**Plane and Spherical Wave Spectra**

There are several ways to express a general solution of the wave equations. The integral equation (G14) is one. A spectrum of plane waves, or spherical waves, are two others. All representations are equally rigorous, and the choice is a matter of convenience for the problem at hand. We again consider the harmonic case, with the exp(jωt) time dependence suppressed.

This section begins with a description of plane wave spectra, and how they are used to solve problems. Then the equations for the important special case of symmetry around the z-axis are derived; this effectively converts the plane waves into cylindrical waves. Spherical wave spectra are introduced, and used to derive the relationship between the plane wave spectrum amplitude and the far-field sound directivity pattern.

We start with a plane wave propagating in the direction of the vector **k**.
The solution to the sound
wave equations (26) and (27) can be obtained from the potential function,
which is a solution of equation
(G4)

Where ɸ_{o} is a constant with the appropriate units, and

Sound wave pressure and velocity are then given by

A general solution is obtained by integrating the potential function over a spectrum of k values

For values of k_{x}, k_{y} > k, k_{z} is imaginary.
For solutions in the region z ≥ 0 the sign of these "evanescent" waves
is chosen such that they are attenuated in the +z direction.

**Solving Problems in Terms of Plane Wave Spectra**

The idea is then to solve for the spectrum amplitude function g(k_{x},k_{y})
for a given problem, and then use equation (W4) to compute the resulting sound
field wherever it is desired. As demonstrated at the end
of this section, the function g(k_{x},k_{y}) has a simple
relationship to the far-field sound directivity pattern. For imaginary values
of k_{z}, g(k_{x},k_{y}) contributes to the near-field,
but not to the far-field.

For z=0 equation (W4) is in the form of a double Fourier transform, so if f(x,y,0)
is known, the solution for g(k_{x},k_{y}) can immediately be
written using the inverse formula

Physically f(x,y,z) is proportional to pressure. For the piston problem considered in another section, it is the velocity that is known at z=0, but a similar equation applies.

For bounded geometries, the plane wave spectrum degenerates into a summation of discrete modes, such as the waveguide modes considered elsewhere. Modes in this case are simply waves that reinforce after multiple reflections. As seen below, spherical waves are modal even in unbounded space.

**Wave Spectrum for Case of Axial Symmetry**

For problems with rotational symmetry around the z-axis, we can write

We now have a cylindrical coordinate system, (ρ, φ, z), where f(ρ,
φ, z) is independent of φ, and in wave-space, g(β ,ξ) is
independent of ξ . (The coordinate ρ is not to be confused with the
air density ρ_{o}). The integration over φ and ξ in equations
(W4) and (W5) can then be performed, yielding

And

The spectrum now consists of axially symmetric cylindrical waves. These waves are solutions to the wave equation in cylindrical coordinates, and could be obtained directly in that manner. With z=0 equations (W7) and (W8) are in the form of a Hankel-transform pair (see for example Korn and Korn, page 230).

The solution for the potential function in spherical coordinates is given by Stratton on page 404.

This solution is for outgoing waves produced by a source distribution of finite size. For incoming waves, the conjugate spherical Hankel function is used. Both of these expansions are infinite at the origin. For an expansion that is finite everywhere, the spherical Bessel function replaces the spherical Hankel function. The m=0, n=0 outgoing spherical wave produced by this potential function is the elementary spherical wave given in another section. As noted above, in the spherical case we have a summation of modes rather than a continuous spectrum. The reason is that k appears in the solution of the wave equation in spherical coordinates as a single value, so integration similar to the plane wave spectrum cannot be used. The index n is restricted to integer values because the Legendre function would otherwise be infinite on the z-axis, and m must be an integer for the field to be a continuous function of φ. If f is known on a spherical surface, the coefficients of this expansion can be determined using the orthogonality property of the Legendre functions. One of my publications is dedicated to the topic of spherical wave expansions, and contains lots of material on this subject.

