SFB said:
Your videos are indeed extremely helpful tutorials for me . Would you please clarify what the idea of plane waves has to do with finite/infinite surface of the slave plate.There can always be some leakage from the waves between the speaker and plate and that may disrupts the plane waves emitted by the slave plate. Is this the problem you are indicating with a finite source?of the over the plate
Well it goes back to what I mentioned before, the radiation pattern of a rigid piston is the Fourier Transform of the shape of its surface. So, a true plane wave has a radiation pattern only in one direction, 0 degrees or broadside. This means that the radiation pattern is a delta function (1 at 0 degrees and 0 everywhere else). The inverse Fourier Transform of a delta function is a function that is 1 at every point. This means that you need to have a rectangular plate with infinite width and length to get a true plane wave. A finite rectangular plate gives you a sinc function and the width of the sinc function is related to the size of the plate. The larger the plate, the narrower the sinc and thus it approaches that of a delta function. So this has nothing to do with your real world implementation but from the fact that no finite radiator or source can ever produce a true plane wave. A similar analysis can be done in electromagnetics (the wave physics is practically identical) and we find that we would need an infinite current sheet.
SFB said:
Now let's place the the source on the wall. One thing to add is that my cavity is open at the top and bottom and enclosed by the side walls.
Then that isn't going to be a real resonating cavity because you will always have large amounts of energy leaking out of the top and bottom. That was related to my suggestion of having a wide chamber. If you can remove the power in the side lobes from the cavity then they will not cause interference and you can put an absorbing layer if it is enclosed or you can just leave it open (although if it is open you do get reflections off of the opening while an ideal absorber will be reflectionless). This is because the cavity is now like a waveguide and it will have an "impedance." The open air will also have an impedance but because it is not in a restricted volume like your waveguide it can be different. This means that at the interface, some of the wave will be transmitted through and some of it will be reflected.
SFB said:
" This is a very large cavity and as such it supports a large number of modes. Thus, the side lobe reflections will easily excite a valid mode and will not die out. If we made a smaller cavity then the side lobe radiation may not excite as many valid modes and less of the energy will persist in bouncing around the cavity. We could also design the cavity to try and minimize the modes that would support them but this would be a complicated feat."
Why a large cavity easily excites more modes? Is this because with larger sidewalls will generate more surface waves during reflection than with smaller side walls of a cavity.
A mode is a field distribution that satisfies the boundary conditions of the cavity. This depends upon the operating frequency and the geometry of the cavity. For example, if we have a cavity of rigid walls, then we have to have the pressure be zero along the walls (since they do not vibrate). So if you excite a wave inside the cavity that does not present a zero pressure along the walls, it will gradually lose energy because it is not being supported by the cavity. It may, however, eventually bounce around until it falls into a wave distribution that is supported. Sometimes though that there is no wave distribution at a given frequency that can support the wave and it will always die out. The larger the cavity is, the easier it is to fit a wave distribution that matches the boundary conditions. So if we have a small cavity, only a few modes can be excited and some of the undesirable wave elements can die out as they bounce around searching for a supported mode. However, again if we are talking about resonant behavior this makes less impact because we are continually feeding in energy. This is because we always replenish the energy that dies out and as long as it eventually finds a mode it will build up energy, just that it will build up energy at a slower rate than the initially supported modes. You might want to read up on how a resonant cavity works.
http://www.amanogawa.com/archive/wavespdf.html
The above has some good notes for electromagnetic waves which I stated before have practically the same physics. Take a look at the "Wave Guide Cavity Resonator" notes.
SFB said:
I tried my setup with a aluminium plate. I mainted a wavelength gap. But when I drive my speaker , I do not feel any movement of the plate. One reason can be the stiffnes of the springs (the plat is connected to the wall where the speaker is flush mounted by 8 springs of same k value ). So may be the reducing the number of springs would help the plate to exhibit some linear motion.
But I was wondering as my source is operating at a frequency of 50 Khz , may be the plate is vibrating also at such a frequency and its too quick to detect or feel. I am taking this as a reason as It seemed to me that when in a frequency within the audnble range , I think the vibrations can be felt more easily. I experienced this before by putting a flat plate over a 4 Khz speaker.
Also I am struggling to select an optimum stifness for the springs. If I want the al plate to move x mm backward and forward , then
pressure difference between both sides of the Al plate = Spring stiffness * x
How do I get the pressure difference as I do not know the average pressure working on the speaker side surface of the Al plate. Is thre any way to calculate it . I know the dimensions and driving voltage of the transducer ?
I never thought of getting so much feedback from this thread.Thank you all for helping beyond my expectations.
I don't know about that, acoustics ain't my bag, that's EM. But springs are going to provide a damped oscillation. You may be driving it in such a manner that the transferred force is too heavily damped. Or, as sophiecentaur mentioned, the rigidity of the plate may not be high enough to perform well at that frequency. Is there any reason why you want to use the plate? The speaker itself may put out a better waveform although it is probably not designed to be directional which will be a problem.
EDIT: I should make a caveat to what I said earlier about the waves being spherical and the source a point source. I was assuming that you were looking at typical audio frequencies and speakers. The extent that the source is like a point source is dependent upon the distance from the source and the size of the source. If the source is small or up to the size of a wavelength, then being maybe around an order of 10 wavelengths away will make it look like a point source. But as the radiator grows in size, we would have to go farther away. This is why my 2 wavelength large source is able to produce slightly non-cylindrical waves at the 10 wavelength length of the cavity. But we can already see some degree of diffraction at the edges that I have been talking about before. This would matter more to you because you are working with 50 Khz and though I do not know the size of your speaker it is probably very large compared to the wavelength. Thus, you may be working in the Fresnel zone or near field as opposed to the far-field. But if your purpose is to get plane wave like radiation via a rigid piston than I think we can see as in my simulation that the large size of your radiator is only going to work to your benefit.
By the way, here is the chamber (slightly undersized though, I forgot to increase the problem size when I increased the size of the absorption layer to allow for longer and cleaner runs) with the sides replaced by an absorber. You can now see, in comparison to the previous video, that the plane wave is able to oscillate in rather a stable manner. However, there is still energy bleed into the absorber over each cycle. This is because the wave is slowly spreading out and leeching into the absorber. This would invariably reduce the Q factor of such a resonant system.