equations
• Fluid dynamics
• Ideal gas law
• Madelung equations
• One-way wave equation
• Pressure
• Thermodynamics
• Wave equation

{{reflist}}

• {{cite book | last=LeVeque | first=Randall J. | title=Finite Volume Methods for Hyperbolic Problems | publisher=Cambridge University Press | date=2002 | isbn=978-0-521-81087-6 | doi=10.1017/cbo9780511791253}}

Category:Acoustic equations