General Solution of the Wave Equation
The general solution of the wave equation provides a means for computing the sound wave pressure (or velocity) anywhere in a region, in terms of values of pressure and velocity on the boundary of the region. The solution itself is completely rigorous. The catch is that to get a rigorous answer, the boundary values must be known exactly everywhere on the boundary. In some cases the boundary values must be approximated. In other cases, the boundary values can be determined numerically. The boundary does not need to be a physical boundary, such as a wall, but can be any surface that encloses the region of interest. The objective here is to obtain the general solution in terms of a boundary integral. It is also possible to express a general solution in terms of wave spectra.
The derivation presented here is one of the most elegant physics solutions that I have encountered. I don't really know it's history, but it appears to be the end product of significant contributions from several physicists and mathematicians. The final equation for sound is called Poisson's formula. There is a version for electromagnetic waves called the Stratton-Chu equations; the derivation closely parallels the solution for the sound equations.
The basis for the derivation is the divergence theorem. The first step is to apply this theorem to derive the properties of a "Green's function." The next step is to again apply the theorem to a cleverly constructed combination of the Green's function and the solution we are looking for. And voilà, the general solution pops out! We next consider the radiation conditions on a surface of infinite radius, discuss Huygens' wavelets, and the near-field and far-field regions of the solution. Finally, as an example of a numerical solution, we calculate the radiation pattern of a flat circular piston. This happens to be a problem where an "exact" solution can be obtained. The solution of piston radiation is related to scattering by a disk or an aperture; see the section on Babinet's principle.
We consider the time-harmonic case, with an assumed time variation ejωt. If desired, the time-domain solution can be obtained via a Fourier transform. This section uses two versions of the Greek letter "phi." In the text they appear as φ and ɸ. My Firefox and Netscape browsers read both of these OK. For more on this see Greek Letters and Special Characters: Microsoft vs. the World. When the term "field" is used here, it means the pressure and velocity components of a sound wave.
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The Divergence Theorem
For a vector function C, the divergence theorem states that:
S is a closed surface bounding the volume V, and n is a unit vector normal to the surface, pointing out of the volume. For a physical process like sound, functions are always very well behaved mathematically - e.g. infinitely differentiable - and the volumes of interest are usually mathematically simple; C can be rather badly behaved, and the volume complex, and equation G1 can still be valid; the details of this are mainly of interest to mathematicians.
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A Green's function looks deceptively simple, but it has rather remarkable properties. Let Rp denote a vector from the origin of a spherical coordinate system to an arbitrary point within the volume V, bounded by the surfaces S1 and S2. See the coordinate system geometry [10.4 kb]. Let r denote a vector from the point Rp to another point within V, and set ρ=|r|. The vector from the origin to this point is then R=r+Rp. The vector Rp is fixed; R and r are variable. The Green's function of interest has the form:
It's gradient is
where ar is a unit vector in the direction of r. For ρ > 0 the function ɸ satisfies
Equation G4 is the basic scalar wave equation, and ɸ is the simplest interesting solution, physically corresponding to a uniform outgoing spherical wave. The complex conjugate function ɸ*, corresponding to a uniform incoming spherical wave, also satisfies G4. More on this below.
We now want to integrate equation G4 throughout the entire volume bounded by the surface S1 and the sphere of infinite radius S2. The volume integration variable is R, which is not restricted to the surface in this case. The integral is zero everywhere except when R = Rp, where ρ = 0. To evaluate the integral we enclose the point Rp with a small sphere of radius ε, centered at Rp. To evaluate the integral inside this volume we break G4 into two parts, and first apply the divergence theorem to ∇ɸ. Since we are evaluating the volume integral inside this small sphere, the normal to the surface points away from the sphere center, in the direction ar.
