The natural lair of Bessel functions, with some interesting lore

Many problems with a centre of symmetry or a line of symmetry can be solved when the appropriate coordinate system is used. Here we look at the latter case, where cylindrical coordinates are the natural choice. It is interesting to solve the wave equation in this case, and the same procedures can be used in other problems. In the Figure, the first line is the simple wave equation, where the phase velocity is c, a number independent of the space coordinates and time. The wave function φ may be taken to be the velocity potential for a sound wave. The expression on the left is the Laplacian of φ, the symbol standing for the proper combinations of partial derivatives, which will be different for different coordinate systems. In rectangular coordinates, it is the sum of the second derivatives with respect to x, y and z. In the second line, φ becomes an amplitude, a function of the space coordinates alone, and an exponential time dependence is chosen, where ω = 2πf is the angular frequency. This throws the wave equation into the form shown on the third line. The physical interpretation of the equation is that the curvature of the amplitude is always proportional to the negative of the amplitude itself, the constant of proportionality being the square of the wave number k, a quantity that is always positive. When k = ω/c is substituted, we find Helmholtz's equation. If the phase velocity c goes to infinity, φ satisfies Laplace's equation, which describes irrotational flow in an incompressible fluid.

Now we write Helmholtz's equation explicitly for cylindrical coordinates r, θ, z, which are defined as shown in the Figure. Note that increasing θ is in a right-handed screw relation with the axis or pole r = 0. The expression for the Laplacian in cylindrical coordinates can be found in many references, or can be derived from first principles by considering the vibration of a circular membrane, for example. The case of φ = φ(r,θ) is included, by simply ignoring the coordinate z. We now substitute a product form for φ and try, successfully, to separate the variables. This is not possible in an arbitrary coordinate system. Cylindrical coordinates is one system in which this works. We first separate Z, by putting all the z-dependence in one term, so that Z"/Z can only be a constant. The constant is taken as -s^{2} purely for convenience; it can be any positive or negative number or zero, as required. Now we can separate Θ"Θ, which has all the θ dependence, and is another constant -n^{2}, again for convenience. This give solutions sin nθ and cos nθ, which have period 2π if n is integral, and this makes the solutions single-valued in θ, which is a requirement if there is indeed axial symmetry. If the range of angles is restricted, this is no longer necessary, and n can have nonintegral values. Finally, the equation for R(r) remains, and it is Bessel's equation, so that R is a linear combination of J_{n} and Y_{n}. It is this connection that led to Bessel's functions being called Cylinder functions as well. If s is not equal to zero, k^{2} - s^{2} replaces k^{2}. We can now write down a family of solutions to the equation, including z-dependence as trigonometrical, hyperbolic or exponential functions, as required. There should be enough variety in these functions that arbitrary initial and boundary conditions can be satisfied by superimposing them in much the same way that we make a Fourier series to represent any function.

Let's try to find the normal modes of the air in a cylinder where the velocities are in planes normal to the axis. We have found the modes for velocities in the z-direction in our study of pipes. The shape of the pipe was of no consequence in those studies. Now, we have to specify the shape, and say that it is a circular cylinder of radius a. The appropriate solutions are of the form φ = J_{n}(kr)cos nθ or J_{n}(kr)sin nθ, where n is integral, from the condition of single-valuedness as a function of angle. The boundary condition at the rigid surface of the cylinder is that the normal velocity must vanish, or [dJ_{n}(kr)/dr]_{r=a} = 0. For n = 0, the roots are 3.832, 7.015, 10.174, 13.324, 16.471, 19.616, .... For n = 1, 1.841, 5.332, 8.536, 11.706, 14.864, 18.016, .... For n = 2, 3.054, 6.705, 9.965, .... For n = 3, 4.201, 8.015, 11.344, .... The lowest mode, with one diametral nodal line, n = 1, has ka = 1.841, or λ = 3.4129a = 1.7d, where d is the diameter. The air rushes from side to side of the cylinder, with maximum velocity on a diameter. If the pipe were a square of side d, it is easy to see that λ = 2d. This mode is *degenerate*, which means that there is another node of the same frequency. This mode depends on the sine instead of on the cosine, and is merely rotated by 90°. The lowest axially symmetric mode, with radial velocities only, has a frequency 2.08 times the frequency of the lowest mode. The modes are not harmonically related, which is not surprising, since systems with harmonic overtones are extremely rare.

