How to calculate radius of curvature of a reflector

[SFB]

Hi


I need to calculate the radius of curvature of a reflector.I have a sound source (Ultrasonic transducer of 40 mm operating at 50 Khz) in air . I am trying to generate a standing wave using this sound source .As curved reflectors can help to amplify the sound pressure (I actually don't exactly understand why ,it would be helpful if you can help me to visualize), I need to calculate the radius of curvature.


As it should be analogous to reflection of light as well, I was wondering if there is a way to calculate desired radius of curvature If I know the wavelength.



Thanks

 

[sophiecentaur]

Having curved reflectors will lessen the amount of sound energy lost at each reflection - hence, more of a standing wave can build up.
The reflectors should, ideally, be parabolic but spherical is easier to make. Ideally you would space them so that their foci are in the same place (i.e. quite close). The focus is about half the radius of curvature away from the back of the reflector.
http://en.wikipedia.org/wiki/Curved_mirror"

The radius of curvature is less important than the actual diameter of the reflectors; the wider they are, the less sound energy is lost and the better 'amplification' you will get.

For some while I used two (only) 1.5m diameter microwave reflectors (facing each other) as a demo in my Physics Lab at School and, with the two dishes closely spaced, the effect was quite disturbing. With them at opposite ends of the room, the 'gain' in sound level with a sound source at one focus and your ears at the other focus was very impressive. I never actually went for a standing wave - wish I had, in hindsight. I did, however, use a cheapo Ultrasound Source and detector and results were excellent.

It may be worth trying to get hold of a couple of knackered satellite dishes from a scrapyard rather than trying to make your own. Sheet metalworking is difficult and a few dents will have no significant effect.

 

[SFB]

Thanks for your reply.Would ou please clarify what did you mean by suggesting that the foci should be ideally in the same place. Unlike you , I have a single sound source at one wall and a reflector on the other wall.

Also ,as the speakers radiates spherical waves , how would it create constructive and destrutive interference unless the reflected waves are coming back in the same direction.

As I understood from light characteristics , the only case when the light will not change its direction in when to transmits through the centre of lens .


It would be great for me if you an kindly clarify how this analogy is drawn.


Thanks

 

[sophiecentaur]

A property of a parabola is that, whatever path you take from the focus to the surface and onwards after reflection, your signal will arrive at the same time in a plane (i.e. a plane wavefront will result). If that wavefront arrives at another dish, the plane waves will again converge, in phase, at the focus. This process would repeat and repeat under multiple reflections between the two dishes.

I guess that, if the resulting waves are plane, then the foci need not actually be in the same place. If you just have one source (against a plane wall?) then the geometry would need to be different and the curved reflector would need to be another shape, in order to deal with a spherical wave. Or something different, if the source is not actually on the wall. This would be because you would be getting the resultant of the direct and wave from the wall behind - producing a phase tilt away from the axis and also an interference pattern with maxes and mins. Even if the source were very directional, the problem would occur with the first-time reflected wave and dissipate the power.

From what you say, I think your ideal shape of (single) reflector would be a sphere of radius equal to the distance from source to dish. Then all paths there and back would be of equal length. However, this arrangement would lose a lot of the reflected energy after just the one return trip, whereas in the two reflector system the energy would make a number of return trips and produce more 'amplification'.

Note: if you use an ellipsoid with source and detector at each focus then all the energy will get transferred. But that would involve the whole room being egg-shaped, reducing its usefulness for anything else !

p.s. I also used my arrangement with a demonstration microwave system and it, not surprisingly, worked equally well. It was a shame that the two dishes ended up in the skip as I had nowhere to store them and I had to vacate the lab.

 

[SFB]

thank you again for your excellent feedback. My ideas on spherical waves are a bit limited and it certainly helped me to think more. Since I have a sound source flush mounted on the wall , I was wondering If I want to have a distance where I can assume that the spherical wavefronts can be considered as planner wavefront , what would that distance be and what would be the other parameters that will influence such approximation.


Moreover , if I use a langevin transducer bolted to a horn (often used for acoustic levitation ),
what kind of wave it basically emits?Do you think it would be a could idea as I am interested to form a plane of nodes and antinodes, it seems to me planar waves would made it a lot easier.


Look forward to your suggestions on this .


Thanks

 

[Redbelly98]

If the reflector's surface is shiny-smooth, you can just see where a distant light source is brought to a focus. The center of the radius of curvature is twice as far from the reflector as the focal length.

 

[sophiecentaur]

Because of the uncertainty of the position of the actual 'source' of microwaves or ultrasound (the spacing between the transducer and the wall will be significant in terms of wavelengths) you will not be able to rely on your wavefront shape, I fear. But does it matter?

