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:: egin{align} ho_0 frac{partial mathbf{v{partial t} + abla p & = 0 qquad ext{(Momentum balance)} \ frac{partial p}{partial t} + kappa~ abla cdot mathbf{v} & = 0 qquad ext{(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, ho_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 = ho_0 c_0^2 ~.

The acoustic wave equation is a combination of these two sets of balance equations and can be expressed as [Douglas D. Reynolds. (1981). Engineering Principles in Acoustics, Allyn and Bacon Inc., Boston.] : cfrac{partial^2 mathbf{v{partial t^2} - c_0^2~ abla^2mathbf{v} = 0 qquad ext{or} qquad cfrac{partial^2 p}{partial t^2} - c_0^2~ abla^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} = ablavarphi. In that case the acoustic wave equation is written as: cfrac{partial^2 varphi}{partial t^2} - c_0^2~ abla^2 varphi = 0 and the momentum balance and mass balance are expressed as: p + ho_0~cfrac{partialvarphi}{partial t} = 0 ~;~~ ho + cfrac{ ho_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: ho left(frac{partial mathbf{v{partial t} + mathbf{v} cdot abla mathbf{v} ight) = - abla p + abla cdotoldsymbol{s} + homathbf{b} where mathbf{b} is the body force per unit mass, p is the pressure, and oldsymbol{s} is the deviatoric stress. If oldsymbol{sigma} is the Cauchy stress, then: p := - frac{1}{3}~ ext{tr}(oldsymbol{sigma}) ~;~~ oldsymbol{s} := oldsymbol{sigma} + p~oldsymbol{mathit{1 where oldsymbol{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 :oldsymbol{s} = mu~left [ ablamathbf{v} + ( ablamathbf{v})^T ight] + lambda~( abla cdot mathbf{v})~oldsymbol{mathit{1 where mu is the shear viscosity and lambda is the bulk viscosity.

Therefore, the divergence of oldsymbol{s} is given by: egin{align} ablacdotoldsymbol{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} ight) ight] + lambda~left [cfrac{partial}{partial x_i}left(cfrac{partial v_k}{partial x_k} ight) ight] 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)~ abla( ablacdotmathbf{v}) + mu~ abla^2mathbf{v} ~. end{align} Using the identity abla^2mathbf{v} = abla( ablacdotmathbf{v}) - abla imes abla imesmathbf{v}, we have: ablacdotoldsymbol{s} = (2mu + lambda)~ abla( ablacdotmathbf{v}) - mu~ abla imes abla imesmathbf{v}~. The equations for the conservation of momentum may then be written as: ho left(frac{partial mathbf{v{partial t} + mathbf{v} cdot abla mathbf{v} ight) = - abla p + (2mu + lambda)~ abla( ablacdotmathbf{v}) - mu~ abla imes abla imesmathbf{v} + homathbf{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: abla imesmathbf{v} = 0 and the momentum equation reduces to: ho left(frac{partial mathbf{v{partial t} + mathbf{v} cdot abla mathbf{v} ight) = - abla p + (2mu + lambda)~ abla( ablacdotmathbf{v}) + homathbf{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: ho left(frac{partial mathbf{v{partial t} + mathbf{v} cdot abla mathbf{v} ight) = - abla p + (2mu + lambda)~ abla( ablacdotmathbf{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: ho left(frac{partial mathbf{v{partial t} + mathbf{v} cdot abla mathbf{v} ight) = - abla 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 (langlecdot angle) that varies in space and a small fluctuating field ( ilde{cdot}) that varies in space and time. That is: p = langle p angle + ilde{p} ~;~~ ho = langle ho angle + ilde{ ho} ~;~~ mathbf{v} = langlemathbf{v} angle + ilde{mathbf{v and: cfrac{partiallangle p angle}{partial t} = 0 ~;~~ cfrac{partiallangle ho angle}{partial t} = 0 ~;~~ cfrac{partiallangle mathbf{v} angle}{partial t} = mathbf{0} ~. Then the momentum equation can be expressed as: left [langle ho angle+ ilde{ ho} ight] left [frac{partial ilde{mathbf{v}{partial t} + left [langlemathbf{v} angle+ ilde{mathbf{v ight] cdot abla left [langlemathbf{v} angle+ ilde{mathbf{v ight] ight] = - abla left [langle p angle+ ilde{p} ight] Since the fluctuations are assumed to be small, products of the fluctuation terms can be neglected (to first order) and we have: egin{align} langle ho angle~frac{partial ilde{mathbf{v}{partial t} & + left [langle ho angle+ ilde{ ho} ight] left [langlemathbf{v} anglecdot abla langlemathbf{v} angle ight] + langle ho angleleft [langlemathbf{v} anglecdot abla ilde{mathbf{v + ilde{mathbf{vcdot ablalanglemathbf{v} angle ight] \ & = - abla left [langle p angle+ ilde{p} ight] end{align}