By construction, Rp is fixed, so in the above equation ∇2 operates with respect to R, or equivalently, with respect to r; . In the limit we get
The volume integral inside the small sphere of the other half of G4
Therefore combining G6 and G7, and now integrating over the entire volume
The combined behavior of G4 and G8 defines a delta function, usually written
A delta function has the amazing property of being zero everywhere except at a single infinitesimal point, but having a non-zero integral. The volume integral of the function G9 multiplied by any continuous function f(R) is simply equal to -4πf(Rρ). And this feature is exactly what is used to obtain our solution.
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The Clever Construction
Now let φ(r) be the solution we are looking for; it can be any solution of the wave equation G4 in the volume V. By definition, V is source-free, all sources being enclosed in S1 (or possibly outside of S2) The clever construction mentioned in the introduction is the function
Applying a vector identity, and the results above,
The General Solution
Then, applying the workhorse divergence theorem again,
This equation is a general solution for any φ(Rp) that satisfies the source-free wave equation everywhere in V, in terms of the value of the solution on the boundary S. To obtain a solution for the acoustic equations (24) and (25) for pressure and velocity, it is easily shown that
provide solutions of the acoustic equations if and only if φ is a solution of the wave equation G4. In G13 ρo denotes the density of air, not to be confused with ρ=|r|. The speed of sound is c, and λ is the wavelength. Substituting G13 into G12 yields the general equation for pressure
Where Zo = ρoc = 420.6 for air. A similar equation can be obtained for u(Rp). Note that the values of both the pressure p and velocity u must be known on the boundary. At a recent Audio Engineering Society meeting one speaker stated that either one or the other was sufficient. This is not correct in general. For the special case of a planar boundary it is true, as discussed in detail below.
In April 2008 I discovered an apparent inconsistency; equations G12 and subsequent equations provide the correct result for a test case, but lead to a bizarre power flow that seemed to indicate that the direction of the unit vector ar as drawn in the geometry figure is reversed. At that time I stated that I thought I had a sign error somewhere. But after doing a painstaking check of all of the equations I am satisfied that they are correct. Details are discussed in the Huygens' wavelets section.
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Surface of Infinite Radius and the Radiation Conditions
For a typical geometry, S consists of a surface S1 enclosing all of the sound sources, plus a sphere of infinite radius S2, as shown in the geometry figure [4.1 kb]. On S2, n=ar, 1/jkρ =0, and the contribution to G14 due to S2 becomes
Where θ and ɸ here denote the usual spherical coordinates [8 kb]. The normal behavior for an outward propagating sound wave far from its sources is to take on the characteristics of a plane wave, meaning it satisfies the "radiation conditions" given by:
For a sound wave satisfying these conditions Equation G15 equals zero, and only S1 makes a contribution to the sound. It is worth emphasizing that this assumes that all sources are enclosed by S1. Also, in general, the surface S2 does make a contribution for the solution based on the conjugate Green's function, as discussed below. Finally, note that if S2 does not make a contribution, any surface between S1 and S2 will also not make a contribution, as long as V is source free.
Properties of the General Solution Integral
We assume mathematically well-behaved boundary values, a smooth surface, and that S1 encloses all of the sources. Then equation G14 has the following interesting and important properties: (1) the solution satisfies the radiation conditions; (2) the sound computed for any point in V is the exact and unique solution for the given set of boundary conditions; (3) in the limit as the point Rp approaches the surface S1 the computed sound value approaches the boundary value; (4) as the point crosses the boundary of S1 and passes out of V, the value provided by G14 discontinuously drops to exactly zero. The last fact is basic to the exact solution for piston radiation derived below. It also provides a very sensitive accuracy test of a computer program that implements equation G14. It should be noted that if the boundary values are approximated by a physically unrealistic function - e.g. one that is not continuous on S - then all bets are off.
The volume V is always empty, but it is allowable to have sources outside of S2. In this case the integral over S2 reproduces the fields of these sources inside S2, and this contribution discontinuously drops to zero as a point crosses the boundary of S2 and passes out of V. For scattering problems the usual situation is an incident plane wave, and a scattering object inside the surface S1. The integral over S1 yields the scattered field, the integral over S2 yields the incident field, and the sum equals the total field.
