A ribbon loudspeaker consists of a rectangular element of aluminum foil suspended within a magnetic gap and clamped at its ends (ie a clamped-clamped beam). Current is applied to the foil, which produces an electromagnetic field that interacts with the magnetic field generated by the permanent magnets. The foil bends in response to this force.

For compression and dynamic cone drivers, they might be approximated as pistons below their fundamental mode. We might assume uniform velocity over the surface of the piston, however we can not make such assumptions with regards to ribbon loudspeakers. Their surface does not move with uniform velocity (maximum at center, 0 at boundaries).

How can we calculate the complex pressure in the exit aperture for a ribbon loudspeaker located in a rectangular duct?

I would be very appreciative if you are able to contribute any knowledge or information (journal articles, texts, etc) regarding this phenomenon.

I'm not sure you'll find a closed-form solution for the pressure in a rectangular duct. However, this problem is of particular interest to myself as well. I have been solving a benchmark problem of an oscillating piston. I realize that the ribbon speakers you're speaking of don't move with constant velocity as my wall boundary does. But, there is a solution for pressure fluctuation of an oscillating piston on a rigid baffle into open air.

I'd be happy to share the solution with you. Aside from that, I could show you some CFD plots at some reduced frequencies. I'm not sure how much it would help, but maybe we could eventually help each other out a little.

That's a toughie. Do you have any kind of expression for the velocity potential at the ribbon? The rectangular duct makes it a bit tougher in that everything I do is round.

The exact solution for an oscillating piston mounted in a rigid baffle is given by:
[tex]
p'(r,\theta,t) = \frac{ik\rho_0 c \pi z^2 U_p e^{i(\omega t - kr)}}{2\pi r} \left[ \frac{2J_1 (kz\sin\theta)}{kz\sin\theta}\right]
[/tex]
Where k is the wave number, c is speed of sound, z is the radius of the piston, U is the maximum piston velocity.

As you can see, this breaks down at small radii so if only valid for the far-field.

I don't have a lot of good looking plots right now, but here is a plot for a piston period of 16. In this case, everything has been nondimensionalized such that c = 1.0. T = 16 gives a reduced wave number kz ~ 3.9, which is just past the first root in the equation:
[tex]
f = \frac{2J_1 (kz\sin\theta)}{kz\sin\theta} [/tex]

That term in the solution gives a directivity and "lobe" relationship. At high frequencies, above kz~3.8 there will be a "lobe" in the solution which theoretically there is no pressure pertubation. As the Bessel functions keeps passing the axis, this number of lobes increases.

It's apparent when looking at the FFT results, which aren't done yet. In fact, I just went on to look at the results and realized that the disk was full and the first run was pretty much completely wasted.

I remember working this problem out in Excel. I will never do that again!

Unfortunately the OP is looking for the near field for sure. That is one of the things that makes this one really tough. The other part about the square ducting doesn't help. The directivity can be tracked down to the relationship of wave number and piston radius. Very low kz, i.e. kz<<1, will produce a symmetrical pattern. Higher kz>1 will go from a double lobed to a single lobed form.

Perhaps a more specific description of my problem would be more appropriate.

A loudspeaker distributes energy throughout the room. We must consider the direct response as well as the power response. The direct response can be visualized as a vector that extends from the acoustic center of the loudspeaker to the ear of the listener. A frequency response measurement of the loudspeaker conveys the amplitude of the direct response. The power response is the total energy radiated by the loudspeaker, which has a great influence on the reverberant field within the room.

Research suggests that consistent power response as well as a flat direct response has a high correlation with sound quality. If power response is consistent with respect to frequency, equalization can be used to correct errors in the resulting acoustic field. However, if the power response is not consistent with respect to frequency, equalization cannot be used.

I am interested in the acoustic response of a line source alignment. The loudspeaker will extend from the floor to the ceiling, essentially creating a quasi-infinite line source (sound cannot expand beyond the floor or ceiling, forcing it to expand in 2 dimensions).

Two solutions are possible to achieve consistent power response and flat direct response with respect to frequency.

1) Use multiple, small ribbons to approach uniform velocity over the height of the line source.

2) Use a single, full length ribbon (non-uniform velocity)

Any thoughts on which would be the better solution?