Assumption 6: Homogeneous medium

Next we assume that the medium is homogeneous; in the sense that the time averaged variableslangle p angle and langle ho angle have zero gradients, i.e.,: ablalangle p angle = 0 ~;~~ ablalangle ho angle = 0 ~. The momentum equation then becomes: langle ho angle~frac{partial ilde{mathbf{v}{partial t} + left [langle ho angle+ ilde{ ho} ight] left [langlemathbf{v} anglecdot abla langlemathbf{v} angle ight] + langle ho angleleft [langlemathbf{v} anglecdot abla ilde{mathbf{v + ilde{mathbf{vcdot ablalanglemathbf{v} angle ight] = - abla ilde{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} angle = 0 . Then the balance of momentum reduces to: langle ho angle~frac{partial ilde{mathbf{v}{partial t} = - abla ilde{p} Dropping the tildes and using ho_0 := langle ho angle, we get the commonly used form of the acoustic momentum equation: ho_0~frac{partialmathbf{v{partial t} + abla 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 ho}{partial t} + abla cdot ( ho mathbf{v}) = 0where ho(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 p angle + ilde{p} ~;~~ ho = langle ho angle + ilde{ ho} ~;~~ mathbf{v} = langlemathbf{v} angle + ilde{mathbf{v and: cfrac{partiallangle p angle}{partial t} = 0 ~;~~ cfrac{partiallangle ho angle}{partial t} = 0 ~;~~ cfrac{partiallangle mathbf{v} angle}{partial t} = mathbf{0} ~. Then the mass balance equation can be written as: frac{partial ilde{ ho{partial t} + left [langle ho angle+ ilde{ ho} ight] abla cdotleft [langlemathbf{v} angle+ ilde{mathbf{v ight] + ablaleft [langle ho angle+ ilde{ ho} ight] cdot left [langlemathbf{v} angle+ ilde{mathbf{v ight] = 0 If we neglect higher than first order terms in the fluctuations, the mass balance equation becomes: frac{partial ilde{ ho{partial t} + left [langle ho angle+ ilde{ ho} ight] abla cdotlanglemathbf{v} angle+ langle ho angle ablacdot ilde{mathbf{v + ablaleft [langle ho angle+ ilde{ ho} ight] cdotlanglemathbf{v} angle+ ablalangle ho anglecdot ilde{mathbf{v= 0

Assumption 2: Homogeneous medium

Next we assume that the medium is homogeneous, i.e.,: ablalangle ho angle = 0 ~. Then the mass balance equation takes the form: frac{partial ilde{ ho{partial t} + left [langle ho angle+ ilde{ ho} ight] abla cdotlanglemathbf{v} angle+ langle ho angle ablacdot ilde{mathbf{v + abla ilde{ ho}cdotlanglemathbf{v} angle = 0

Assumption 3: Medium at rest

At this stage we assume that the medium is at rest, i.e., langlemathbf{v} angle = 0 . Then the mass balance equation can be expressed as: frac{partial ilde{ ho{partial t} + langle ho angle ablacdot ilde{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}{d ho} = cfrac{gamma~p}{ ho} ~;~~ gamma := cfrac{c_p}{c_v} ~;~~ c^2 = cfrac{gamma~p}{ ho} ~. 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}{d ho} approx cfrac{ ilde{p{ ilde{ ho ~;~~ cfrac{p}{ ho} approx cfrac{langle p angle}{langle ho angle} ~;~~ c^2 approx c_0^2 = cfrac{gamma~langle p angle}{langle ho angle} ~. where c_0 is the speed of sound in the medium.

