Standing waves by a speaker

1. Sep 26, 2010

SFB

Hi

I am actually confused on the relation of standing waves and resonance.

I know that standing waves are fromed when two waves of equal amplitude and frequency (one travelling and the other reflected or both travelling waves ) are superimposed in each other. This is the case when we also say that we have resonance. From here it seems that resonance is actually the formation of standing waves

Then again resonance is the situation when driving frequency mathes the system frequency. So can I have resonance even if there are no standing waves. Whats the exact relation.

Moreover , I have always seen this figure of stanging waves where two sine waves (one reflected and on travelling ) superimpose on each other and crates nodes and antinodes.

How does this thing takes place in case of a speaker. A speaker generates spherical waves and when such waves meets with a reflected wave of equal amplitude , do they form nodal and antinodal planes. Since the waves are sphere , I wonder how can it forms plane of uniform pressure .Is it the spherical that we call nodal plane or its s plane perpendicular to horizontal axis?

2. Sep 26, 2010

Born2bwire

You do not necessarily have resonance when you have standing waves. A simple case is the reflection off in one direction and open space in the opposite. In this case, if we send a wave towards the reflection (say a wall) then we can tune the distance from the wall so that the reflection and the source waves create a standing wave pattern. However, this is not a resonance since the reflected wave just keeps travelling off in the other direction. I would describe resonance as the behavior of a system to readily receive energy from an excitation and continually store it, barring dampening effects.

That is, if we continually input energy into a system at resonance, the system's energy continually increases. So we can see with the above example that even if we keep driving the source, the energy density remains the same despite the standing wave. If we were to place another reflector (assuming a 1D system) then we could trap the energy in the system but we need to excite the waves properly to satisfy the boundary conditions otherwise the waves will die out. As for resonance requiring standing waves... Hmmmm... I guess that is a reasonable assumption. Resonance isn't always stringently defined. It's not a mathematical quantity (though we can help quantify it with things like the Q factor) and so I guess there may be a counterexample but I cannot imagine what it would be.

Speakers are not point sources really. They are more like a piston though it is not meant to be as simple as that. However, a proper setup is one where the listener is in the far-field. 1 KHz has a wavelength of around 1 ft and 100 Hz thus has a wavelength of 10 ft. So a large portion of the audio signal is going to have a wavelength on the order of a few feet and less which means that most listening situations can be done in the far field. In this case, the source does look like a point-source that is directionally scaled (that is the wave is spherical but the amplitude will depend on direction since the source is directed). However, in the far-field, a small portion of the wavefront appears like a plane wave. So given the size of our ears (or a typical receiver) and the distance from the source, the intercepted wave is approximately a plane wavefront. As for the planes of uniform pressure, a spherical wave by definition has its isobars as spherical shells. So if two spherical waves intersect, the intersection of two spherical shells is a circle or ellipsoid (something to that effect) and thus that would the line over which we could see nodes form.

3. Sep 26, 2010

SFB

Thank you very much for your reply. It was really helpful. But I still have some confusion and it would be great if you can kindly provide the clarifications

I did not understand what you meant by the open space opposite to the reflector as I thought I have a speaker on that side.

As far as I understand, the source wave and reflector wave should form a standing wave and thus trap energy -then again as there is a driving source (a speaker ), the speaker is always adding energy to the system and thus the amount of trapped energy is continually increasing.

When the spherical waves intersect , do I have nodes only on the line of the comon circle or on any point on that circular plane.

I am basically interested to form for vertical planes of nodes and antinodes insude a rectangular cavity by using a speaker.

Some people suggested me to use a curved reflector to get better focusing but I am not aware of hot it helps to from standing waves since the reflected waves would then be directed towards the focal point and may not superimpose on the incident waves.

Thank you again for helping me out.

4. Oct 6, 2010

Born2bwire

This would only be true if the source acted as a perfect reflector as well. This is not generally assumed though it would of course depend upon the actual physical structure of the source. The wave is going to spread out as it propagates and thus the reflected wavefront is going to be larger than the original radiator. This means that if the radiator is reflective then not all of the energy is going to be reflected back. So just having a source and a single reflector can produce a standing wave, but it is not going to always create a resonant cavity (or one with a high Q) since it will be leaky.