Knowing the position of the 'phase centre' of a horn is hard. Google that and you'll see what I mean. It's very important when designing high quality microwave dish feeds and there is a lot written about the problem. I actually think that there may be no need for the horn, in the first place (unless it is integral to the transducer). A flat plate with the transducer embedded in it will launch a reasonable wave and the phase centre will, I think, be at the surface. But even if it's not, why should that matter? It's only an 'end effect' and you will still get a standing wave as long as the spacing between the two reflectors is appropriate. (That's an important factor which will be hard to deal with using the whole room)


I have a feeling that, if you want to produce a good standing wave that can be measured / observed, you would be better to have a pair of flat reflectors, spaced by only a few wavelengths with the source mounted flush in the surface of one. The energy will spread outwards, of course, and spill out from the edges but the standing wave pattern should be pretty strong near the middle. (The sharpness will basically depend upon the number of reflections you get.) You could use a probe on a stick, poking though a hole in the 'other' reflector and examine maxes and mins at your leisure.
Doing the experiment with room sized equipment (involving reflections from 'clutter' in the room, too) would cause more losses and much less distinct standing wave (multiple extra interference patterns).
I can understand the attractiveness of a whole-room demonstration but I think it may just not work well enough compared with a smaller system. I could point out that you could use audible frequencies which would allow students (? or you) actually to Hear the standing wave effect as they move about!

@Redbelly
Yes - even a not-too-shiny / smooth reflector will show the focusing effect. And the wavelength involved means that the optics aren't too critical.

 

[sophiecentaur]

btw, you can only rely on a spherical wavefront if your source is a point, out in the middle of nowhere. Second best would be on the end of a long stick and firing out along the axis. That would give you an inverse square law for intensity with distance.

 

[SFB]

I actually used a flat reflector at the very beginning and tried to measure the nodes ith a microphone. But one confusion I had is if I insert the microphone in the sound field , won't it interrupt with the sound field inside the cavity . What would be a good reason for this question. I have seen some papers which suggested standing wave visualization rather than using a microphone as it may destroy the sound field. Then again , there are numerous papers that used microphone and successfully find nodes and antinodes.





Thanks

 

[SFB]

Moreover , I am also confused with the waves emitting from the source. As I am using a ultrasonic spekaer (it has a vibrating membrane that vibrates when alternating voltage is applied ), I am assuming that I am getting spherical waves. What would be the reflected wave pattern if my reflector is a flat wall? Is it spherical or plane wave? In what manner would it interfere with the incident spherical wave(source wave ). What would be the difference in interference behavior between plane -spherical wave and spherical -spherical waves.


It seems to me the only case when I can get a plane of standing wave is when both my source and reflected waves are plane waves.Am I assuming right?

 

[sophiecentaur]

You seem to be over-concerned about the actual shape of the wavefronts. Only from a 'true' point source will you get perfectly spherical waves - or when you are at a great distance (in which case the radius is so huge that they look like plane waves to an observer)

The wave which is launched from a horn is fairly flat near the axis because the horn is not a point source. If this is reflected from a nearby, flat reflector, back to another flat reflector (which the horn pokes through), the phases of the multiple reflections near the axis of the horn will also be well enough behaved to expect a fair standing wave. (i.e the waves will be fairly plane) You are right in implying that odd shaped wavefronts can spread out / fill in / degrade a good standing wave pattern but it shouldn't be such a problem on-axis.
The worst perturbation can be the microphone you stick in the way. That's why I talked about a 'probe' , which could be a narrow tube protruding from the reflector. The actual transducer can be on the end of the tube, outside the resonant region. It will affect the waves inside minimally, that way. It means that you need to be taking as little energy out as poss to keep the resonance high.
You will only get a standing wave at all if the separation is the right number of exact half wavelengths, remember - just like standing waves on a guitar string. For any other separation the interference is just destructive. (Hope that's not telling my Grandmother how to suck eggs; sorry if it is) The better the potential standing wave is, in fact, the more precise you will need to be.

Visualising - or producing a visible demonstration of standing waves- is possible using a light powder (Lycopodium Powder, classically) in a cylindrical glass tube with a sound source one end and a movable piston in the other. There is an ancient demo with the delightful name of Kundt's Tube which works after a fashion, in the same way and causes 'much mirth' amongst students; I can't think why. But a small enough (and sensitive enough) probe should work well enough.

 

[SFB]

Thank you very much for such an excellent feedback. Your last reply clarified a lot of the ambiguities that I am facing with my experiment. Would it possible to suggest me some literatures that you believe can help me to develop a conceptual framework on spherical wave propagation and its dependence on the geometry of the cavity.


Thank you again for all your help.