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Acoustic theory is the field relating to mathematical description of sound waves. It is derived from fluid dynamics. See acoustics for the engineering approach.## Derivation of the governing equations

The derivations of the above equations for waves in an acoustic medium are given below.
### Conservation of momentum

The equations for the conservation of linear momentum for a fluid medium are
#### Assumption 1: Newtonian fluid

In acoustics, the fluid medium is assumed to be Newtonian. For a Newtonian fluid, the deviatoric stress tensor is related to the velocity by
#### Assumption 2: Irrotational flow

For most acoustics problems we assume that the flow is irrotational, that is, the vorticity is zero. In that case
#### Assumption 3: No body forces

Another frequently made assumption is that effect of body forces on the fluid medium is negligible. The momentum equation then further simplifies to
#### Assumption 4: No viscous forces

Additionally, if we assume that there are no viscous forces in the medium (the bulk and shear viscosities are zero), the momentum equation takes the form
#### Assumption 5: Small disturbances

An important simplifying assumption for acoustic waves is that the amplitude of the disturbance of the field quantities is small. This assumption leads to the linear or small signal acoustic wave equation. Then we can express the variables as the sum of the (time averaged) mean field ($langlecdotrangle$) that varies in space and a small fluctuating field ($tilde\{cdot\}$) that varies in space and time. That is
#### Assumption 6: Homogeneous medium

Next we assume that the medium is homogeneous; in the sense that the time averaged variables
$langle\; p\; rangle$ and $langle\; rho\; rangle$ have zero gradients, i.e.,
#### Assumption 7: Medium at rest

At this stage we assume that the medium is at rest which implies that the mean velocity is zero, i.e. $langlemathbf\{v\}rangle\; =\; 0$. Then the balance of momentum reduces to
### Conservation of mass

The equation for the conservation of mass in a fluid volume (without any mass sources or sinks) is given by
#### Assumption 1: Small disturbances

From the assumption of small disturbances we have
#### Assumption 2: Homogeneous medium

Next we assume that the medium is homogeneous, i.e.,
#### Assumption 3: Medium at rest

At this stage we assume that the medium is at rest, i.e., $langlemathbf\{v\}rangle\; =\; 0$. Then the mass balance equation can be expressed as
#### Assumption 4: Ideal gas, adiabatic, reversible

In order to close the system of equations we need an equation of state for the pressure. To do that we assume that the medium is an ideal gas and all acoustic waves compress the medium in an adiabatic and reversible manner. The equation of state can then be expressed in the form of the differential equation:
## Governing equations in cylindrical coordinates

If we use a cylindrical coordinate system $(r,theta,z)$ with basis vectors $mathbf\{e\}\_r,\; mathbf\{e\}\_theta,\; mathbf\{e\}\_z$, then the gradient of $p$ and the divergence of $mathbf\{v\}$ are given by
### Time harmonic acoustic equations in cylindrical coordinates

The acoustic equations for the conservation of momentum and the conservation of mass are often expressed in time harmonic form (at fixed frequency). In that case, the pressures and the velocity are assumed to be time harmonic functions of the form
#### Special case: No z-dependence

In the special case where the field quantities are independent of the z-coordinate we can eliminate $v\_r,\; v\_theta$ to get
## References

## See also

The propagation of sound waves in a fluid (such as air) can be modeled by an equation of motion (conservation of momentum) and an equation of continuity (conservation of mass). With some simplifications, in particular constant density, they can be given as follows:

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- $kappa\; =\; rho\_0\; c\_0^2\; ~.$

The acoustic wave equation is a combination of these two sets of balance equations and can be expressed as

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We make several assumptions to derive the momentum balance equation for an acoustic medium. These assumptions and the resulting forms of the momentum equations are outlined below.

- $boldsymbol\{s\}\; =\; mu~left[nablamathbf\{v\}\; +\; (nablamathbf\{v\})^Tright]\; +$

Therefore, the divergence of $boldsymbol\{s\}$ is given by

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The equations for the conservation of momentum may then be written as

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nablalangle p rangle = 0 ~;~~ nablalangle rho rangle = 0 ~.The momentum equation then becomes

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- $frac\{partial\; rho\}\{partial\; t\}\; +\; nabla\; cdot\; (rho\; mathbf\{v\})\; =\; 0$

The equation for the conservation of mass for an acoustic medium can also be derived in a manner similar to that used for the conservation of momentum.

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If we neglect higher than first order terms in the fluctuations, the mass balance equation becomes

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nablalangle rho rangle = 0 ~.Then the mass balance equation takes the form

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= 0

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For small disturbances

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Therefore,

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= c_0^2 qquad implies qquadcfrac{partialtilde{p}}{partial t} = c_0^2 cfrac{partialtilde{rho}}{partial t} The balance of mass can then be written as

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Dropping the tildes and defining $rho\_0\; :=\; langlerhorangle$ gives us the commonly used expression for the balance of mass in an acoustic medium:

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The equations for the conservation of momentum may then be written as

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The equation for the conservation of mass can similarly be written in cylindrical coordinates as

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p(r,theta) = R(r)~Q(theta)we can write the partial differential equation as

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Last updated on Friday June 27, 2008 at 22:14:45 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday June 27, 2008 at 22:14:45 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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