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fluctuating

Acoustic theory

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

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:

begin{align} rho_0 frac{partial mathbf{v}}{partial t} + nabla p & = 0 qquad text{(Momentum balance)} frac{partial p}{partial t} + kappa~nabla cdot mathbf{v} & = 0 qquad text{(Mass balance)} end{align} where p(mathbf{x}, t) is the acoustic pressure and mathbf{v}(mathbf{x}, t) is the acoustic fluid velocity vector, mathbf{x} is the vector of spatial coordinates x, y, z, t is the time, rho_0 is the static mass density of the medium and kappa is the bulk modulus of the medium. The bulk modulus can be expressed in terms of the density and the speed of sound in the medium (c_0) as
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

cfrac{partial^2 mathbf{v}}{partial t^2} - c_0^2~nabla^2mathbf{v} = 0 qquad text{or} qquad cfrac{partial^2 p}{partial t^2} - c_0^2~nabla^2 p = 0

The acoustic wave equation (and the mass and momentum balance equations) are often expressed in terms of a scalar potential varphi where mathbf{v} = nablavarphi. In that case the acoustic wave equation is written as
cfrac{partial^2 varphi}{partial t^2} - c_0^2~nabla^2 varphi = 0

and the momentum balance and mass balance are expressed as
p + rho_0~cfrac{partialvarphi}{partial t} = 0 ~;~~ rho + cfrac{rho_0}{c_0^2}~cfrac{partialvarphi}{partial t} = 0 ~.

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
rho left(frac{partial mathbf{v}}{partial t} + mathbf{v} cdot nabla mathbf{v}right) = -nabla p + nabla cdotboldsymbol{s} + rhomathbf{b}

where mathbf{b} is the body force per unit mass, p is the pressure, and boldsymbol{s} is the deviatoric stress. If boldsymbol{sigma} is the Cauchy stress, then
p := -tfrac{1}{3}~text{tr}(boldsymbol{sigma}) ~;~~ boldsymbol{s} := boldsymbol{sigma} + p~boldsymbol{mathit{1}} where boldsymbol{mathit{1}} is the rank-2 identity tensor.

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.

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
boldsymbol{s} = mu~left[nablamathbf{v} + (nablamathbf{v})^Tright] +
lambda~(nabla cdot mathbf{v})~boldsymbol{mathit{1}}

where mu is the shear viscosity and lambda is the bulk viscosity.

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

begin{align} nablacdotboldsymbol{s} equiv cfrac{partial s_{ij}}{partial x_i} & = mu left[cfrac{partial}{partial x_i}left(cfrac{partial v_i}{partial x_j}+cfrac{partial v_j}{partial x_i}right)right] + lambda~left[cfrac{partial}{partial x_i}left(cfrac{partial v_k}{partial x_k}right)right]delta_{ij} & = mu~cfrac{partial^2 v_i}{partial x_i partial x_j} + mu~cfrac{partial^2 v_j}{partial x_ipartial x_i} + lambda~cfrac{partial^2 v_k}{partial x_kpartial x_j} & = (mu + lambda)~cfrac{partial^2 v_i}{partial x_i partial x_j} + mu~cfrac{partial^2 v_j}{partial x_i^2} & equiv (mu + lambda)~nabla(nablacdotmathbf{v}) + mu~nabla^2mathbf{v} ~. end{align}

Using the identity nabla^2mathbf{v} = nabla(nablacdotmathbf{v}) - nablatimesnablatimesmathbf{v}, we have
nablacdotboldsymbol{s} = (2mu + lambda)~nabla(nablacdotmathbf{v}) - mu~nablatimesnablatimesmathbf{v}~.
 
