11b: Surface Resonances

If you fix a string at both ends in free space with no resonating body attached to it the string does not make very much sound. Connecting the string to a surface that vibrates allows the energy of the string to move to the surface and cause it to vibrate. A large surface can move much more air, resulting in a much louder sound.

You might think this violates conservation of energy; a vibrating string in air doesn’t produce much sound but a string attached to a surface does. But conservation of energy is a fundamental law of physics that can’t be broken. So what is happening? If you time how long a free string vibrates and compare it to how long it will vibrate if attached to a surface you find something interesting. When attached to a surface the string’s vibrations die away much more rapidly. In other words, because the energy is being used to create lots of sound, it dissipates much faster. A free string can vibrate longer because it doesn’t dissipate its energy making sound.

Vibrating strings have resonances with different numbers of nodal points, places on the string which do not vibrate. These depend on the driving frequency. A similar thing happens with surfaces; there are resonance frequencies which result in places where not much vibration occurs. These locations are linked together in nodal lines which depend on the shape and thickness of the surface. One way to see these lines is to drive the surface with an oscillator and put powder or salt on the surface. The powder will not move from the nodal line but will be thrown off of the anti-nodal regions, as shown in the following example.

Video/audio examples:

Chladni plate. Notice that as the frequency is increased different nodal lines occur, just like different nodes occurred on the string at different frequencies. If you listen carefully to this video you can hear that there is more sound when the plate reaches a resonance. This is because the amplitude is larger at the resonance frequency (as expected). You will also notice that the overtones are not harmonic.


Flat plates of various shapes called Bell plates, tuned to specific frequencies, have long been used as inexpensive substitutes for bells.

We know that stringed instruments have harmonic frequencies which are multiples of the fundamental. This is the case because the string is a fixed length; the longest wave that can exist on the string has a wavelength that is half the length of the string. The next wavelength that can fit is the exact length of the string; the next wavelength that will fit is 1.5 times the length of the string and so on as we saw previously in the chapter on stringed instruments. Wavelengths in between these would not have a node at both ends and so can’t exist on the string. Each of these different ways of vibrating is labeled by the number n; the fundamental is labeled n = 1, the second harmonic is labelled n = 2, etc. This number, called the mode number indicates how many anti-nodes are on the string.

Surfaces of various shapes (round, rectangular, square) however, are two dimensional and so will require two mode numbers, n and m, to label each mode of vibration. For a rectangular surface fixed at the edges we can label the two dimensions as x and y and there are sine wave shapes in the x-direction and in the y-direction with nodes at the edges, just like a string, as shown in the following simulation.

Simulation exercise 11B (turn in answers on a separate sheet of paper): Square Membrane.

The following are a few pictures of nodal lines by Thomas Erndl from a web site with a discussion and collection of nodal surfaces for guitars and violins. Notice that they are similar to the case of a square plate (in the video above) but because of the shape of the guitar and irregular thickness of the surface, the patterns are not exactly symmetric.

nodalsurfaces

As was the case for a string, surfaces have many different resonance frequencies. For the string these other resonances are all multiples of the fundamental. However if you look carefully at the Chladni patterns of the plate and the guitar and violin you will notice that these resonance frequencies are not all harmonics (multiples) of the fundamental. These higher frequencies are called overtones to distinguish them from harmonics. Strings and tubes (discussed in the next chapter) have overtones that are harmonic while surfaces typically do not.

The Chladni plate resonance frequencies tend to be sharp or high Q-factor resonances, as are the circular and rectangular modes in the various simulations linked from this page. Because the bodies of most musical instruments do not have symmetric shapes and are not of uniform thickness (due to supporting structures inside), the resonances of musical stringed instruments are not sharp. As mentioned in Chapter 4 on resonances, the Q-factor is a measure of how wide the resonance curve is. In general, stringed instruments have surfaces with low Q-factor, in other words, broad resonance curves. The trade-off is that, as previously mentioned, a low Q-factor also means more oscillations before being damped out. This means the sound will last longer before dissipating.

There is a second way to see the vibrational modes of a surface, called holographic interferometry. In the chapter on wave behavior we found that if two waves arrive at a point and are out-of-phase they undergo destructive interference and cancel out. If they are in-phase they interfere constructively. Suppose we reflect laser light off of the vibrating surface of a guitar into a camera. Now shine a second, reference laser beam directly into the camera. If the reflected beam comes off a surface that has moved half a wavelength (at the instance the camera shutter opens) the two beams will be out of phase and not expose the film. If the surface has moved a whole wavelength (or two or three, etc.) the reflected beam will be in-phase with the reference beam and expose the film. The result is a contour map of the deflections of the surface from equilibrium.

In the following picture (from the BBC report In Pictures: Stringed Theory) the concentric lines show regions where the guitar surface is vibrating in and out. The center of each region of concentric circles is where the movement is a maximum. A dark center indicates movement that is out of phase (inward) from a center that is light colored (outward). It is also possible to make successive pictures into a video of the surface as it changes over time.

guitarholagram

Because different parts of the surface vibrate for each frequency range, the direction of sound emission is affected. For example, a violin in the frequency range 200 Hz to 500 Hz emits sound pretty much equally in all directions. But in the frequency range 550 Hz to 700 Hz more sound is emitted to the left and right of the performer than straight ahead or behind. In the 800 Hz range the sound is emitted forward and left and right but not behind. For frequencies between 1000 Hz and 1250 Hz more sound is emitted at an angle of about 70 degrees forward and to the right of the performer.

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