Since the field in V is uniquely determined by the value of the pressure on S, and the velocity component normal to S, this proves that these two factors are the necessary and sufficient boundary conditions for boundary value problems.
The usual physical interpretation of equation G14 is that Huygens' wavelets, which are described in detail in this subsection, emanate from each surface point. Assuming that all sources are inside S1, and that they generate an outgoing wave (meaning energy flows out of the volume enclosed by S1), the Huygens' wavelets propagate outwards from each point on S1, as one would intuitively expect. The wavelets from the surface S2 also propagate outward, and do not make any contribution to the fields in the volume V. So S2 and it's wavelets can be ignored.
This simple and intuitively satisfying picture does not apply to all situations. As noted above, the Green's function of equation G2, or its complex conjugate, each provide equally valid general solutions. Another situation is where the "sources" inside S1 are actually "sinks," where energy flows into the volume enclosed by S1. Or the volume enclosed by S1 can be be empty, or contain a non-absorbing scatterer, so no net energy flows in or out of V. Thus there are a number of different cases: various source fields and energy flows; regular or conjugate Green's functions. In several of these cases the surface S2 does contribute to the fields in V, and in some cases the surfaces suck in the Huygens' wavelets, or generate standing wavelets, rather than radiate the wavelets outward. Several specific examples of these cases are described in the Huygens' wavelet subsection referenced above.
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Near and Far Field Regions
The solution of equation G14 behaves differently depending on how far the point Rp is from the sources. The engineering definition of the "far-field" is |Rp|>2D2/λ, where D is the diameter of the smallest sphere that can enclose all of the sound sources. In this case approximations can be made such that G14 can be integrated analytically in some cases. Details are left as "an exercise for the student" as my old Caltech professors were fond of saying.
When I started in engineering in the dark ages of 1962 the far-field approximations were usually necessary to obtain a numerical result, since evaluating G14 in the near field was hopeless in most cases; now any PC can evaluate G14 under almost any conditions you would be interested in. The main interest now is because sound behaves differently in three regions (the dividing lines are approximate); in the far-field the radiation conditions, equation G16, are a good approximation. In the "near field" where |Rp|>λ/2π but smaller than the far field distance, the pressure and velocity begin to develop a significant relative phase shift, and in the extreme near field where |Rp|< λ/2π the magnitudes also diverge from the plane wave relationship.
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Sound Radiation From a Flat, Circular Piston
I recently (Oct 1999) rediscovered the fact that there is a simple exact solution for this problem. I am retaining my original approximate solution, immediately below, because it can be adapted to any surface, whereas the exact solution is only valid for a planar surface. The exact solution is then presented.
Actually the "exact" solution is not truly exact. In all cases the piston has velocity, but the piston excursion is assumed to be zero, or more precisely, negligible compared to a wavelength. This is a virtually universal assumption, since the analysis is intractable otherwise. It is quite a good approximation for the questions of interest here. A numerical solution which includes the piston excursion is given in another section.
Approximate Solution for pistons that are large compared to the wavelength
For a flat circular piston of radius a>λ, we make the following approximations:
(1) The volume V is infinite and empty (no reflections).
(2) The pressure and velocity are zero everywhere on the boundary except on the face of the piston (and on the boundary at infinity where the radiation conditions are assumed to hold).
(3) On the face the velocity u is equal to the velocity of the piston, and the pressure p0 = Z0|u|. By this approximation, either pressure or velocity is sufficient to determine the answer. But in general pressure and velocity are not simply related by the impedance.
Then in the far-field region equation G14 can be evaluated analytically, resulting in (coordinates from the same spherical coordinate system [8 kb], but with the piston at z=0 and Rp in the +z direction). [Note: the sign of this equation was corrected on 10/8/99. Originally I forgot that the unit vector normal to the surface in equation (G14) points out of the volume, and thus in the negative z direction here].
