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Keywords: cylindrical 3D resonator, standing wave modes, rotating wave modes, Dirichlet, Neumann, Flash animation, technical animations, student science experiment
Contents of this posting 

Standing waves and rotating waves in three dimensional cylindrical resonators
1. Resonant modes of a cylindrical resonator with Neumann boundary conditions
Animations of standing wave and rotating wave resonances in a cylinder with Neumann boundary conditions.
These animations show a color mapping of the wave parameter at the surfaces of the cylindrical resonator. A real example of these resonances would be an excited acoustical resonance inside a cylindrical container filled with air or some other fluid.
Neumann boundary conditions specify that the derivative of the wave parameter normal to the surface is zero. Dirichlet boundary conditions which specify that the wave parameter is zero would not produce a visible representation in this type of display because the wave parameter would be a constant zero everywhere shown. To see the action, mouse over an image, mouse off to suspend it and click on it to restart it. All four of these these images are for the 2,2,1 mode (these indices are for the φ, r and z directions). These particular images show two complete sinusoidal cycles in the φ direction (around the curve of the cylinder), one half of a sinusoidal cycle in the z direction (along the axis of the cylinder) and represent the second solution in the r direction (in a direction straight away from the cylindrical axis). The mathematics explaining these modes is below.  
⇐ Figs. 1a and 1b. Twin modes: a mathematically independent pair of standing wave modes. These are basically the same mode, but one is rotated a quarter cycle in the φ direction (45 degrees for this mode).
To see the action, mouse over an image, mouse off to suspend it and click on it to restart it.  
⇐ Figs. 2a and 2b. Rotating wave modes. These are basically the same rotating mode, but one is rotating in the positive φ direction (counter clockwise) and the other in the opposite direction. Each of these modes is the sum of the two above standing wave modes with a 90 degree shift in temporal phase (either positive shift or negative shift) of one standing wave mode compared with the other standing wave mode. We often label these modes as the the +2,2,1 and −2,2,1 modes with the minus sign indicating rotation in the negative φ direction.  
⇐ Figs. 2c and 2d. These two images show the 2,2,1 mode with the cylinder sliced into two pieces. We pretend that the other half of the cylinder is still there but show only the surface of the back half. We see half of the top disk with its field shown in color, and the fields of the cut through the center of the cylinder. The left image shows the standing wave mode and the right image shows the positive rotating mode.
Interestingly, the time dependence of the fields in the slice are the same in both images (going red, then blue, then red, etc.). The fields on the top surface show the difference between standing wave modes and rotating modes. To see the action, mouse over an image, mouse off to suspend it and click on it to restart it. 
More modes in same type of representation.  
⇐ Figs. 3a and 3b. Modes 0,0,1 and 1,1,0. The 1,1,0 mode is the lowest frequency mode (see the plot in Fig. 6 below. The 0,0,1 is the next lowest frequency with the 1,1,1 mode after that (see Fig. 5c below).
To see the action, mouse over an image, mouse off to suspend it and click on it to restart it.  
⇐ Figs. 4a and 4b. Modes 0,1,0 and 4,1,0. The mode with a large φ index such as shown in in Fig. 4b is often called a whispering gallery mode. Such modes cling to the curved surfaces of their cylindrical containers. 
⇓ Figs. 5a through 5f. A series of rotating modes with the same r and z indices with varying φ indices.
To see the action, mouse over an image, mouse off to suspend it and click on it to restart it.  
Mode = 1,1,1. 
Mode = 0,1,1. 
Mode = +1,1,1. 
Mode = +2,1,1. 
Mode = +3,1,1. 
Mode = +4,1,1. 
The last blog concerned resonances on a circular surface, a two dimensional wave field. This blog concerns resonances in a volume, a three dimensional cylindrical space. An example of such cylindrical waves might be the acoustical waves inside an acoustically excited gas or liquid filled cylindrical container. For acoustical wave fields we normally first solve for the acoustical pressure field. The acoustical oscillating pressure field would have Neumann conditions at the container walls. This is because the acoustical pressure drives acoustical gas velocity and the gas cannot penetrate the walls of the container; thus the gradient perpendicular to the walls at the walls must be zero.
