Resonance and conservation of energy

[Alexander83]

Hello,
We're studying sound waves in Physics and my instructor did a quick demo showing resonance with a tuning fork. He showed that when a tuning fork is struck and then connected to a resonating cavity, the sound is substantially amplified. I understand that resonance arises due to sound waves bouncing back and forth in the cavity and amplifying themselves, but I'm having a hard time understanding the greater intensity of sound waves that my ear perceives in terms of energy conservation.

When the tuning fork is not connected to the resonant cavity, the sound is relatively tinny. When it is connected to the cavity, the sound is much greater. Doesn't the latter scenario seem to imply that the tuning fork is somehow giving off energy at a greater rate to produce the greater intensity of sound when connected to the resonant cavity?
Ultimately, whether the tuning fork is connected to a cavity or not, the energy that it gives off as sound waves must come from the deformation of the tuning fork (which is initially elastic potential energy coming from striking and deforming the tuning fork... right?) from the initial strike. It seems to me that the entire object (tuning fork + cavity) is giving off much more energy when struck in this case, but I can't see how resonance would account for this. What am I missing here?

Thanks,

Alexander.

 

[Bobbywhy]

Alexander83, Welcome to Physics Forums!

When a tuning fork is vibrating in air it radiates sound waves in all directions. This is called “spherical spreading”. The energy radiated is dissipated omnidirectionally.

The tuning fork does not increase its acoustic energy output when in contact with the cavity. The resonant cavity stores each incoming wave in phase and therefore it stores the vibrational energy. This is heard as an increases the sound amplitude.

“In physics, resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system's resonant frequencies, or resonance frequencies. At these frequencies, even small periodic driving forces can produce large amplitude oscillations, because the system stores vibrational energy.
http://en.wikipedia.org/wiki/Resonance

Cheers,
Bobbywhy

 

[Alexander83]

Bobbywhy,
Thanks for the response. A couple of quick follow up questions if you have time.

i) The cavity can't trap all of the incoming energy as some of the energy must still propagate outwards from it for it to be perceptible to our ears. The way I understand it, when the waves bounce back and forth in the cavity, a portion of each wave reflects from the open end of the cavity (this is the portion that increases the amplitude of the waves in the cavity?) and a certain portion is transmitted (which is what our ears perceive?). Is this correct?

ii) You state that the standalone tuning fork radiates energy omnidirectionally. Does the cavity end up concentrating some of this energy into particular dissipation directions? Is this part of the reason for the perceived greater intensity... that some of the energy that would be directed omnidirectionally is preferentially re-directed in particular directions?

Thanks again for your reply!

Alexander

 

[AlephZero]

If you hit the tuning fork in the same way with the same force, you always put the same amount of energy into the fork.

A tuning fork vibratiing in free air is very inefficient at transferring the energy from the fork into the air, partly because the size of the fork is small compared with the wavelength of the sound in air, and also because the two prongs vibrate in opposite directions so the vibrations in the air caused by each prong almost cancel out. If you hold the fork close to your ear, it takes a long time to transfer all the energy into the air and for the sound to stop.

When you put the fork at the end of a tube that can resonate at the frequency of the fork, the energy transfer into the air in the tube is much more efficient, so you get a louder sound (i.e. more sound energy per second) but for a shorter time.

 

[Bobbywhy]

When a tuning fork is vibrating and its handle is placed on a resonator box, sound waves are sent into the volume of air inside the box. The waves reflect off the closed end of the box and travel back to the opening. The reflected waves will combine with new waves coming from the tuning fork. Resonance occurs if the reflected waves and the new waves are in step with each other (constructive interference) to create a standing wave. The amplitudes of the reflected and new waves combine and a louder sound with a definite frequency is heard.

Here are three excerpts from Wikipedia:

“One characteristic of the tuning fork shape is that, when it vibrates in its principal mode, the handle vibrates up and down as the prongs move apart and together. There is a node (point of no vibration) at the base of each prong. The handle motion is small, allowing the fork to be held by the handle without damping the vibration, but it allows the handle to transmit the vibration to a resonator (like the hollow rectangular box often used), which amplifies the sound of the fork.

Without the resonator (which may be as simple as a table top to which the handle is pressed), the sound is very faint. The reason for this is that the sound waves produced by each fork prong are 180° out of phase with the other, so at a distance from the fork they interfere and largely cancel each other out. (which AlephZero has already pointed out above)

If a sound absorbing sheet is slid in between the prongs of a vibrating fork, reducing the waves reaching the ear from one prong, the volume heard will actually increase, due to a reduction of this cancellation.”
http://en.wikipedia.org/wiki/Tuning_fork

Cheers,
Bobbywhy

 

[Low-Q]

A tuning fork is very inefficient when it comes to transfering soundwaves through air. The area of the fork is not big enough, so the fork generate a pressure wave that is acoustically short cuircuit as the pressure waves wants to travel towards the sub pressure wave on the other side of it. The wavelength of 440Hz is about 0.7m, the fork is maybe 0.5cm wide, so the short distance cancel out the sound easily.
However, when the fork is attached to a big wooden board, sheet of paper, wall etc. the energy that is stored in the fork is transferred directly to the surface of the board. The board will then start to pick up the viberations, and its area will in greater extent transfer the soundwaves you hear. The system has become more efficient when it comes to sound transportation through the air. That said, the bord surface must face your ears for you to hear the sound as loud as possible.

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