# How does sound resonance work?

1. Sep 11, 2006

### Chaos' lil bro Order

Here is an interesting application of sound resonance. I was sitting in my computer room when I noticed that if I sat forward in my chair and settled in a particular spot, the high pitched 'buzz' coming from my computer sounded extra loud. I concluded that I was listening to a resonance created by this high pitched 'buzz' being reflected by my room's walls. The extra loud sound (or frequency resonance) that I heard was only particularly noticeable in my left ear, so we will ignore my right ear for now. The area of resonance had a planar area of about 5x5cm(paralell to the ground) and I didn't test for z-axis (height) variations. In other words, if my left ear was anywhere within this 5x5cm resonance zone, I heard the extra loud 'buzz'.

Let me setup of the parameters of my computer's spacing with respect to the dimensions of my room now:

Room: a square room with 4 walls, each wall is 4 meters long

Computer location in the Room: in a corner of the room, located 0.5 meters from one wall and 1 meter from the other wall. The computer's height from the ground is equal to the height of my ear's from the ground.

So I guess my question would be can you calculate the frequency of my computer's 'buzz' given what I told you about the resonance zone I found and the dimensions of my room?

I'm guessing the answer has to do with the Wavelength = speed of sound/ frequency formula. And perhaps also to do with how sound behaves in a pipe with two closed ends (like my room's walls close off my room). If memory serves me correctly and I don't think it does here, the formula is something like f = 2L (that doesn't look right does it).

PS. Physics 101 says that if waves constructively interefere with one another so that their respective peaks overlap, a resonance is formed where the waves overlap to form a larger amplitude. But I must admit that I always thought of resonance as a sharp peak that is either ON or OFF, but not somewhere in the middle. If I want to get picky, the resonance zone that was 5x5cm had an even smaller region inside of it 1x1cm which was where the absolute peak sound amplitude was heard. This smaller region was uniform in sound amplitude and I could not distinguish any minute movement within it (aka. it all sounded the same).

2. Sep 11, 2006

### franznietzsche

First thing: Your hearing is logarithmic with power, meaning that what you hear as uniform (linear) increases in volume, are exponential increases in power.

Second: The sum amplitude of the reflected sound waves should be sinusoidal in 3 dimensions. Something like sin(ax+by+cz) (not exactly that, but something of that form). So near the peak, a sin function is nearly flat, so you wouldn't notice the difference with your logarithmic hearing once you got within a certain distance of the peak. So that is expected.
resonance occurs when you have two waves given by

$$sin(n x +\phi)$$ and $$sin(nx + \theta)$$ where phi and theta are some multiple of $$2\pi$$ apart from each other ($$\phi - \theta$$ = 2 n \pi[/tex]). If you change them slightly then the overall volume goes down, until they get to $$\phi-\theta = (2n+1)\pi$$ which is when you will hear nothing. to get the frequency of the sound you need to use something like

$$sin( k x' + \omega t)$$ as the equation of the wave, where x' is a coordinate that is different for each path from the computer, off the walls to your ears. The frequency is $$\frac{\omega}{2 \pi}$$, k is constant for all the paths because

$$\omega = k c_s$$ where $$c_s$$ is the sound speed. So you find k based on the geometry of the room (its not quite as simple as you suggested, because there are an infinite number of paths the sound could take) such that you get a maximum volume at the point you described. Then with the sound speed you have the frequency.

3. Sep 11, 2006

### rcgldr

Resonance is due to an object being sensitive to a range or ranges of frequences that cause the object to resonant when these frequencies are presented on the object.

Interference, is when sound waves combine, and their sum produces a waveform with larger peaks.

Focus is what happens when sound waves are reflected to converge near a specific area.

From what you're describing, it seems that the sound waves are reflecting off the walls and roof, and you're getting a combination of interference and focusing effects.

There is a movie theater with a spherically domed ceiling over an eating area in the lobby. The eating area is usually filled, and it's pretty noisy. I noticed that I could clearly hear the conversation of a couple on the opposite side of the eating area from where I stood under the domed ceiling. I then had my wife stand on one side under the dome, and then went to the other side under the dome and freaked her out when I could talk just above a whisper and she could clearly hear it, in spite of the very noisy crowd between us. This was a case of the sound waves being focused.

4. Sep 11, 2006

### Chaos' lil bro Order

Franz
'Second: The sum amplitude of the reflected sound waves should be sinusoidal in 3 dimensions. Something like sin(ax+by+cz) (not exactly that, but something of that form). So near the peak, a sin function is nearly flat, so you wouldn't notice the difference with your logarithmic hearing once you got within a certain distance of the peak. So that is expected.'

Excellent answer, this perfectly explains why I perceived the sound amplitude within that 1x1cm area to be uniform.

Jeff
Your food court example was interesting too. I know that orhestral rooms also use curved ceilings to make sure that the sounds you hear reflecting off of various walls all combine to reach your ears at the same time. Thus preserving the 'crisp' sound you hear, rather than a fuzzy muffled sound.

PS As a side question, can sound waves be focussed by, say, a glass magnifying glass? Or is this medium only appropriate for the transmission and focussing of light?

5. Sep 11, 2006

### franznietzsche

Sound waves are pressure waves through air. if you want to focus sound waves (lensing) you would have to mess with the properties of the air (changing sound speeds, essentially, because that is how light is bent by a lens, due to the differing speed of light in the two media) in a very particular manner. It would be easier to do in a material than in air (where you might be able to arbitrarily alter the density with changing composition).

Caveat: I am semi-speculating. Not entirely sure that is correct, but its what comes to mind off the top of my head.

6. Sep 11, 2006

### Chaos' lil bro Order

'Sounds' good to me ;)

I would only say that sound is a pressure wave found not only in air, but in any medium that can be compressed and stretched (eg. water, oil, metal, etc.). With this in mind, I'm wondering if a person's eye contact lens with its shape and flexibility could be milled to amplify a sound wave.

7. Sep 11, 2006

### franznietzsche

Amplify no, focus maybe. Depends entirely on the materials and the sounds speeds in them. Actually, since sound will be faster in most materials I believe you would want what would be a concave lens, rather than the convex shape usually use in optics.

Are you thinking hearing aids?

I don't think it would work, unless you can find a material that is transparent to sound in the human hearing range (just as contact lense are transparent to light in the human visual range).

8. Sep 11, 2006

### Chaos' lil bro Order

Yes, focus.

Well I'm not sure you need a material transparent to sound since the material will vibrate when the sound waves hit it whether or not the sound waves are transmitted or reflected.

9. Sep 11, 2006

### Gokul43201

Staff Emeritus
Amplify, no. Focus, also no. Not with glass; not with a contact lens.

Heard of acoustic mismatch? You'll end up reflecting more sound than you transmit.

10. Sep 12, 2006

### rcgldr

To focus the sound, you need a shape for the sound waves to bounce off on and converge to a specific location. You can do a web search for "whispering gallery sound dome" for other instances of domes that focus sound from one side to the other.

11. Sep 12, 2006

### Chaos' lil bro Order

Nope, haven't heard of it, but can't the small portion that does get transmitted be focussed by lensing?