- Acoustic theory
Acoustic theory is the field relating to mathematical description of
sound waves. It is derived from fluid dynamics. See acousticsfor the engineeringapproach.
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:: where is the acoustic pressure and is the acoustic fluid velocity vector, is the vector of spatial coordinates , is the time, is the static mass density of the medium and is the bulk modulusof the medium. The bulk modulus can be expressed in terms of the density and the speed of sound in the medium () as:
acoustic wave equationis 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.] :The acoustic wave equation(and the mass and momentum balance equations) are often expressed in terms of a scalar potentialwhere . In that case the acoustic wave equationis written as:and the momentum balance and mass balance are expressed as:
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:where is the body force per unit mass, is the pressure, and is the deviatoric stress. If is the Cauchy stress, then:where 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 :where is the shear
viscosityand is the bulk viscosity.
Therefore, the divergence of is given by:Using the identity , we have:The equations for the conservation of momentum may then be written as:
Assumption 2: Irrotational flow
For most acoustics problems we assume that the flow is irrotational, that is, the
vorticityis zero. In that case:and the momentum equation reduces to:
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 () that varies in space and a small fluctuating field () that varies in space and time. That is:and:Then the momentum equation can be expressed as:Since the fluctuations are assumed to be small, products of the fluctuation terms can be neglected (to first order) and we have:
Assumption 6: Homogeneous medium
Next we assume that the medium is homogeneous; in the sense that the time averaged variables and have zero gradients, i.e.,:The momentum equation then becomes:
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. . Then the balance of momentum reduces to:Dropping the tildes and using , we get the commonly used form of the acoustic momentum equation:
Conservation of mass
The equation for the conservation of mass in a fluid volume (without any mass sources or sinks) is given by:where is the mass density of the fluid and 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:and:Then the mass balance equation can be written as:If we neglect higher than first order terms in the fluctuations, the mass balance equation becomes:
Assumption 2: Homogeneous medium
Next we assume that the medium is homogeneous, i.e.,:Then the mass balance equation takes the form:
Assumption 3: Medium at rest
At this stage we assume that the medium is at rest, i.e., . 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 statefor the pressure. To do that we assume that the medium is an ideal gasand all acoustic waves compress the medium in an adiabaticand reversiblemanner. The equation of state can then be expressed in the form of the differential equation::where is the specific heatat constant pressure, is the specific heatat constant volume, and is the wave speed. The value of is 1.4 if the acoustic medium is air.
For small disturbances:where is the
speed of soundin the medium.
Therefore,:The balance of mass can then be written as:Dropping the tildes and defining gives us the commonly used expression for the balance of mass in an acoustic medium::
Governing equations in cylindrical coordinates
If we use a
cylindrical coordinate systemwith basis vectors, then the gradientof and the divergenceof are given by:where the velocityhas been expressed as .
The equations for the
conservation of momentummay then be written as:In terms of components, these three equations for the conservation of momentumin cylindrical coordinatesare:
The equation for the
conservation of masscan similarly be written in cylindrical coordinatesas:
Time harmonic acoustic equations in cylindrical coordinates
The acoustic equations for the
conservation of momentumand the conservation of massare often expressed in time harmonicform (at fixed frequency). In that case, the pressures and the velocity are assumed to be time harmonic functions of the form:where is the frequency. Substitution of these expressions into the governing equations in cylindrical coordinatesgives us the fixed frequency form of the conservation of momentum: and the fixed frequency form of the conservation of mass:
Special case: No z-dependence
In the special case where the field quantities are independent of the z-coordinate we can eliminate to get:Assuming that the solution of this equation can be written as:we can write the partial differential equation as:The left hand side is not a function of while the right hand side is not a function of . Hence, :where is a constant. Using the substitution:we have:The equation on the left is the
Bessel equationwhich has the general solution:where is the cylindrical Bessel functionof the first kind and are undetermined constants. The equation on the right has the general solution:where are undetermined constants. Then the solution of the acoustic wave equation is:Boundary conditions are needed at this stage to determine and the other undetermined constants.
* law of gases
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