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Two parabolas, facing each other

  1. Aug 19, 2014 #1
    Hi, all.

    I was wondering what happens to sound if you've got two identical parabolas facing each other.

    Apparently, from what I can read, a parabola concentrates all the incoming sound into its focal point. Once at the focal point, it bounces back to the parabola, where it is sent out in straight (would you call it coherent?) parallel lines away from the parabola.

    Now, if that sound coming off Parabola A is recaptured by Parabola B (and again concentrated at its focal point, then bounced off Parabola B toward Parabola A again)... would the sound continue bouncing back and forth? Would it degenerate into heat? What happens?

    Now imagine those two parabolas inside a cylindrical container... and that you're continually adding more sound pulses (say, from an exhaust pipe on an engine) through a hole in the center of Parabola A (but far enough back from its focal point that the flow isn't impeded by a reverse pressure wave traveling back up the pipe)... and you're exhausting the gasses around the outer edges of Parabola B.

    Does the sound pressure just keep building up between Parabola A and Parabola B? Does it somehow slip out along with the exhaust gasses past the outer edges of Parabola B? What happens?

    This is a thought exercise in trying to come up with the smallest, lightest, quietest muffler possible for a small single cylinder engine. My thoughts are to divorce the gas flow from the sound pulses somehow, make the sound pulses bounce around until they convert to heat, and extract the gas flow sans the sound.
    Last edited: Aug 19, 2014
  2. jcsd
  3. Aug 19, 2014 #2


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    A parabola focusses parallel waves to a point called the focus.

    If two parabolas face each other, with you at the focus of one and a noise source at the focus of the other, then you will hear the noise source very well.

    Energy cannot be built up between two parabolas, or a parabola and a flat wall. That is because the focus does not reflect the energy that passes it. Using a tube does not help.

    A sphere with a noise source at the centre, will build up energy at wavelengths that are a multiple of the radius.
  4. Aug 19, 2014 #3
    Wait, I thought once the sound was "concentrated" in the focal point of the parabola, it passed through that focal point (continuing on in whatever direction it was going to begin with after being bounced off the parabola toward that focal point) to the diametrically opposed side of the parabola and was re-reflected back out in a straight line off the face of the parabola. Am I mistaken?
  5. Aug 19, 2014 #4


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    Well, you didn't originally say you had a surface to reflect the sound from at the focal point back to the parabola.
  6. Aug 19, 2014 #5


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    BTW, the geometric shape you are looking for is the ellipse. And yes, theoretically sound would bounce from one foci to the other forever, while in reality they will dissipate as heat.

    With parabolas, rays crossing the focus outbound won't all go back to the other focus. Turning the shape into an ellipse fixes that mostly by rounding off the area where the join.
  7. Aug 19, 2014 #6
    Thanks, Russ. Is there an equation to use to determine the proper shape of the ellipse such that all the sound waves are kept within the "chamber" of the two ellipses or do all ellipses regardless of shape do pretty much the same thing? The outer diameter would be an input, since I already know the maximum outer diameter that'll fit on the engine.

    I'm assuming that your suggestion of turning it into an ellipse means the gas would be extracted around the perimeter of the shape, exactly in the center between the two curves (would they still be called parabolas individually?), yes? This should be where the sound is least intense?
  8. Aug 19, 2014 #7


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    Unless a single cylinder engine runs at only one speed, any resonator will increase the sound output at some RPM. The problem is that shapes such as the three corner reflector, the ellipse and the parabola have a constant path lengths.

    To quieten the exhaust you need many varied path lengths.

    That can be achieved with a directional coupler, maybe by counterflow in two parallel conical tubes connected by many small holes.
    Another way is to have many conical baffles in a tube, each with a small hole. The exhaust flows through the hole while the acoustic energy is confused by multiple reflections within the chambers between the baffles. By making the baffles with curved surfaces the baffled chambers will have diverse path lengths and the baffles will be more rigid.
  9. Aug 21, 2014 #8
    If the ellipse were designed such that its resonant frequency fell outside the range of that created by the engine under all its operating speeds, would that work? Like I said, I'm trying to come up with a shape that'll contain all the sound within itself and dissipate it as heat, while allowing me to exhaust the gas sans the sound.

    So for this engine, the pulses would be from about 13.33 Hz to 76.66 Hz. If the resonant frequency of the ellipse were, say, 80 Hz, that'd prevent resonance and the resultant sound output increases, yes?
  10. Aug 21, 2014 #9


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    No! The exhaust valve opening produces sharp pulses, not fundamental sinewaves alone. 80 Hz is the second harmonic of 40 Hz, the 3rd of 26.66 Hz, 4th of 20 Hz etc. Sound also comes from the sides of the exhaust components.

    To get significant quietening you need a low pass filter. That needs to be made without significant resonances. Each pulse needs to be spread out in time to average out the peak pulse flow to a continuous quiet gas flow.

    If you want to sneak up on the enemy, you need a very good exhaust silencer;
    Attached is an advert from 1947, for one developed during WW2.

    Attached Files:

  11. Aug 21, 2014 #10


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    They are still in business, if you want to buy one :smile: http://www.laycockprecision.co.uk/
  12. Aug 22, 2014 #11
  13. Aug 23, 2014 #12
    Looks like someone has already done extensive research on this topic, a Dr. Akhilesh Mimani and staff:

    Their design gave ~22 dB of wideband attenuation without any other means of attenuation used (no absorptive material, etc., just the elliptical shape itself), and very low backpressure.

    So I think I'm on the right track here... separate the sound pulses from the gas flow, let the sound pulses convert their energy to heat, extract the gas flow sans the sound pulses... adding in sound absorbing material before and after the elliptical shape should make it attenuate even more.

    Gotta find an open source acoustical modeling program.
  14. Aug 23, 2014 #13


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    That paper is modelling a cylinder of elliptic section.
    It is a multi-path system with multiple outputs, it is certainly not a focussed resonant system.

    22 dB? Notice that at 1500 Hz there is no attenuation at all.
    You should be able to get a minimum of 40 dB broadband attenuation with a properly designed muffler.
    Last edited: Aug 23, 2014
  15. Aug 24, 2014 #14
    It's still an elliptical shape, which is a start. Once I find modeling software, I'll be able to tweak the design to achieve the stated design goals. I think ADEM might fill the bill, but I've got to contact the author to find out how much it costs. Can't find any that are open source that'd work.

    40 dB attenuation would be equivalent to a Super Critical Grade muffler... has muffler technology advanced that much in recent years? It'd be great if I could achieve that level of sound attenuation on this small engine... that'd make the engine exhaust at full bore as quiet as it is now when idling.
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