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 flow velocity by
![{\displaystyle {\boldsymbol {\tau }}=\mu ~\left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{T}\right]+\lambda ~(\nabla \cdot \mathbf {u} )~{\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffceb4db528b36e739f8ff1695e993b51c5f1da9)
where
is the shear viscosity and
is the bulk viscosity.
Therefore, the divergence of
is given by
![{\begin{aligned}\nabla \cdot {\boldsymbol {\tau }}\equiv {\cfrac {\partial s_{{ij}}}{\partial x_{i}}}&=\mu \left[{\cfrac {\partial }{\partial x_{i}}}\left({\cfrac {\partial u_{i}}{\partial x_{j}}}+{\cfrac {\partial u_{j}}{\partial x_{i}}}\right)\right]+\lambda ~\left[{\cfrac {\partial }{\partial x_{i}}}\left({\cfrac {\partial u_{k}}{\partial x_{k}}}\right)\right]\delta _{{ij}}\\&=\mu ~{\cfrac {\partial ^{2}u_{i}}{\partial x_{i}\partial x_{j}}}+\mu ~{\cfrac {\partial ^{2}u_{j}}{\partial x_{i}\partial x_{i}}}+\lambda ~{\cfrac {\partial ^{2}u_{k}}{\partial x_{k}\partial x_{j}}}\\&=(\mu +\lambda )~{\cfrac {\partial ^{2}u_{i}}{\partial x_{i}\partial x_{j}}}+\mu ~{\cfrac {\partial ^{2}u_{j}}{\partial x_{i}^{2}}}\\&\equiv (\mu +\lambda )~\nabla (\nabla \cdot {\mathbf {u}})+\mu ~\nabla ^{2}{\mathbf {u}}~.\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb3148adbff7f31faa4d7322919173d52a0a650)
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 vorticity is 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
![\left[\langle \rho \rangle +{\tilde {\rho }}\right]\left[{\frac {\partial {\tilde {{\mathbf {u}}}}}{\partial t}}+\left[\langle {\mathbf {u}}\rangle +{\tilde {{\mathbf {u}}}}\right]\cdot \nabla \left[\langle {\mathbf {u}}\rangle +{\tilde {{\mathbf {u}}}}\right]\right]=-\nabla \left[\langle p\rangle +{\tilde {p}}\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40512e54af694d92ba1dc0c19b69f850ce1ea79)
Since the fluctuations are assumed to be small, products of the fluctuation terms can be neglected (to first order) and we have
![{\begin{aligned}\langle \rho \rangle ~{\frac {\partial {\tilde {{\mathbf {u}}}}}{\partial t}}&+\left[\langle \rho \rangle +{\tilde {\rho }}\right]\left[\langle {\mathbf {u}}\rangle \cdot \nabla \langle {\mathbf {u}}\rangle \right]+\langle \rho \rangle \left[\langle {\mathbf {u}}\rangle \cdot \nabla {\tilde {{\mathbf {u}}}}+{\tilde {{\mathbf {u}}}}\cdot \nabla \langle {\mathbf {u}}\rangle \right]\\&=-\nabla \left[\langle p\rangle +{\tilde {p}}\right]\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be0f83ab1e8686dfcd71be55d066dc7c6477a13b)
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
![\langle \rho \rangle ~{\frac {\partial {\tilde {{\mathbf {u}}}}}{\partial t}}+\left[\langle \rho \rangle +{\tilde {\rho }}\right]\left[\langle {\mathbf {u}}\rangle \cdot \nabla \langle {\mathbf {u}}\rangle \right]+\langle \rho \rangle \left[\langle {\mathbf {u}}\rangle \cdot \nabla {\tilde {{\mathbf {u}}}}+{\tilde {{\mathbf {u}}}}\cdot \nabla \langle {\mathbf {u}}\rangle \right]=-\nabla {\tilde {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e1d9870083de7a2ec74355c955f7731902ab95)
Assumption 7: Medium at rest
At this stage we assume that the medium is at rest, which implies that the mean flow 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 flow 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
![{\frac {\partial {\tilde {\rho }}}{\partial t}}+\left[\langle \rho \rangle +{\tilde {\rho }}\right]\nabla \cdot \left[\langle {\mathbf {u}}\rangle +{\tilde {{\mathbf {u}}}}\right]+\nabla \left[\langle \rho \rangle +{\tilde {\rho }}\right]\cdot \left[\langle {\mathbf {u}}\rangle +{\tilde {{\mathbf {u}}}}\right]=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ac647bd1b51d88e3628e2b280a540e6de99ec5)
If we neglect higher than first order terms in the fluctuations, the mass balance equation becomes
![{\frac {\partial {\tilde {\rho }}}{\partial t}}+\left[\langle \rho \rangle +{\tilde {\rho }}\right]\nabla \cdot \langle {\mathbf {u}}\rangle +\langle \rho \rangle \nabla \cdot {\tilde {{\mathbf {u}}}}+\nabla \left[\langle \rho \rangle +{\tilde {\rho }}\right]\cdot \langle {\mathbf {u}}\rangle +\nabla \langle \rho \rangle \cdot {\tilde {{\mathbf {u}}}}=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b23d73e230210fa880b4568f1b55cc74d64c90e4)
Assumption 2: Homogeneous medium
Next we assume that the medium is homogeneous, i.e.,

Then the mass balance equation takes the form
![{\frac {\partial {\tilde {\rho }}}{\partial t}}+\left[\langle \rho \rangle +{\tilde {\rho }}\right]\nabla \cdot \langle {\mathbf {u}}\rangle +\langle \rho \rangle \nabla \cdot {\tilde {{\mathbf {u}}}}+\nabla {\tilde {\rho }}\cdot \langle {\mathbf {u}}\rangle =0](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3960d8d72513773489d66b2ba76ea4b0bc95658)
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
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:

where
is the specific heat at constant pressure,
is the specific heat at 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 sound in 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 system
with basis vectors
, then the gradient of
and the divergence of
are given by

where the flow velocity has been expressed as
.
The equations for the conservation of momentum may then be written as
![\rho _{0}~\left[{\cfrac {\partial u_{r}}{\partial t}}~{\mathbf {e}}_{r}+{\cfrac {\partial u_{\theta }}{\partial t}}~{\mathbf {e}}_{\theta }+{\cfrac {\partial u_{z}}{\partial t}}~{\mathbf {e}}_{z}\right]+{\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](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b1aae42ee36c0cc3e9cbc7855beeff5bd385aa)
In terms of components, these three equations for the conservation of momentum in cylindrical coordinates are

The equation for the conservation of mass can similarly be written in cylindrical coordinates as
![{\cfrac {\partial p}{\partial t}}+\kappa \left[{\cfrac {\partial u_{r}}{\partial r}}+{\cfrac {1}{r}}\left({\cfrac {\partial u_{\theta }}{\partial \theta }}+u_{r}\right)+{\cfrac {\partial u_{z}}{\partial z}}\right]=0~.](https://wikimedia.org/api/rest_v1/media/math/render/svg/14e0e691b30273e708b9c5c2f113da1a3682c99e)
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 flow 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 coordinates gives 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 equation, which has the general solution

where
is the cylindrical Bessel function of 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
![p(r,\theta) = \left[A_\alpha~J_\alpha(k~r) + B_\alpha~J_{-\alpha}(k~r)\right]\left(C_\alpha~e^{i\alpha\theta} + D_\alpha~e^{-i\alpha\theta}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8de433e62804358d7a32e6cc88196e28cf149a62)
Boundary conditions are needed at this stage to determine
and the other undetermined constants.