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Spherical harmonics
Keywords: spherical harmonics, tilted spherical harmonics, rotating waves, standing waves, Flash animation, technical illustration, separation of variables
Contents of this posting
 Introduction to spherical harmonics and the Helmholtz equation
 Spherical harmonics
 Table 2a. Spherical Harmonics  real form
 Table 2b. Spherical Harmonics  complex form
 Time dependence
 Real rotating modes
 Table 3. Illustrations of standing wave spherical harmonics
 Relating twin modes to rotating wave modes
 Table 4. Animations of time varying standing wave spherical harmonics
 Table 5. Animations of time varying rotating wave spherical harmonics
 An acoustic spherical resonator  details of separation of variables
 Special functions in physics
 A symmetry dilemma  axis of rotation in a spherically symmetric resonator
 Student experiment  vibrational modes of a spherical balloon
 Spherical radiation
1. Introduction to spherical harmonics and the Helmholtz equation  

Spherical harmonics represent the angular part of a solution of the Hemholtz equation in cases of spherical symmetry. Because the Helmholtz equation governs most simple waves, spherical harmonics can be applied to various cases of spherical symmetry involving waves such as spherical acoustic resonators and quantum mechanical electron waves around an atom. Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves.

Fig. 4. Cartesian and spherical coordinate system used in Tables 3, 4 and 5 as mapped on a surface of a sphere. We show the φ and θ coordinates of point P. 
Relating twin modes to rotating wave modes
Note also that for a given ℓ value and m > 0 we can consider the modes in pairs. For example as shown in Table 3 above, for ℓ = 2, the m = +2 spherical harmonic is almost the same as the ℓ = 2, m = −2 spherical harmonic. The only difference is that the m = −2 spherical harmonic has been rotated in the φ direction such that its maxima (antinodes) are located where the the other spherical harmonic has its minima (nodes), i.e. rotated by an angle of φ_{rotated}= π/2m . Because the spherical harmonics can be paired up +m with −m both being practically the same, many references only show the spherical harmonics with positive m values. This is the case with the reference given above. At the same time the two twin modes in a pair are distinctly mathematically independent (linearly independent).
A rotating wave mode is simply a sum or difference of two twin standing waves with an added π/2 temporal phase shift between the twin modes. The math behind this was explained in earlier postings: link1, link2. At its core is the trig manipulation of the φ dependence of a spherical harmonic and the time dependence: cosmφ cosωt + sinmφ sinωt = cos(mφ−ωt).
There are many online references that show spherical harmonics as three dimensional objects. These show the values at various angles as colored projections. While these may be appropriate for the use of spherical harmonics when combined with a radial dependence, such as for solving for the electron wave function in a hydrogen atom, they are misleading in that they imply that spherical harmonics by themselves involve a radial component. We believe coloring the surface of a sphere correctly implies the purely angular nature of spherical harmonics.
Table 4. Animations of time varying standing wave spherical harmonics  

ℓ = 0, m = 0 
These animations show the time dependence added to the spherical harmonics as per Equation (5) above.
Mouse over each image to see the animation, mouse off to suspend it and click on it to restart it. Viewing these animations requires a free adobe flash viewer on your computer. Many computers already have this viewer installed.  
ℓ = 1, m = −1 
ℓ = 1, m = 0 
ℓ = 1, m = +1  Note that the three ℓ modes are the same except they are oriented in three different directions, along the x, y and z axes.  
ℓ = 2, m = −2 
ℓ = 2, m = −1 
ℓ = 2, m = 0 
ℓ = 2, m = +1 
ℓ = 2, m = +2 
Table 5. Animations of time varying rotating wave spherical harmonics  

