Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
Contents
- Derivation of the governing equations
- Conservation of momentum
- Assumption 1 Newtonian fluid
- Assumption 2 Irrotational flow
- Assumption 3 No body forces
- Assumption 4 No viscous forces
- Assumption 5 Small disturbances
- Assumption 6 Homogeneous medium
- Assumption 7 Medium at rest
- Conservation of mass
- Assumption 1 Small disturbances
- Assumption 2 Homogeneous medium
- Assumption 3 Medium at rest
- Assumption 4 Ideal gas adiabatic reversible
- Governing equations in cylindrical coordinates
- Time harmonic acoustic equations in cylindrical coordinates
- Special case No z dependence
- References
Propagation of sound waves in a fluid (such as water) can be modeled by an equation of continuity (conservation of mass) and an equation of motion (conservation of momentum) . With some simplifications, in particular constant density, they can be given as follows:
where
If the flow velocity field is irrotational,
where we have used the vector Laplacian,
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
where
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
where
Therefore, the divergence of
Using the identity
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 (
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
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 flow velocity is zero, i.e.,
Dropping the tildes and using
Conservation of mass
The equation for the conservation of mass in a fluid volume (without any mass sources or sinks) is given by
where
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.,
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
For small disturbances
where
Therefore,
The balance of mass can then be written as
Dropping the tildes and defining
Governing equations in cylindrical coordinates
If we use a cylindrical coordinate system
where the flow velocity has been expressed as
The equations for the conservation of momentum may then be written as
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
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
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
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
where
we have
The equation on the left is the Bessel equation, which has the general solution
where
where
Boundary conditions are needed at this stage to determine