Propagation and interference of sound from waves on a string. (a) Explain why on
ID: 1770564 • Letter: P
Question
Propagation and interference of sound from waves on a string. (a) Explain why only certain frequency multiples are able to contribute significantly to the spectrum of standing waves on a string. (That is, explain why resonance occurs harmonically) the arriving frequency is the same as the naturaJ (b) Consider a string fixed at both ends vibrating in the fundamental transverse e mode (lowest frequeney resonance). Plot theoretical sound intensity (power per area) vs transverse distance from the center of the string. Pay particular attention to the functional dependence close to the string and far from the string (c) Explain what would be different in your plot for the previous question if the vibration energy is the same but the frequency is twice the fundamental frequency (d) Repeat for a frequency that is triple the fundamental frequeney, etc. Make a general statement regarding the influence of the harmonic order (multiple of fundamental frequency) on the behavior of sound intensity vs transverse distance from the string Wave energyExplanation / Answer
a)
If a medium is bounded such that its opposite ends can be considered fixed, nodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one antinode in the middle. This is half a wavelength. To make the next possible standing wave, place a node in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another node. This gives us one and a half wavelengths. It should become obvious that to continue all that is needed is to keep adding nodes, dividing the medium into fourths, then fifths, sixths, etc. There are an infinite number of harmonics for this system, but no matter how many times we divide the medium up, we always get a whole number of half wavelengths (12, 22, 32,…, n2).
There are important relations among the harmonics themselves in this sequence. The wavelengths of the harmonics are simple fractions of the fundamental wavelength. If the fundamental wavelength were 1 m the wavelength of the second harmonic would be 12 m, the third harmonic would be 13 m, the fourth 14 m, and so on. Since frequency is inversely proportional to wavelength, the frequencies are also related. The frequencies of the harmonics are whole-number multiples of the fundamental frequency. If the fundamental frequency were 1 Hz the frequency of the second harmonic would be 2 Hz, the third harmonic would be 3 Hz, the fourth 4 Hz, and so on.
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