1.3 Acoustics and the Helmholtz Equation
The governing equations
in acoustics need to be prepared for the application
of the boundary element method. For example the time-dependent sound
pressure in the original wave equation model is
replaced by a potential in the Helmholtz equation and
the BEM is applied to the latter. However, eventually
the results from the BEM need to be interpreted in
terms of physical acoustic properties - not just the sound
pressure but also the sound power, radiation ratio and also
often on the decibel scale.
In this Section it is shown how the time-dependent
wave equation governing the acoustic field can be simplified
to the Helmholtz equation when harmonic solutions are considered.
For further background to acoustic properties and acoustic
modelling see texts such as Morse (1962) or Pierce (1970).
The three classes of acoustic problem that form the subject of
this text are formally described. The content of this Section
assumes more importance towards the end of Chapters 4-6
where test problems and applications of the methods are considered.
Background knowledge of
vector calculus ,
partial differential equations and
Fourier analysis is required.
1.3.1
The Wave Equation and the Helmholtz Equation
The acoustic field is assumed to be present in the
domain of a homogeneous isotropic fluid.
Whatever the shape and nature of the domain, the acoustic field
is taken to be governed by the linear wave equation
|
Ñ2 Y(p, t) = |
1
c2
|
|
¶2
¶t2
|
Y(p,t) |
| (1.1) |
where Y(p,t) is the scalar time-dependent velocity potential related to the
time-dependent particle velocity by
and c is the propagation velocity (p and t are the spatial and
time variables). The time-dependent sound pressure Q(p,t) is given
in terms of the velocity potential by
where r is the density of the acoustic medium.
Only periodic solutions to the wave equation are considered, thus
the time-dependent velocity potential Y(p,t)
can be reduced to a sum of components each of the form
where w is the angular frequency (w = 2 ph,
where h is the
frequency in hertz) and f(p) is the (time-independent) velocity
potential. The substitution of expression (1.4) into (1.1) reduces
it to the Helmholtz (reduced wave) equation:
where k2 = [(w2)/( c2)] and k is the wavenumber. It
follows that the wavenumber and the frequency of an acoustic
medium are connected by the equation
In order to carry out a complete solution, the wave equation
is written as a series of Helmholtz problems, through
expressing the boundary conditions as a Fourier series with components of
the form (1.4). For each wavenumber
and its associated boundary condition, the Helmholtz equation
is then solved. The time-dependent sound pressure y(p,t)
can then be constituted from the separate solutions. In practical
situations, such as that considered in the example of the analysis of
engine noise in Section 5.7, the wave equation is resolved into a large series
of the order of hundreds or thousands of frequency components.
The sound pressure p(p) at the point p in the
acoustic domain is related to the velocity potential by
the formula
Often sound levels are measured on the decibel scale.
The magnitude in decibels of the sound pressure can be
found by the expression
where p* is the reference pressure which is taken to
be 2.0 ×20-5.
The phase of the signal is also important. The phase
is equivalent to the angle about the origin that is
made by plotting the point corresponding to the
sound pressure in the Argand plane and it can be
found using the arctan or tan-1 function. In Fortran
the generalised arctangent function
ATAN2 is most useful for this and the phase is
given by an expression of the form
|
ATAN2(AIMAG(p),DBLE(p)) , |
| (1.9) |
which will return a result in the range [-p, p].
The expression (1.9) gives the phase in radians. The phase in
degrees can be found by multiplying this by 180/p.
1.3.2 Other Acoustic Properties (Exterior Problems)
At any frequency of enquiry, the sound pressure at all points in the
acoustic domain describe the solution. However other acoustic
properties are often of interest in practical situations.
For example the acoustic intensity on the surface is
at each point p, where the * denotes the complex
conjugate.
The sound power is given by
or equivalently by
|
W = |
1
2 rc
|
|
ó
õ
|
SF
|
p*(q) p(q) dSq |
| (1.12) |
where SF is a hypothetical closed surface in the far-field that
encloses S.
The radiation ratio is given by
|
s RAD = |
W
|
|
1
2
|
rc |
ó
õ
|
S
|
v*(q) v(q) dSq |
|
. |
| (1.13) |
1.3.3 Acoustic Domains and Conditions
In this text a Chapter is devoted to
each of three distinct types of acoustic problems:
the interior problem, the exterior problem and the
interior modal analysis problem.
The three distinct cases of
two-, three- and
axisymmetric three-dimensional
problems are covered within each Chapter.