We describe waves in terms of planes of constant phase. So when we envision a wavefront as a spherical wave we do so with the understanding that the phase of the wave is consant over this spherical shell. Nodal/antinodal points only arise when we have full destructive/constructive interference. This can only occur at places where two wavefronts meet and are always 180 degrees (or 0 degrees) out of phase. Thus, nodal and antinodal points can only occur along the intersections of wavefronts. If we are in the far field and have spherical waves, then the intersection of these waves will be a set of singular and separated points. At sufficient distance though, these spherical waves will look like plane waves over a given area. Thus, instead of having points you may have a line or even a plane that represents a nodal point (though you have to allow for a bit of tolerance since you still cannot get a true line or plane of nodal points). It all depends, but if you are talking about getting nodal points inside a cavity then you should expect to only get nodal points.

Getting a nodal plane will be a tall order and only exists in the limit of infinity. You would have to create a true plane wave (requires an infinite radiator) and a true plane reflector (also infinite). Of course you can always get a plane wave over a small area by setting your source and reflector far enough away, but you still get some power that will have to bounce off the tops and sides of the cavity and you will need a very large cavity to make these reflections minor so that it looks like an open space to the speaker and reflector.

A curved reflector would not be good because as you state it is meant to focus a large wavefront into a small point (and vice-versa).

5. Oct 6, 2010

SFB

"The wave is going to spread out as it propagates and thus the reflected wavefront is going to be larger than the original radiator. This means that if the radiator is reflective then not all of the energy is going to be reflected back. So just having a source and a single reflector can produce a standing wave, but it is not going to always create a resonant cavity (or one with a high Q) since it will be leaky."

The reflected wavefront will be larger than the speaker and there may be losses (leakage ) if the reflected wavefront can travel freely over the speaker. But if the radiator is flush mounted on a wall , should not this effect be minimized as relflections from the wall will prevent such leakage.

As plane wave can ensure nodal and anti nodal plane , isnt there anyway to convert the spherical wave to plane wave by putting something flat (that can move linearly and generate plabne waves.The plate can be connected to the wall by some elastic material to ensure forward-backward movement) near the face of the speaker .Then again as there would be an air gap between the flat plate and the speaker, how much of the wve energy would actually be transferred from the radiator to the flat plate. Would there be losses due to impedance difference between air and the plate.

Also what would cause the plate to move linearly.Does it depend on how much force the incident wave is giving on the plate or on how much of the wave is transmitted.

What is a Q factor ?

In a curved reflector ,the sound beam is more focussed and does intensities near on-axis should be higher than the that of the incident waves. Can there ever be constructive or destructive interference ( as constructive or destructive interference requires waves of same amplitude /energy to impose on each other ).

Last edited: Oct 6, 2010
6. Oct 6, 2010

sophiecentaur

For a good resonance to occur, you need a lot of reflections (back and forth), so that the energy builds up. The 'Q' factor is a measure of how many of these waves contribute significantly to the energy within the resonator - you get a good resonance on a guitar string because relatively little energy is lost per cycle of oscillation. Q for a Quartz crystal is in the order of millions, for a tuning fork it's a bit less, for a guitar string it could be thousands. With sound waves in the system you describe there will be a lot of energy lost so you won't expect a 'good' resonance; the Q could well only a few tens.

Of course, the phasefront shapes are ultimately important and the best way to control this problem is to have source and reflector consisting of flat parallel planes. Then you should have well defined nodes on axis and not lose too much energy in the middle - allowing you to detect a fair standing wave which will demonstrate your resonance.

Putting an appropriate speaker at the centre of a sphere could work well - true but irrelevant as you have neither! A tiled bathroom works quite well; It does wonders for my voice but the actual standing wave pattern (many different modes) is a bit too confused for any serious measurement.

7. Oct 6, 2010

SFB

I was thinking of putting a thin metal plate infront of my speaker.The plate can be connected to the wall by springs (or some other elastic means ). In that way , the oscillation from the speaker can cause linear motion of the plate and it can generate plane waves in the air ahead of it.These plane waves , when reflected by the opposite walls can create nodal planes.

But I am not sure what would be the optimum air gap between the speaker and the metal plate . As the wave is changing medium (air to metal on one side and then metal to air on the other side of the metal plate ), how much of loss it may incur.Should I choose the air gap in a way so that the there is a pressure antinode on the speaker side of the metal plate (as it would create the maximum pressure difference between air on on both sides of the speaker and thus will cause maximum movement of the plate.

I can understand that I have losses as the reflected wavefront gets bigger.But that can happen with any transmitter .So why would there still be so much loss in the system.Would you please clarify what are factors degrading the Q value for my system. If the tuning can have have such a high Q , should not I be able to acheive that once I get plane waves. I apologize if I am asking for something real simple.Since it has only been a few months that I am working on this area , my conception of this topic is yet to be strong.