The equations for the conservation of momentum may then be written as
rho left(frac{partial mathbf{v}}{partial t} + mathbf{v} cdot nabla mathbf{v}right) = -nabla p + (2mu + lambda)~nabla(nablacdotmathbf{v}) - mu~nablatimesnablatimesmathbf{v} + rhomathbf{b}

Assumption 2: Irrotational flow

For most acoustics problems we assume that the flow is irrotational, that is, the vorticity is zero. In that case
nablatimesmathbf{v} = 0

and the momentum equation reduces to
rho left(frac{partial mathbf{v}}{partial t} + mathbf{v} cdot nabla mathbf{v}right) = -nabla p + (2mu + lambda)~nabla(nablacdotmathbf{v}) + rhomathbf{b}

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
rho left(frac{partial mathbf{v}}{partial t} + mathbf{v} cdot nabla mathbf{v}right) = -nabla p + (2mu + lambda)~nabla(nablacdotmathbf{v})

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
rho left(frac{partial mathbf{v}}{partial t} + mathbf{v} cdot nabla mathbf{v}right) = -nabla p

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
p = langle prangle + tilde{p} ~;~~ rho = langlerhorangle + tilde{rho} ~;~~ mathbf{v} = langlemathbf{v}rangle + tilde{mathbf{v}}

and
cfrac{partiallangle p rangle}{partial t} = 0 ~;~~ cfrac{partiallangle rho rangle}{partial t} = 0 ~;~~ cfrac{partiallangle mathbf{v} rangle}{partial t} = mathbf{0} ~.

Then the momentum equation can be expressed as
left[langlerhorangle+tilde{rho}right] left[frac{partialtilde{mathbf{v}}}{partial t} + left[langlemathbf{v}rangle+tilde{mathbf{v}}right] cdot nabla left[langlemathbf{v}rangle+tilde{mathbf{v}}right]right] = -nabla left[langle prangle+tilde{p}right]

Since the fluctuations are assumed to be small, products of the fluctuation terms can be neglected (to first order) and we have
begin{align} langlerhorangle~frac{partialtilde{mathbf{v}}}{partial t} & + left[langlerhorangle+tilde{rho}right]left[langlemathbf{v}ranglecdotnabla langlemathbf{v}rangleright]+ langlerhorangleleft[langlemathbf{v}ranglecdotnablatilde{mathbf{v}} + tilde{mathbf{v}}cdotnablalanglemathbf{v}rangleright] & = -nabla left[langle prangle+tilde{p}right] end{align}

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.,
 nablalangle p rangle =  0 ~;~~ nablalangle rho rangle = 0 ~.

The momentum equation then becomes
langlerhorangle~frac{partialtilde{mathbf{v}}}{partial t} + left[langlerhorangle+tilde{rho}right]left[langlemathbf{v}ranglecdotnabla langlemathbf{v}rangleright]+ langlerhorangleleft[langlemathbf{v}ranglecdotnablatilde{mathbf{v}} + tilde{mathbf{v}}cdotnablalanglemathbf{v}rangleright] = -nablatilde{p}

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
langlerhorangle~frac{partialtilde{mathbf{v}}}{partial t} = -nablatilde{p}

Dropping the tildes and using rho_0 := langlerhorangle, we get the commonly used form of the acoustic momentum equation
rho_0~frac{partialmathbf{v}}{partial t} + nabla p = 0 ~.

Conservation of mass

The equation for the conservation of mass in a fluid volume (without any mass sources or sinks) is given by
frac{partial rho}{partial t} + nabla cdot (rho mathbf{v}) = 0
where rho(mathbf{x},t) is the mass density of the fluid and mathbf{v}(mathbf{x},t) is the fluid velocity.

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.

Assumption 1: Small disturbances

From the assumption of small disturbances we have
p = langle prangle + tilde{p} ~;~~ rho = langlerhorangle + tilde{rho} ~;~~ mathbf{v} = langlemathbf{v}rangle + tilde{mathbf{v}}

and
cfrac{partiallangle p rangle}{partial t} = 0 ~;~~ cfrac{partiallangle rho rangle}{partial t} = 0 ~;~~ cfrac{partiallangle mathbf{v} rangle}{partial t} = mathbf{0} ~.