There can also be more complicated wave fields inside cylindrical containers such as shear waves and electromagnetic waves. These involve transverse vector fields and the boundary conditions are often a mix of the Dirichlet and Neumann types
2. Mathematical derivation of the modes in a cylindrical resonator
Deriving the math requires some simple changes to that of the previous blog[link].
The new wave equation in three dimensional cylindrical coordinates is:with the additional term shown in red. This becomes:
in cylindrical coordinates.
We proceed with separation of variables by assuming a product solution of the form:
where R(r) is only a function of radius r, Φ(φ) is only a function of the azimuthal angle φ, Z(z) is only a function of z, and T(t) is only a function of time t. Substituting (52) into (51) and rearranging we get:
which has the form of being only a function of r, φ, and z on the left side of the equals sign and only a function of time t on the right side. The only way this is true for the whole space is if both sides equal a constant, which we label as − ω^{2} (the traditional label and yes, this constant does turn out to equal minus the angular frequency squared).
The right side of (52a) is a simple differential equation of T(t), i.e. T " = − ω^{2}T with sinusiodal solutions such as T = A cos ωt where A is a constant.
The left side of (52a) can be rearranged into:
which is arranged so that it is only a function of r and z on the left side and only a function of φ on the right side (the middle expression in (52b) ). Again we argue that this requires both sides to equal a constant, which we label as m^{2}. The right side yields the simple differential equation Φ " = − m^{2}Φ with simple sinusoidal solutions such as Φ = B cos mφ where B is a constant.
Rearranging a third time we have:
where the left side is purely a function of radius r and the right side is purely a function of z which means that both sides equal a constant. We label this constant as k_{z}^{2} . The right equation can be rewritten as Z " = − k_{z}^{2} Z . This has sinusoidal solutions such as z = C cos k_{z}z where C is a constant.
Looking at (52a), (52b), and (52c) we have four differential equations, each of only one variable:
where ω, m, and k_{z} are separation constants. The first three of the equations, (53a), (53b), and (53c), are very simple and have the standard sinusoidal solutions such as (written here as complex exponentials):
or the equivalent sine and/or cosine sinusoids.
The geometries we will be discussing include the whole space around the origin ( 0 ≤ φ ≤ 2π ) and require that m be an integer in order that the function Φ(φ) and its derivative are continuous between φ = 2π and 0 .
The last equation in (53) can be recast as:
or as:
where u is defined as u = k_{r}r is a normalized radius. The radial wavenumber k_{r} is typically defined as:
Note that ω/v_{p} = k is a standard equation for waves where k is the free space wavenumber for the frequency and media being considered. So applying this to (54d) we get k^{2} = k_{r}^{2} + k_{z}^{2} . We can interpret k_{r} and k_{z} as two vector wavenumber components in the r and z directions while k is their vector sum, the total wavenumber. The k_{r} and k_{z} wavenumbers are usually determined by the size of the resonator and the boundary conditons which we will discuss in the next section. Once k_{r} and k_{z} are determined, the frequency of a mode can then be gotten using a rearrangement of (54d):
Eqs (54b) and (54c) are two forms of Bessel's equation and have Bessel function solutions, J_{m}(k_{r} r) and Y_{m}(k_{r} r), as discussed in the previous blog[link]. We will not use the Y_{m}(k_{r} r) form here because it is infinite at the origin and thus is not useful for resonators which include the origin as ours do.
If we substitute the appropriate solutions for standing waves in a cylinder into (52) we have:
where we have chosen Z = cos k_{z}z instead of the sine form so that the derivative of the function would be zero at z = 0 as appropriate if the cylinder had a Neumann boundary at this location.
The mathematically independent mode which is the twin of (55a) would then be:
Summing (55a) and (55b) with a positive or negative 90 degree phase shift in time creates a rotating wave moving in the positive or negative φ direction:
The animations at the beginning of this section put these equations into action. Note that the rotating wave in (55c) has the φ and time dependence together inside the argument of the exponential, similar to the x and time dependence which are together in the equation for a traveling wave, e.g. Acos(κx − ωt) . In a standing wave as in (55b) the φ and time dependence are arguments in separate functions (sine and exponential in this case) that are multiplied together.