It is easy to extend our results to the normal modes of a cylindrical can of height L. Since dφ/dz = 0 at z = 0 and L (taking the origin at one end of the can), Z = cos (mπz/L), m = 1,2,3,.... This gives the longitudinal modes we have studied elsewhere. If we now introduce transverse motions, we have k^{2} = (mπ/L)^{2} + α^{2}, where α is one of the roots of J_{n} that we used above.

If we are concerned with radiation problems, we will be interested in the behaviour of the solutions for large values of kr. We are looking for solutions that contain a factor like e^{-ikr} at large distances. Relton gives the following asymptotic forms:

J_{n}(x) = (2/πx)^{1/2}(P cos u - Q sin u)

Y_{n}(x) = (2/πx)^{1/2}(P sin u + Q cos u)

where u = x -(n + 1/2)π/2, and P and Q are asymptotic series. This means that they do not converge, but the error when all terms past a certain one are neglected becomes zero at large values of x. The leading term in P is 1, and in Q it is (n^{2} - 1/4)/2x. Each succeeding term is multiplied by x^{-2}. The combination J - iY = (2/πx)^{1/2}e^{-iu}(P - iQ), which has the required form, if x = kr. It depends on the inverse square root of r, which is appropriate for a cylindrical wave, where the intensity spreading is proportional to 1/r, not to 1/r^{2}, as for a wave diverging from a point source. There is also an interesting phase shift of π/4, or λ/8. These linear combinations of J and Y are called Hankel functions.

Huyghens' construction is familiar in discussions of wave motion. This idea assumes that the wavefront at a later time is the envelope of the elementary wavelets originating from all points of an earlier wavefront. This gives a quantitative explanation of reflection and refraction, and a qualitative explanation of some of the phenomena of diffraction. Fresnel showed that Hughens' construction had a basis in the mathematical theory of wave propagation, and that the disturbance at any point could be expressed as the superposition of waves coming from every point of an earlier wavefront. This principle gives a quantitative explanation of the phenomena of diffraction.

To demonstrate how this worked, Fresnel showed how to divide a spherical wavefront into half-period zones, in which the distance from the point under consideration to the earlier wavefront increased by half a wavelength. Then, the disturbance from adjacent zones nearly cancelled, and the sum of all the contributions was equal to half the effect of the first zone. The only problem was that the phase of the wave was retarded by a quarter of a wavelength with respect to the wavelet coming from the nearest point on the wavefront. This meant that the wavelet had to be emitted with a phase advance of a quarter of a wavelength in order to give the correct result. This rather clearly shows that we are dealing with a mathematical model, not a physical process.

In the present case, we are dealing with a cylindrical wavefront that does not divide into half-period zones as neatly as the spherical wavefront. We can, however, t reat it as a two-dimensional problem, and sum up the wavelets from elementary line sources on the wavefront, instead of from elementary point sources. Now the necessary advance is an eighth of a wavelength, not a quarter.

Diffraction from a straight edge can be treated either in the Fresnel manner, summing the contributions from a cylindrical wavefront, or as a geometrical shadow and an edge wave, which is a line source of rather unusual angular dependence . Nevertheless, the λ/8 advance is present.

Rayleigh uses φ for the velocity potential of an incident wave, and ψ for the velocity potential of the scattered wave. The velocity is the gradient of his velocity potential, not the negative gradient, but this is a small matter that is easily accommodated. For an incident wave φ = cos[2π(ct + x)/λ] the wave scattered by a rigid cylinder of radius a is, when kr is large and ka small, ψ = -[2π^{2}a^{2}/r^{1/2}λ^{3/2}](1/2 + cos θ) cos [2π(ct - r - λ/8)/λ]. r is the closest distance from the point of observation to the cylinder, and we see that an extra eighth of a wavelength is added to it. The factor (a/r)^{1/2} is the cylindrical spreading, and the factor (a/λ)^{3/2} is the wavlength dependence. The scattered intensity is inversely proportional to the cube of the wavelength, instead of as the fourth power with a sphere. By integrating over the angles and averaging the cosines, we can find the total energy scattered by the cylinder per unit length. This is best expressed as a linear cross section, a width that intercepts the same amount of energy in the incident beam. This width is 3π^{4}(a/λ)^{3} times the diameter of the cylinder. For a wavelength of 30 cm (about f = 1 kHz) and a cylinder 1 cm in diameter, the cross-section is only 0.135 mm, so very little energy is scattered. The cross-section only becomes equal to the diameter when (a/λ) is about 0.15, or ka about unity, so the formula is near its limit of validity.

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Composed by J. B. Calvert

Created 4 July 2000

Last revised