Then the mass balance equation can be written as
frac{partialtilde{rho}}{partial t} + left[langlerhorangle+tilde{rho}right]nabla cdotleft[langlemathbf{v}rangle+tilde{mathbf{v}}right] + nablaleft[langlerhorangle+tilde{rho}right]cdot left[langlemathbf{v}rangle+tilde{mathbf{v}}right]= 0
 
If we neglect higher than first order terms in the fluctuations, the mass balance equation becomes
frac{partialtilde{rho}}{partial t} + left[langlerhorangle+tilde{rho}right]nabla cdotlanglemathbf{v}rangle+ langlerhoranglenablacdottilde{mathbf{v}} + nablaleft[langlerhorangle+tilde{rho}right]cdotlanglemathbf{v}rangle+ nablalanglerhoranglecdottilde{mathbf{v}}= 0
 

Assumption 2: Homogeneous medium

Next we assume that the medium is homogeneous, i.e.,
 nablalangle rho rangle = 0 ~.

Then the mass balance equation takes the form
frac{partialtilde{rho}}{partial t} + left[langlerhorangle+tilde{rho}right]nabla cdotlanglemathbf{v}rangle+ langlerhoranglenablacdottilde{mathbf{v}} + nablatilde{rho}cdotlanglemathbf{v}rangle
   = 0
 

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
frac{partialtilde{rho}}{partial t} + langlerhoranglenablacdottilde{mathbf{v}} = 0
 

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:
cfrac{dp}{drho} = cfrac{gamma~p}{rho} ~;~~ gamma := cfrac{c_p}{c_v} ~;~~ c^2 = cfrac{gamma~p}{rho} ~.

where c_p is the specific heat at constant pressure, c_v is the specific heat at constant volume, and c is the wave speed. The value of gamma is 1.4 if the acoustic medium is air. 

For small disturbances

cfrac{dp}{drho} approx cfrac{tilde{p}}{tilde{rho}} ~;~~ cfrac{p}{rho} approx cfrac{langle p rangle}{langle rho rangle} ~;~~ c^2 approx c_0^2 = cfrac{gamma~langle prangle}{langle rho rangle} ~.

where c_0 is the speed of sound in the medium.

Therefore,

cfrac{tilde{p}}{tilde{rho}} = gamma~cfrac{langle p rangle}{langle rho rangle}
    = c_0^2 qquad implies qquad
cfrac{partialtilde{p}}{partial t} = c_0^2 cfrac{partialtilde{rho}}{partial t}

The balance of mass can then be written as
cfrac{1}{c_0^2}frac{partialtilde{p}}{partial t} + langlerhoranglenablacdottilde{mathbf{v}} = 0
 
Dropping the tildes and defining rho_0 := langlerhorangle gives us the commonly used expression for the balance of mass in an acoustic medium:
frac{partial p}{partial t} + rho_0~c_0^2~nablacdotmathbf{v} = 0 ~.
 

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
begin{align} nabla p & = cfrac{partial p}{partial r}~mathbf{e}_r + cfrac{1}{r}~cfrac{partial p}{partial theta}~mathbf{e}_theta + cfrac{partial p}{partial z}~mathbf{e}_z nablacdotmathbf{v} & = cfrac{partial v_r}{partial r} + cfrac{1}{r}left(cfrac{partial v_theta}{partial theta} + v_rright) + cfrac{partial v_z}{partial z} end{align}

where the velocity has been expressed as mathbf{v} = v_r~mathbf{e}_r+v_theta~mathbf{e}_theta+v_z~mathbf{e}_z.

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

rho_0~left[cfrac{partial v_r}{partial t}~mathbf{e}_r+cfrac{partial v_theta}{partial t}~mathbf{e}_theta+cfrac{partial v_z}{partial t}~mathbf{e}_zright] + cfrac{partial p}{partial r}~mathbf{e}_r + cfrac{1}{r}~cfrac{partial p}{partial theta}~mathbf{e}_theta + cfrac{partial p}{partial z}~mathbf{e}_z = 0

In terms of components, these three equations for the conservation of momentum in cylindrical coordinates are
rho_0~cfrac{partial v_r}{partial t} + cfrac{partial p}{partial r} = 0 ~;~~ rho_0~cfrac{partial v_theta}{partial t} + cfrac{1}{r}~cfrac{partial p}{partial theta} = 0 ~;~~ rho_0~cfrac{partial v_z}{partial t} + cfrac{partial p}{partial z} = 0 ~.