3. Fitting a mode to the resonator at hand
A. Neumann Boundary Conditions
Neumann boundary conditions specify that at the edge of the space in question (at the resonator's walls) the gradiant of the wave function perpendicular to the edge be zero. In terms of a cylindrical resonator this means that on the top and bottom surfaces dZ(z)/dz = 0 , i.e. Z' = 0 at z = 0 and at z = h where 0 and h are z coordinates of the top and bottom surfaces. Choosing Z = C cos k_{z} z satisfies this requirment for z = 0 . For the z = h requirement, we must restrict our choices of k_{z} to:
k_{z} = p π/h , (56)
where p is and integer, i.e. p = 0, 1, 2, 3, ... .
To satisfy the Neumann condition on the curved cylindrical surface, we need dR/dr = 0 at r = a where a is the radius of the cylindrical resonator. Since R(r) is proportional to J_{m}(k_{r} r) we need to have J_{m}' = 0 at r = a. This occurs at the roots of the derivative of the Bessel functions, values which are readily found in references. This is to say, we need k_{r}a = u'_{m,n} where u'_{m,n} is the nth root of J_{m}' . Rearranging yields:
k_{r} = u'_{m,n }/a , (56a)
where n = 0, 1, 2, 3, ... and also m = 0, 1, 2, 3, ... .
Note that the n = 0 root is only used with m = 0. Fig. 3a above shows this mode. Trying to use the n = 0 index with m > 0 indices results in a discontinuity at the origin. On the other hand, the m = 0 index can be used with a variety of other modes such as shown in Figs. 4a and 5b.
Figures 1 through 5 above are animations of modes with Neumann boundary conditions.
B. Dirichlet Boundary Conditions
Dirichlet boundary conditions require the wave parameter be zero at the resonator's walls. To satisfy this requirement we change the cos k_{z} z in (55a) to a sine, and use the roots of the Bessel function instead of the roots of the derivative of the Bessel function. Thus as a Dirichlet solution we have:
where (56) above still holds, and (56a) goes to:
k_{r} = u_{m,n }/a , (57a)
where n = 1, 2, 3, ... and m = 0, 1, 2, 3, ... .
As to a easy example of a cylinder with Dirichlet boundary conditions, consider having water or other dense fluid contained in a light weight flexible cylindrical container. The high acousitic impedance of the water would mean that the container walls would be pushed around by the acoustical pressure oscillations forcing the acoustical pressure there to be almost zero.
4. Spectra of resonant frequencies
A. Spectra with Neumann boundaries
Fig. 6. Still images of the eight resonant modes of lowest resonant frequencies for a cylinder with Neumann boundaries, height of 100mm and radius of 70mm. Animations of most of these modes are found above in Figs 1 through 5. Generally, modes that are orientated to fit the number of wavelengths prescribed by the mode numbers in a longer distance per spatial cycle have lower frequencies.  
1,1,0 
0,0,1 
1,1,1 
2,1,0 
2,1,1 
0,1,0 
3,1,0 
0,0,2 
We use (54d) to find the resonant frequencies of cylindrical resonators with Neumann boundary conditions. We substitute (56) and (56a) into (54d) to yield:
where the indices of each mode are shown in red. Also v_{p} is the phase velocity (the p here stands for "phase" and is not the index p).
As to the allowed indicies, we allow any combination of m = 0, 1, 2, 3, ... , n = 1, 2, 3, ..., and p = 0, 1, 2, 3, ... .
Also there is an additional set of low order modes m = 0, n = 0, p = 1, 2, 3, ... . Setting the index n = 0 requires m = 0 in order that there not be a discontinuity along the axis of the cylinder. This additional set of modes is similar to the organ pipe modes studied in elementary physics but for pipes closed at both ends. An example of these would be the 0,0,1 mode shown in Figs. 3a and 6 above and also the 0,0,2 mode in Fig. 6.
B. Spectra with Dirichlet boundaries
⇐ Fig. 8b. Animations of resonant modes of a cylinder with Dirichlet boundary conditions. Shown are the standing wave 2,2,1 mode and the rotating wave 2,2,1 mode. The half disk hovering above the half cylinder shows the fields in a horizontal slice where the fields are maximum. The lower half cylinder is similar to the animations in Figs. 2c and 2d and represents the fields on the top and sides of the cylinder, as well as the fields on a vertical slice down the middle. It is assumed that the whole cylinder is present for supporting the mode but the front half is not shown. With Dirichlet boundary conditions, fields are zero at the top surfaces of the cylinders as is shown in the animations.
To see the action, mouse over the image, mouse off to suspend it and click on it to restart it. 