The complex pressure in the exit aperture is required under both circumstances to determine the resulting response.

The dimensions of the full length ribbon are 2286mm (height) x 2.83mm (width). The dimensions of the duct are 2286mm (height) x 2.83mm (width) x 3mm (depth). The duct is located on an infinite, rigid baffle. However, it is constrained by the floor and ceiling boundaries (located at the top and bottom of the line source). Would the aforementioned alignment suffer from a lobing response?

It is obvious that a piston with uniform velocity distribution will exhibit lobing artifacts beyond ka=3.83, however the vibroacoustic response of a beam in bending is quite dissimilar to that of a piston.

The researchers here did test using various boundary conditions. One was a rigid piston moving at constant velocity, one was a gaussian movement, and one was sinusodial. This gave them one set with continuous displacement and velocity, one with only continuous displacement, and then of course the rigid where both disp and velocity are discontinuous.

The rigid piston gave them the largest lobes. They weren't spot on with accuracy, so I'm not sure how much you can pull out of it, but either way, it seems as though the piston not moving with constant velocity gives the "smoothest" results.

Would it be easier to solve for the fundamental resonance of a tensioned membrane within a particular duct geometry?

The texts I have available (Mechanical Vibrations and Roark's Formulas for Stress and Strain) make reference to rectangular plates, but do not combine plate stiffness and tension stiffness. Does an analytical solution exist for such a problem?

How could I combine the reactive/resistive forces imparted by the duct geometry?

For a circular flat plate, per (http://solidmechanics.org/text/Chapter10_7/Chapter10_7.htm) the natural frequencies is given by the solution to:
[tex]
J_n (\omega_{(m,n)}r\sqrt{\rho t/T_0}) = 0
[/tex]
Where the index m corresponds to the number of circumferential lines and n corresponds to the number of diametral lines that have zero displacement.

So the first fundamental mode will be m=1,n=0, or a zeroth order Bessel Function of the First Kind.

However, I think that the solution is simply too complex to form a closed-form solution to. The interaction between the waves with a real geometry I believe is simply too much get a "real" solution.

Moreso, I am finding that with "real" pistons there can be a 2E harmonic directly above the piston with a much smaller cone angle. But, that all needs to be taken with a grain of salt.

The addition of a duct adds complications of wave guide phenomena to the situation. Not all frequencies will propagate and will be a function of mode number.

I'm interested in building a DIY ribbon loudspeaker in my free time over Christmas break and would like to drive it directly. Following the resolution of some meetings with faculty in the engineering department, I might be able to gain access to the Universities Laser Vibrometer and Anechoic chamber for measurements (and comparison with the RAAL 140-15D)

My primary goal for the project is reproducing a Dirac Pulse/Square Wave, so impulse response will be weighted with the highest significance.

The ribbon will initially launch a plane wave, which will transform into a spherical wave with respect to time. As the wave propagates down the rectangular duct, an abrupt transformation of the wavefront will occur once it reaches the end of the duct. Diffraction and Higher Order Modes (reactive forces) will result from this I believe (read: BAD). This is what is causing me trouble with regards to optimization.

For flat pistons, they are affected by the global pressure over the surface. However, ribbons, being elastic membranes, are affected by local pressure over the surface. I haven't done any simulations, but the reactive nature of the aperture may have an effect on the ability of the ribbon to reproduce an impulse.

A small roundover should provide a linear rate of expansion and can provide a vector that is orthogonal to the initial vector, however this seems like an obvious solution.

If the width of the exit aperture of the waveguide is restricted to 1/2 wavelength of the highest reproduced frequency (8.5mm for 20khz) and the depth of the waveguide is restricted to 1/4 wavelength (4.25mm for 20khz), I believe the acoustic impedance would be primarily resistive. The optimal aperture should minimize reactance and allow resistance to approach a constant value, I believe. Could generalizations be made in this case? Would the shape of the waveguide be significant with regards to the response of the tensioned membrane if the following restrictions are assumed?

If the duct was engineered to provide a primarily resistive impedance above a particular frequency, where the uncoupled membranes fundamental resonance may lie, wouldn't the ducts effects (coupled membrane-duct) be reduced to a broadening of the Q of the fundamental, rather than a shift of the fundamental?