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

cfrac{partial p}{partial t} + kappaleft[cfrac{partial v_r}{partial r} + cfrac{1}{r}left(cfrac{partial v_theta}{partial theta} + v_rright) + cfrac{partial v_z}{partial z}right] = 0 ~.

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
p(mathbf{x}, t) = hat{p}(mathbf{x})~e^{-iomega t} ~;~~ mathbf{v}(mathbf{x}, t) = hat{mathbf{v}}(mathbf{x})~e^{-iomega t} ~;~~ i := sqrt{-1}

where omega is the frequency. Substitution of these expressions into the governing equations in cylindrical coordinates gives us the fixed frequency form of the conservation of momentum
cfrac{partialhat{p}}{partial r} = iomega~rho_0~hat{v}_r ~;~~ cfrac{1}{r}~cfrac{partialhat{p}}{partial theta} = iomega~rho_0~hat{v}_theta ~;~~ cfrac{partialhat{p}}{partial z} = iomega~rho_0~hat{v}_z

and the fixed frequency form of the conservation of mass
cfrac{iomega hat{p}}{kappa} = cfrac{partial hat{v}_r}{partial r} + cfrac{1}{r}left(cfrac{partial hat{v}_theta}{partial theta} + hat{v}_rright) + cfrac{partial hat{v}_z}{partial z} ~.

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
frac{partial^2 p}{partial r^2} + frac{1}{r}frac{partial p}{partial r} + frac{1}{r^2}~frac{partial^2 p}{partialtheta^2} + frac{omega^2rho_0}{kappa}~p = 0

Assuming that the solution of this equation can be written as
   p(r,theta) = R(r)~Q(theta)

we can write the partial differential equation as
cfrac{r^2}{R}~cfrac{d^2R}{dr^2} + cfrac{r}{R}~cfrac{dR}{dr} + cfrac{r^2omega^2rho_0}{kappa} = -cfrac{1}{Q}~cfrac{d^2Q}{dtheta^2}

The left hand side is not a function of theta while the right hand side is not a function of r. Hence,
r^2~cfrac{d^2R}{dr^2} + r~cfrac{dR}{dr} + cfrac{r^2omega^2rho_0}{kappa}~R = alpha^2~R ~;~~ cfrac{d^2Q}{dtheta^2} = -alpha^2~Q

where alpha^2 is a constant. Using the substitution
tilde{r} leftarrow left(omegasqrt{cfrac{rho_0}{kappa}}right) r = k~r

we have
tilde{r}^2~cfrac{d^2R}{dtilde{r}^2} + tilde{r}~cfrac{dR}{dtilde{r}} + (tilde{r}^2-alpha^2)~R = 0 ~;~~ cfrac{d^2Q}{dtheta^2} = -alpha^2~Q

The equation on the left is the Bessel equation which has the general solution
R(r) = A_alpha~J_alpha(k~r) + B_alpha~J_{-alpha}(k~r)

where J_alpha is the cylindrical Bessel function of the first kind and A_alpha, B_alpha are undetermined constants. The equation on the right has the general solution
Q(theta) = C_alpha~e^{ialphatheta} + D_alpha~e^{-ialphatheta}

where C_alpha,D_alpha are undetermined constants. Then the solution of the acoustic wave equation is
p(r,theta) = left[A_alpha~J_alpha(k~r) + B_alpha~J_{-alpha}(k~r)right]left(C_alpha~e^{ialphatheta} + D_alpha~e^{-ialphatheta}right)

Boundary conditions are needed at this stage to determine alpha and the other undetermined constants.

References

See also

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