ℓ = 0, m = 0 
These animations show the time dependence added to the spherical harmonics as per Equation (6) above.
Mouse over each image to see the animation, mouse off to suspend it and click on it to restart it. Viewing these animations requires a free adobe flash viewer on your computer. Many computers already have this viewer installed.  
ℓ = 1, m = −1 
ℓ = 1, m = 0 
ℓ = 1, m = +1  
ℓ = 2, m = −2 
ℓ = 2, m = −1 
ℓ = 2, m = 0 
ℓ = 2, m = +1 
ℓ = 2, m = +2 
2. An acoustic spherical resonator  details of separation of variables
The wave equation for the acoustic pressure p (the oscillating component of the pressure due to an acoustic resonance) is:
where c is the speed of propagation of a plane wave front in the medium under consideration. The Laplacian (i.e. the ∇^{2}) in spherical coordinates makes this:
Following the separation of variables method, we assume a product solution:
where R(r) is only a function of r, Θ(θ) is only a function of θ, Φ(φ) is only a function of φ and T(t) is only a function of t.
Extracting the time dependence:
When we substitute (12) into (11) and divide by R(r)⋅Θ(θ)⋅Φ(φ)⋅T(t) we get:
We see that the function on the right depends only on time t while the function on the left depends on the other variables and not on time. This requires that both sides equal a constant, a separation constant which, in hindsight, is negative the wavenumber squared, i.e. −κ^{2}. So we have two equations now:
and
The second equation has sinusoidal solutions of ωt such as:
where ω = cκ is the angular frequency.
Extracting the φ dependence:
Equation (14) can be rearranged as:
which again can be separated with a separation constant of −m^{2} and yields the equations:
Equation (19) has sinusoidal solutions of mφ such as:
where B is a constant (the amplitude). We discuss this solution above.
Extracting the θ dependence:
Equation (18) can be rearranged as:
with the left side only being a function of r and the right side only a function of θ. We use the separation constant ℓ(ℓ + 1) to yield the following two equations:
Equation (23) can be rearranged as:
If we let x = cosθ , so that sin^{2}θ = 1  cos^{2}θ = 1 − x^{2} and dx = −sinθ dθ then our equation becomes:
which yields:
which is the same form as the general Legendre equation which has solutions of associated Legendre polynomials Θ(x) = P_{ℓ}^{m}(x) or written as:
as noted above.
Extracting the radial or r dependence  spherical Bessel functions:
Turning our attention to the radial dependence, Eqn. (22) can be rewritten as:
According to Wolfram, the "transformed" version of Bessel's equation is:
⇑ Fig. 6. First three spherical Bessel functions, j_{n}(x) for n = 0, 1, 2. Wikipedia. 
⇑ Fig. 7. First three spherical Neumann functions, y_{n}(x) for n = 0, 1, 2. 
and has solutions of:
where q is defined as:
We can discard the second term in (30) because of Y's singularity at the origin which is included in our spherical resonator. Also if we compare (28) and (29) we see that in our case: p = ½, a = κ, r = 1, β = i√(ℓ(l + ℓ)), and q = √(¼ + ℓ(ℓ + 1)) = ½√((2ℓ + 1)^{2}) = ℓ + ½.
Spherical Bessel functions:
This makes our solution for R(r):
where D is a constant and the j_{ℓ}(κr) are "spherical Bessel functions" defined as:
As (33) suggests, spherical Bessel functions can be considered as a special subclass of regular Bessel functions having half integer orders. Alternatively, they can be related to sinusoids in a much simpler fashion than normal integerorder Bessel functions can be. The relations between the first few spherical Bessel functions and sinusoids are:
Putting the whole solution together:
Back to the larger problem of the solutions for acoustical waves inside a spherical enclosure, we assemble the parts above to give:
where ω/κ = c and c is the speed of sound. Eqn. (35) specifies a set of solutions with ℓ being any positive integer or 0 and m being any positive or negative integer or 0 in the range of −ℓ ≤ m ≤ +ℓ . The wavenumber κ can be adjusted to fit the boundary condition at the interface with the spherical container which will affect the resonant frequency ω via the equation ω/κ = c. The amplitude E can be adjusted to fit the intensity of the captured waves. A general solution will involve sums of these solutions. A general solution may also substitute other sinusoids in for the cosmφ and cosωt.
Resonant frequencies of a spherical resonator
To calculate the frequencies of the resonant modes of our acoustic spherical resonator we need to adjust the radial wavenumber κ so that the boundary conditions are met. This means that if we have Dirichlet boundaries we need the R(r) factor in (35) to be zero at the resonator's radius. Since R(r) is proportional to j_{ℓ}(κr) this means j_{ℓ}(κa) = 0 at r = a where a is the radius of the resonator. This means κa = u_{ℓ,n} or:
κ = u_{ℓ,n} /a , (36)
where u_{ℓ,n} is the n^{th} root of j_{ℓ} .
Similarly for Neumann boundaries, the derivative of j_{ℓ}(κr) with respect to r must be zero at r = a. Thus:
κ = u'_{ℓ,n} /a , (37)
where u'_{ℓ,n} is the n^{th} root of the derivative of j_{ℓ} .
We might note that in Equations (28) through (37) nowhere is the index m refered to. This means that the resonant frequencies do not depend on m and only depend on ℓ and n. So when we examine Tables 4 and 5 above, we should remember that all the modes with the same ℓ will have the same resonant frequencies as long as the n index is also the same. Thus, except for the top most mode of the chart (ℓ = 0), all the modes are degenerate with other modes. Since there are 2ℓ + 1 modes for a given ℓ, we should expect a 2ℓ + 1 degeneracy. This can be broken if we introduce a perturbation to make the resonator a little asymmertric.
3. Special functions in physics 