Fig. 9. Frequencies of a cylindrical resonator with Dirichlet boundaries and with radius a = 70mm and height h = 100mm. All the frequencies have been divided by (i.e. normalized by) the lowest frequency with the Neumann boundary conditions (see Fig. 7 above). The mode indices (m, n and p for the φ, r, and z directions) are shown next to each corresponding spectral line. Note that the Dirichlet boundaries result in higher frequencies for the lowest mode and the lack of certain modes which were present with the Neumann boundary conditions. 
With Dirichlet boundary conditions, we use roots of the Bessel functions instead of roots of the derivative of the Bessel functions. So (58) becomes:
where allowed are any combination of m = 0, 1, 2, 3, ... , n = 1, 2, 3, ..., and p = 1, 2, 3, ... . Note that some of the modes are missing that were present with Neumann boundary conditions.
5. Experimentally verifying these equations
The existence and frequencies of the above modes can be experimentally verified with very modest laboratory equipment and are suitable for student lab experiments. Neumann boundaries are easiest to work with. Experimenting with these resonance will give students insight into resonant modes in general and they are of large comfortable sizes.
Fig. 10. Experimental setup for measuring frequency and shape of acoustical modes in a rigid cylindrical container with Neumann boundaries. 
A large soup pot with a lid is a suitable cylinder for Neumann boundaries. Provisions must be made for inserting a small microphone into the pot with the lid tightly sealed and getting the microphone signal out to an oscilliscope. The nicest arrangement would be to attach the microphone to a stiff wire so it could be moved around to sample the acoustic strength in various places while the cylinder is excited.
Excitation can be provided with an electronic sine wave generator (signal generator) scannable over low kilohertz frequencies attached to a small speaker. An amplifier may or may not be needed depending on the signal generator output strength and the requirements of the speaker. The speaker could be placed inside the soup pot in a place where the gradient of the resonant mode of interest is strong. Or the speaker could be attached to the outside of the pot adjacent to a small hole in the pot allowing the sound to enter the pot. Clay can be used to seal around the wires going into the pot and to seal unused holes. Clay is available as "duct seal" at hardware stores in the electrical section. If one does not want to drill holes in the pot, a piece of laminated wood (as used for doityourself book cases) could used as a lid and suitable holes drilled in it. Weights on the lid will help it seal to the pot better. The microphone should be placed a reasonable distance from the speaker to pick up the response of the acoustic resonance and not the direct speaker output.
Fig. 11. Sketch of expected experimental results plotted in red. The data is collected one frequency at a time. The amplitude of the microphone signal is plotted vertically and the frequency horizontally. The blue lines represent the theoretically expected frequencies with the mode indices in black. If one reduces leaks and other losses the resonance peaks will be sharper (see ref. 1, ref. 2, and ref. 3 ). 
With the microphone in place and its output monitored on the oscilliscope, one slowly varies the exciting frequency noting at what frequencies the strongest microphone signals are detected. These are to be compared with frequencies predicted by (58). The shape of a particular mode can be ascertained by setting the exciting frequency to the measured resonant frequency of the mode of interest and moving the microphone around to see where the mode has maximums and minimums. This shape can be compared with the figures above. If one were to carefully record the acoustic strength versus z, r, or φ they could compare the results with the equations above for Z(z), R(r), or Φ(φ).
Exciting and observing rotating modes (see P. Ceperley, The American Journal of Physics, 60, 93842 (1992) or Fig. 2 above and (55c) above) require additions to the basic setup shown in Fig. 10. Exciting these waves can be done with two speakers located to excite the two twin standing wave modes such as those in Figs. 1a and 1b. The signals driving the exciting speakers must also be phased so that one is 90 degrees different in time. This requires some skill in electronics, or ability to use Octave (or the equivalent) to produce a signal in stereo with the two signals properly phased. An alternative method is to slightly split the resonant frequencies of the two twin modes with a "perturbation" and use a single speaker as is discussed in US patent 4686407 Split mode traveling wave ringresonator. To detect the rotating mode, one might send one of the driving signals to an oscilliscope and also send the microphone output to the same oscilliscope. If the microphone is moved around in the φ direction one should notice a shifting of the phase of the microphone signal as compared with the speaker signal. This is in contrast to a standing wave mode in which all fields are in phase or 180 degrees out of phase.
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