As noted above, it is assumed that a > λ, and in general the greater the difference the more accurate G17 will be. Loudspeakers do not satisfy this condition over most of their frequency range; however as seen shortly, the result is close to the exact solution anyway. This is quite surprising because, as shown in another section, for small ka the pressure distribution at the surface doesn't look anything like the velocity distribution.
Exact Solution for a piston of any size
This derivation follows one given in the text by Pierce. The boundary surface of integration we will consider consists of a planar surface at z=0, closed by a hemisphere of infinite radius in the region z>0. We assume that the far-field conditions are satisfied, so the only contribution to equation G14 will be from the planar part of the boundary. This eliminates the possibility of any sources outside the surface S2.
As noted above, when the point Rp lies inside the boundary, the integral equals the sound pressure at that point. If it is outside the boundary, the value of equation G14 is identically zero. So we define two points Rp1 and Rp2 to be exact mirror images of each other, with respect to the planar surface, as shown in this illustration. We can write two equations that look exactly like G14, where one is equal to the value of the pressure at point Rp1, and the other at Rp2 equals zero. The sum of the two equations is then the value of p(Rp1) at the inside point. By construction, ρ is the same in the two equations. The pressure p(r) and u(r) on the planar surface, and the unit vector normal to the surface, are also the same. The only difference is that the direction of the unit vector ar is exactly opposite in the two integrals. Therefore the term proportional the pressure p(r) in one integral exactly cancels the corresponding term in the other integral. The term proportional to the velocity u(r) is the same in each integral and the velocity term simply doubles, resulting in the equation
Thus we now have an exact result in terms of the velocity alone on the surface. By subtracting the two equations in this construction one can obtain a result in terms of just the pressure on the surface.
The result for the piston problem is then the same as equation G17, except that the 1+cosθ term is replaced by a factor of 2. Here the approximate relation between velocity and pressure on the piston surface was not needed, and the result is exact for any piston size. Equation G18 is exact, other than neglecting piston excursion, but there is another qualification regarding the "exact" characterization of the solution. The assumed velocity distribution is discontinuous on S. One can derive equations for u(Rp) similar to equations G14 and G18. In the limit as Rp approaches the surface the resulting velocity will not precisely agree with the postulated distribution in this case. Mathematically this could be fixed by slightly rounding off the corners of the assumed velocity on S, so this is not really troublesome.
The essential feature of the above derivation is the possibility of selecting a pair of image points so all of the terms in the two integrals are the same except for a change of sign in the unit vector ar. This only works for a planar surface. Note that while equation G14 is valid everywhere, equation G18 is valid only inside the volume.
Sample Sound Radiation Pattern
My Vifa 5-inch midrange has an effective cone radius of about 5.4 cm. At a typical crossover frequency of 3kHz, this is still less than one-half wavelength. In order to get an interesting looking pattern, we will go to 6.4 kHz where a=λ. The far-field angular pattern [23.2 kb] shows a main lobe of sound radiation, and a sidelobe. The black curve is computed using equation G17 as is; the blue curve is for the exact solution for the flat piston. The green curve is a numerical solution for a more realistic cone-shaped driver as described elsewhere. The horizontal axis is the angle of the listening position with respect to the pointing direction of the speaker. The vertical axis is the relative sound level. The main lobe is down 3dB from its peak at an angle of 15 degrees. Twice this angle is the "beamwidth" of the lobe. In general, the beamwidth in degrees is approximately 29λ/a. There is an angular nulls at about 37 degrees. If you happened to be sitting at this angle relative to the pointing direction, you would not hear a 6.4 kHz tone.
Equation G14 is valid at any distance from the speaker. Near-field plots are presented in the section on Sound Radiated by a Circular Piston in an Infinite Baffle.
And that's the end of this part of the story. Note: this section was revised July 2004 to add the section on Babinet's principle, and to clarify the text, particularly concerning the derivation of equation G18. Revised again June 2006 to move Babinet's principle entirely to its own section, and again in 2008 to clarify the derivation. Also in 2008 the question regarding the direction of the radial unit vector was hopefully laid to rest, and much material was added on Huygens' wavelets.
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