In the far-field

Whereas in the other limit

Enclosing one of these waves in a sphere of radius ε → 0, both the pressure and velocity diverge, but the surface integral over the sphere, equation (G14), remains finite for points exterior to the sphere, since it must of course simply equal the value of the wave. For the m=0 modes on the z-axis in the far-field, the relative contributions of the pressure and velocity terms in the surface integral are shown in the table below, for the sphere of radius ε → 0. As the size of the sphere increases, the relative contributions approach equality.

Surface Integral Contributions

n |
Pressure |
Velocity |

0 |
0 |
1 |

1 |
1/3 |
2/3 |

2 |
2/5 |
3/5 |

This result demonstrates that the strength of a source cannot always be characterized
solely by the surface integral of the normal velocity component. Also very complex
radiation patterns can be produced by an infinitesimal source. (Both of these
facts are contrary to statements made in the popular text by Kinsler
et al. on pages 164-165). This also seems counter-intuitive, since one normally
thinks of maxima of radiation patterns as directions where all parts of the
source add in phase, or nearly in phase. For the higher order spherical waves,
the surface integral comes very close to perfect cancellation in __all__
directions, and the maxima are simply the directions where the cancellation
is less perfect.

The spherical wave expansion that we need next is the expansion of a plane wave. This is also given by Stratton on page 409. The wave direction is arbitrary, but the expansion is relative to the direction of propagation, so only m=0 waves are present

**Plane Wave Spectrum Amplitude and the
Far-field Pattern**

Equation (W12) is a rigorous solution of the wave equation for any value of
N. The interesting feature of equation (W12) that we need is: within a sphere
of radius N/k it produces a virtually perfect plane wave. Outside this sphere,
in the plane perpendicular to **k,** the field decays quite rapidly. In other
words, it closely approximates a uniformly illuminated planar aperture of radius
a=N/k. We can make the radius a as large as we want, still keeping N finite.
Then we can let r approach infinity, and use the asymptotic form of the spherical
Bessel function. This is valid as long as kr > > N. The result is

Equation (W13) is accurate for r< N/k. For large N equation (W13) has the
character of a delta-function, having significant value only close to the direction
of the propagation vector **k**. Exactly in this direction, and for N>>1,
the value of (W13) is closely approximated by

Where a is the effective radius of the plane wave segment, as discussed above. This is exactly the same as the result of direct aperture integration given by equation (G17).

The relationship between the variables k_{x}, k_{y} and spherical
coordinate angles θ, φ of physical space is given by

So comparing to equation (W6) we have

At this point, we have shown that we can produce a plane wave within a spherical region centered at the origin, of radius as large as we want. In the far-field this plane wave segment will radiate essentially only in the direction perpendicular to its phase front, which corresponds to the angle θ in the equation above. The same sound pressure is produced for any direction in space. As N becomes larger, we approach a plane wave as in the spectrum of equation (W4). Therefore at a given angle θ the sound pressure is directly proportional to the value of the spectrum amplitude function at the corresponding value of β. However there is an additional angular weighting function. To determine this, we consider the potential function for a sound source that radiates a spherical wave with equal pressure in all directions. In spherical coordinates,

The potential function for this wave is, in cylindrical coordinates

Where

Plugging equation (W18) into equation (W8), with z=0, yields a Hankel transform that is given by Korn and Korn, Table D-5, page 769

Since this pattern has directivity of unity by definition, in general the directivity pattern of the sound radiation must be

So equation (W21) provides the general relationship between the spectrum amplitude function and the far-field pattern.

As footnote to this derivation, if we tried to start with a plane wave occupying all space, we would quickly get into hot water mathematically. For openers, you can't get to the far-field of an aperture of infinite radius. If one correctly evaluates the exact integral expression, equation (G14), for this case you simply recover the field of the plane wave - no delta function. Carving out a section of finite radius with a cookie-cutter would result in a discontinuous field that doesn't satisfy the wave equations. So using the spherical wave solution is much sounder mathematically.