Therefore,: cfrac{ ilde{p{ ilde{ ho = gamma~cfrac{langle p angle}{langle ho angle} = c_0^2 qquad implies qquad cfrac{partial ilde{p{partial t} = c_0^2 cfrac{partial ilde{ ho{partial t} The balance of mass can then be written as: cfrac{1}{c_0^2}frac{partial ilde{p{partial t} + langle ho angle ablacdot ilde{mathbf{v = 0 Dropping the tildes and defining ho_0 := langle ho angle gives us the commonly used expression for the balance of mass in an acoustic medium:: frac{partial p}{partial t} + ho_0~c_0^2~ ablacdotmathbf{v} = 0 ~.

Governing equations in cylindrical coordinates

If we use a cylindrical coordinate system (r, heta,z) with basis vectors mathbf{e}_r, mathbf{e}_ heta, mathbf{e}_z, then the gradient of p and the divergence of mathbf{v} are given by: egin{align} abla p & = cfrac{partial p}{partial r}~mathbf{e}_r + cfrac{1}{r}~cfrac{partial p}{partial heta}~mathbf{e}_ heta + cfrac{partial p}{partial z}~mathbf{e}_z \ ablacdotmathbf{v} & = cfrac{partial v_r}{partial r} + cfrac{1}{r}left(cfrac{partial v_ heta}{partial heta} + v_r ight) + cfrac{partial v_z}{partial z} end{align} where the velocity has been expressed as mathbf{v} = v_r~mathbf{e}_r+v_ heta~mathbf{e}_ heta+v_z~mathbf{e}_z.

The equations for the conservation of momentum may then be written as: ho_0~left [cfrac{partial v_r}{partial t}~mathbf{e}_r+cfrac{partial v_ heta}{partial t}~mathbf{e}_ heta+cfrac{partial v_z}{partial t}~mathbf{e}_z ight] +cfrac{partial p}{partial r}~mathbf{e}_r + cfrac{1}{r}~cfrac{partial p}{partial heta}~mathbf{e}_ heta + 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: ho_0~cfrac{partial v_r}{partial t} + cfrac{partial p}{partial r} = 0 ~;~~ ho_0~cfrac{partial v_ heta}{partial t} + cfrac{1}{r}~cfrac{partial p}{partial heta} = 0 ~;~~ ho_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_ heta}{partial heta} + v_r ight) + cfrac{partial v_z}{partial z} ight] = 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~ ho_0~hat{v}_r ~;~~ cfrac{1}{r}~cfrac{partialhat{p{partial heta} = iomega~ ho_0~hat{v}_ heta ~;~~ cfrac{partialhat{p{partial z} = iomega~ ho_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}_ heta}{partial heta} + hat{v}_r ight) + 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_ heta 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}{partial heta^2} + frac{omega^2 ho_0}{kappa}~p = 0 Assuming that the solution of this equation can be written as: p(r, heta) = R(r)~Q( heta) 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^2 ho_0}{kappa} = -cfrac{1}{Q}~cfrac{d^2Q}{d heta^2} The left hand side is not a function of heta 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^2 ho_0}{kappa}~R = alpha^2~R ~;~~ cfrac{d^2Q}{d heta^2} = -alpha^2~Q where alpha^2 is a constant. Using the substitution: ilde{r} leftarrow left(omegasqrt{cfrac{ ho_0}{kappa ight) r = k~r we have: ilde{r}^2~cfrac{d^2R}{d ilde{r}^2} + ilde{r}~cfrac{dR}{d ilde{r + ( ilde{r}^2-alpha^2)~R = 0 ~;~~ cfrac{d^2Q}{d heta^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( heta) = C_alpha~e^{ialpha heta} + D_alpha~e^{-ialpha heta} where C_alpha,D_alpha are undetermined constants. Then the solution of the acoustic wave equation is: p(r, heta) = left [A_alpha~J_alpha(k~r) + B_alpha~J_{-alpha}(k~r) ight] left(C_alpha~e^{ialpha heta} + D_alpha~e^{-ialpha heta} ight) Boundary conditions are needed at this stage to determine alpha and the other undetermined constants.

References

ee also

* Aeroacoustics
* Transfer function
* Sound
* Acoustic impedance
* Acoustic resistance
* law of gases
* Frequency
* Fourier analysis
* Music theory
* Voice production
* Formant
* Speech synthesis
* Loudspeaker acoustics
* Lumped component model


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