During the last few centuries mathematicians and physicists created many special functions (2nd ref) such as the above mentioned spherical Bessel functions. These are documented in classic references such as Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables and many other places. These are mathematical functions, each of which is the solution to a special differential equation governing some physical process in nature. The best known of these are trigometric functions (sine, cosine, etc) and exponentials (e^{x}). Each of these functions have all sorts of theory and mathematical relationships, generated by mathematicians and physicists down through the years. The above reference by Abramowitz and Stegun lists many of these relationships. The trick for casual users is not to get too caught up in the heavy overhead of these special functions, but just to consider each a solution to an equation that has been provided to you as a gift by reseachers in the past. My philosophy is to absorb as much or as little information as required on the special function under consideration to provide a solution to a particular physics problem at hand and then move on. 
4. A symmetry dilemma  axis of rotation in a spherically symmetric resonator  

Above we have been considering the solutions to waves inside a spherical resonator. Such a resonator is totally spherically symmetric with no distinguishing axis or direction. The confusion is: why do most of the solutions have a distinct axis and are not spherically symmetric? Answers:

5. Student experiment  vibrational modes of a spherical balloon 

Using the setup described in an earlier posting (Fig. 48a) one can excite spherical harmonics on a spherical balloon. Inflate the balloon and suspend it by a string above and a little off center of the loud speaker. Lines drawn on the balloon will help in seeing the motion. Heavyweight rubber balloons, full but not inflated to the max, seem to work best. An example of a heavy weight spherical balloon.
This balloon experiment will allow students to see the spherical harmonic modes in real life that are mathematically described above and see the succession of modes as one scans through the frequencies. Next we briefly show that the Helmholtz equation governs motion of the surface of a balloon meaning that we should expect spherical harmonic modes in its vibrations. A little math to show the relevance of spherical harmonics to the surface of a balloon:In this discussion, we assume that most of the wave action is due to the balance between the outward pressure force and inward tensional force of the stretched balloon. We neglect any wave resonance of the air inside the balloon because such resonance would be in the kilohertz range, instead of the Hertz range that we experience. Similar to the waves on a string case, the inward tensional force is proportional to the curvature of the balloon's surface.

6. Spherial radiation fields:
Figs. 10 and 11. Flash animation and video of spherical radiation pattern for the ℓ = 4, m = 2 mode as prescribed by (50) below. One quadrant has been sliced from the sphere to show the radiation waves propagating outwards to the surface. It is assumed that the waves are either absorbed at the surface or continue to travel outwards unseen past the surface.
This animation uses the color black for the near zero areas of the sphere similar to that used in Fig. 8 above. Mouse over the image in Fig. 10 to activate it, mouse off to suspend it, and click on it to restart it. Fig. 11 is started by clicking on the arrow. Because of its complexity the flash animation runs slow on slower computers. The video in Fig. 11 will run faster but is a much larger file to download. 
6. Spherical radiation
Radial radiation fields
The radiation patterns for waves in three dimensions coming from a small source are described by spherical harmonics. These waves are similar to the two dimensional case previously discussed. In the three dimensional case, for radiation propagating straight away from the source we use the following equation:
Note that we use both j_{ℓ}(κr) and y_{ℓ}(κr) since the origin is not included in our space (it must be excluded because a radiation field will be infinite there since radiation fields fall off as 1/r). It is the interplay of the two kinds of spherical Bessel functions that mathematically provide for the outward propagation of the waves (similar to the interplay of J_{ℓ}(κr) and Y_{ℓ}(κr) in two dimensional radiation).
An animation of the ℓ = 4, m = 2 radiation mode is shown in Figs. 10 and 11 above (flash and mp4 video versions). Note that to create particular radiative mode such as as the one shown, the launching "antenna" must be properly shaped for that one mode. Alternately, a phased array with its radiative elements covering the surface of a sphere could also be programmed to a lauch particular radiative mode.
Spiral radiation fields
The radiation equivalent of rotating waves is spiral radiation, which uses this equation:
Such radiation is similar to the two dimensional equivalent from the earlier posting. An animation of the ℓ = 4, m = 2 spiral radiation mode is shown in Figs. 12 and 13 (flash and mp4 video versions).
Figs. 12 and 13. Flash animation and video of spherical radiation pattern for the ℓ = 4, m = 2 mode as prescribed by (51). One quadrant has been sliced from the sphere to show the radiation waves propagating outwards to the surface. It is assumed that the waves are either absorbed at the surface or continue to travel outwards unseen past the surface. Note that on the surface of the sphere, the field pattern rotates about the z axis instead of oscillating in place as in Figs. 10 and 11. Also we can see the spiral radiation arm in the base of the removed quadrant.
This animation uses the color black for the near zero areas of the sphere similar to that used in Fig. 8 above. Mouse over the image in Fig. 12 to activate it, mouse off to suspend it, and click on it to restart it. Fig. 13 is started by clicking on the arrow. Because of its complexity the flash animation runs slow on slower computers. The video in Fig. 13 will run faster but is a much larger file to download (4KB vs. 5MB). 