The solution of interior and exterior boundary value problems
is determined by the boundary condition and the shape
of the boundary. Some classes of problem, for example
the scattering problem,
also involve an incident acoustic field.
The boundary
condition specifies either the velocity potential f
(that is related closely to the sound pressure by (1.7))
or the normal velocity v (the derivative of
f with respect to the normal to the boundary) or some
relationship between them at the boundary points.
For the boundary value problems, the boundary condition
takes the following form:
|
a(p) f(p) + b(p) v(p) = f(p) (p Î S) |
| (1.14) |
where a, b and f are complex-valued functions
defined on S. For the modal analysis problem the boundary
condition is the homogeneous form of (1.14).
In a pure
radiation problem, for example, v will be specified on
the boundary and there will be no incident field.
In a pure scattering problem the boundary condition
may be specified such that it is perfectly reflecting
(v = 0), for example, or that it has an impedance boundary
condition of the form f = gv where g is a constant.
The general form of the boundary condition (1.14) allows
different forms of condition to be specified at different
regions on the boundary.
1.3.4 Condition at Infinity for Exterior Problems
For the exterior problems it is necessary to introduce a condition
at infinity. This ensures the physical requirement that all
scattered and radiated waves are outgoing. This
is termed the Sommerfeld radiation condition:
|
|
lim
r ® ¥
|
r1/2 ( |
¶f(p)
¶r
|
- i k f(p) ) = 0 in two dimensions and |
| (1.15) |
|
|
lim
r ® ¥
|
r ( |
¶ f(p)
¶r
|
- i k f(p) ) = 0 in three dimensions, |
| (1.16) |
where r is the distance from a fixed origin to a general field point.
1.3.5 Resonant Frequencies and Mode Shapes
An enclosed volume of fluid exhibits resonances in the
same way as a structure does. At the acoustic resonances
any excitation generally leads to a (theoretically)
infinite response. In practical situations the resonance
frequencies inform us of the frequencies at which the
acoustic response appears to be significantly magnified.
Knowledge of the modal properties can be of great value.
For example a particular car design may be such that
that its structural resonant frequencies coincide with the
acoustic resonances of the car interior,
having the apparent effect of magnifying the noise.
Re-designing the car so that the
two sets of resonant frequencies do not coincide
could significantly reduce the interior car noise.
In acoustic problems the eigenvalues of the Helmholtz equation
correspond to the resonant frequencies and the corresponding
eigenfunctions to the mode shapes. They are the
wavenumbers k* and the mode shapes f*
that satisfy the Helmholtz equation
|
Ñ2 f*(p) + (k*)2 f*(p) = 0 (p Î D) |
|
subject to a homogeneous boundary condition of the form
|
a(p) f(p) + b(p) v(p) = 0 (p Î S) . |
|
The solutions of the
eigenvalue problem are related to the shape of the domain and
the nature of the boundary condition.
The numerical determination of the resonant frequencies
and mode shapes of an enclosed fluid are considered in Chapter 6.
1.3.6 Units of Measurement
The units of the acoustic properties considered in this text are
not always stated. Table 1.B shows the standard units
that generally apply.
In the case of the velocity, the velocity potential and the sound pressure,
the quantities are complex-valued. These are phasar quantities;
their complex value expresses both their magnitude and relative phase.
|
Table 1.B: Standard Units
| Property | Unit
| mass | kg
| distance | m
| density | kg/m3
| sound pressure | N/m2 or Pa
| velocity | m/s
| velocity potential | no units
| power | W
| intensity | W/m2
| frequency | Hz
| wavenumber | no units | | | | | | | | | | | |
1.3.7 Acoustic Media: Air and Water
In linear acoustics, different acoustic media are simply reflected
in the change in scaling factors that relate quantities such as
wavenumber and frequency.
The scaling factors can be found from tabulated values of
the density r of the fluid
and the speed at which sound propagates through the fluid
c. Note also the density of the fluid generally varies
with other conditions such as the temperature of the medium.
The two most important acoustic media are those of air and water.
Typically air at 20° celcius and one atmosphere
has a density of 1.29 kg/m3
and speed of sound of 334 m/s. Water at 4° celcius
has a density of 1000 kg/m3 and the speed of sound is 1524 m/s.
From Section 1.3, equation (1.6) it can be deduced that
n = 53.2 k for air and
n = 242.6 k for water under the stated conditions.
From equation (1.7) it can be deduced that
p = i 431 k f for air and
p = i 1524 ×103 k f for water under same conditions.
Return to boundary-element-method.com/acoustics
Return to boundary-element-method.com