Thank you again for your feedbacks.

8. Oct 6, 2010

sophiecentaur

Not a bad idea if you can actually 'tune' the plate and suspension to resonate at a suitable frequency. Not sure what frequency you want to work with but that may not be easy to get the plate to move when coupled to the speaker this way (i.e. just by air coupling, which will involve two serious 'mis-matches').
You may have better luck if you just attach the plate to the speaker cone (brute force and ignorance). It will ruin the performance of the unit. of course but, as you would only be using a single frequency, the response wouldn't matter. It will not be efficient, of course, but that will not be a bad thing as the plate will also be a better reflector than the speaker cone. The plate could have a much bigger diameter than the cone and provide a more ideal source of plane waves near the axis. You would need to hunt for a suitable frequency in the range of interest, at which the plate moves most. The distant reflector would need to be as large as possible, rigid and of high mass (a concrete wall?). The spacing must be small, too. Only a few wavelengths at the most. That way you should get a standing wave which can be measured with a tiny microphone.
Why won't you get a good Q? The waves will not be truly plane - there will be a huge amount of energy lost around the edges and out of the gap because of diffraction. Remember, if you only lose 1% of the power flow in the cavity this way, the Q will only be 100.
Did you mention what sort of frequency you wanted to use?

I just thought. This plate would need to be fairly light in order to get it vibrating and it needs to be rigid. You might try some honeycomb structure or make it ridged - as long as the ridges were fine enough - less than the wavelength in air.

Last edited: Oct 6, 2010
9. Oct 6, 2010

SFB

If I have a honeycomb structure, then waves from radiating speaker may pass through the holes. Would not it interrupt the plain waves created by the honeycomb.

I am using a 500 micron thick aluminum plate.The air gap is one wavelength so that I can form a pressure antinode on the aluminum plate.The pressure difference with air pressure on the other side of the plate is maximum and that causes the plate to move.

10. Oct 6, 2010

Studiot

This is already done in commercial applications, speaker manufacturers call it a bass resonator or passive bass amplifier or some such advertising nonsense.

The only problem with any such arrangement is that usually the standing wave you get is determined by the column of air between the driven plate (speaker cone) and the passive slave plate, not by the input signal frequency.

11. Oct 6, 2010

SFB

@Studiot Thanks for the info. Would you please elaborate the phenomena .

If I use the flat plate , should not I try to drive it at its resonance frequency.How do I calculate it.

12. Oct 6, 2010

sophiecentaur

I meant a honeycomb behind to make it rigid. This will avoid surface waves on the plate - the reason why many loudspeaker drivers sound so bad on transients. I think 500mu will not be rigid enough on its own. It will form standing waves all over itself, I am sure. because it will not be excited co-phasally (after all, if it were, then you wouldn't need it! It could be useful to paint it with a bitumen to stop these surface waves.

13. Oct 6, 2010

Studiot

In amongst all the stuff others have already given you is the crux of the difference. Resonance is the ability of a system to accept periodic energy of certain frequencies.
These frequencies are determined by the physical characteristics of the system.

Periodic Energy at other than the resonant frequency will be quickly dispersed by the system.

Continued supply of energy of the resonant frequency to a system may result in the vibration amplitude growing without limit until the system is destroyed, or it may be self limiting.

The formation of standing waves is one mechanism for providing the self limiting.

Consider a rope lying on the ground.
Pick up one end, leaving the other free, and supply a forcing function (waggle it up and down).
You will be able to generate waves in the rope of frequency equal to your waggling i.e. any frequency.
Now tie the other end down.
You will find the only waves you will now be able to generate are frequencies whose wavelength are a whole multiple/fraction of the distance between your arm and the fixed end.
The rope will now exhibit standing waves.

This experiment also demonstrates another characteristic of some systems. The ability to transform the output frequency to its resonant frequency for a wide range of input frequencies.
Whistles and organ pipes are both acoustic examples of this phenomenon. This is also what I mean about your speaker - plate system.

Let the spacing between the speaker cone and the plate be s. Then there is a 'column' of air, of length s, available between the cone and the plate. This column will have a normal column resonance and the output will be this frequency for a wide range of input frequencies.
This phenomenon is the bane of speaker designers.

14. Oct 6, 2010

SFB

@Studiot ...What did you mean by periodic energy.Is it the energy coming from the driving source? I heard resonance occurs when driving freq matches the natural freqeuncy of the system. As I understand , when these two frequency match , there is periodic supply of energy from the speaker. Is this what you meant by periodic energy.

@ sophiecentaur... The transmitter I have resonates at 50KHz. Any suggestions on tuning the frequency of the speaker with the metal plate and suspension.....

15. Oct 7, 2010

Born2bwire

Here's the problem with your setup. I did some simulations and generated some videos to better illustrate the points I made out before.

Let's say we have our plate resonator, the problem is that it is a finite surface and thus will not create a perfect plane wave. Only in a small area in the middle will there be a good approximation of a plane wave. That's all fine and dandy if this was in open space but now you want to enclose this inside a resonant cavity. This means that the non-plane wave elements of the wave will be trapped inside. They will bounce around and interfere with the desired plane wave and thus eventually it will completely disrupt things.

So, let's look at a cavity of 10 wavelengths by 10 wavelengths and we place our speaker flush against a wall but assume the speaker is transparent. The source is two wavelengths in size and so it is on the large size.

Now let's place the the source on the wall.

Notice that in these pulsed simulations we do get a decent plane wave but the resulting side lobe reflections disrupt the plane wave the more it reflects off the walls.

Now let's continually feed the source to see the resonant behavior.

At first we can see that the cavity sets up a pretty good set of nodal planes (they flip back and forth but the scale is saturated enough to make it easy to pick out). But as time progresses, these nodal planes start shifting around and breaking down as the side lobe reflections interfere more and more with the desired plane waves.

Now there could be ways that we could reduce this. 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. You could try to place a very good absorber along the side walls of the cavity. This would absorb the side lobe energy and help kill off the reflections. I could run a simulation that shows that since I have an absorbing boundary layer that I can turn on. We could also try to make the cavity very wide so that the side lobe radiation gets directed down off to the sides while the plane waves keep oscillating back and forth in a straight line. However, this will not work unless we place an absorber along the other parts of the cavity because eventually, in a closed cavity, the radiation that leaked off the sides has to go back in given enough time it will fully permeate the cavity and we will have the same problem as before.

One thing to note is that when it comes to a rigid piston, the radiation pattern in the far-field will look like the Fourier Transform of the piston. So a rectangular piston will have a radiation pattern similar to a sinc function. You can work this out with paper and pencil though I have forgotten how to do so (been a long time since I did acoustics), but it should indicate to you the existence of the sidelobes that can cause problems.

Last edited by a moderator: Sep 25, 2014
16. Oct 7, 2010

SFB

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 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.

" 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 beacause with larger sidewalls will generate more surface waves during reflection than with smaller side walls of a cavity.

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.

Last edited: Oct 7, 2010
17. Oct 7, 2010

Studiot

You have a very ambitious project here.
I suggest you have a chat to your tutor about mechanical resonances, since I think you are missing the true source of this.
Why is your plate mounted on springs, rather than edge clamped?
In such cases the resonance will be controlled by the springs, not the plate, where merely acts as a connector between them.

Oh and periodic energy = energy input at regular time intervals by for instance mechanical force or electrical voltage. This may or may not be sinusoidal, the point is that it repeats every time period.
If you have studied wave motion, vibrations and oscillations mathematically it is the driving function or forcing function on the right hand side.

18. Oct 7, 2010

Born2bwire

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.

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.

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.

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.

Last edited by a moderator: Sep 25, 2014
19. Oct 7, 2010

SFB

@Born2bewire

Thanks for your email .I wanted to use a flat plate rather than a speaker because its difficult to get a nodal planes with spherical wave unless your cavity is large enough to make a planar wave approximation of the large spherical wavefronts.

If my sidewalls are open (exposed to air ) , why would I still have reflections as there will be no impedance mismatch when the sides are open (the medium is all air ).

@Studiot

It seemed to me that springs, if properly selected, would behave more elastically rather than some other connections. I tried to edge clamp the plate using some elastomers (the air gap was 2 mm in this case with other dimensions remaining the same )but it did not work as the elastomers absorbs a consderable amount of the vibration.

20. Oct 7, 2010

Born2bwire

Take a look at my edit. I did not realize you were working at such a high audio frequency. I was thinking like 1 KHz human auditory noises. I don't know how big your speaker or plate is, but chances are it is many wavelengths since the wavelength is on the order of millimeters. So it is not going to be point source like under these circumstances but it becomes more complicated to find out what it actually looks like.