Standing Waves

For certain frequencies the interference produces a

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Unformatted text preview: encies, the interference produces a standing wave pattern (or oscillation mode) with nodes and large antinodes like those shown in Figure 3. Such a standing wave is said to be produced at resonance, and the string is said to resonate at these certain frequencies, called resonant frequencies. If the string is oscillated at some frequency other than a resonant frequency, a standing wave is not set up. Then the interference of the right-going and leftgoing traveling waves results in only small (perhaps imperceptible) oscillations of the string. Let a string be stretched between two clamps separated by a fixed distance L. To find expressions for the resonant frequencies of the sting, we note that a node must exist at each of its ends, because each end is fixed an cannot oscillate. The simplest pattern that meets this key requirement is the first wave shown in Figure 3, which shows the string at both its extreme displacements. There is only one antinode, Figure 3 A string, stretched between two which is at the center of the string. Note that half a clamps, is made to oscillate in a standing wave wavelength spans the length L, which we take to be the pattern. string’s length. Thus for this pattern L=λ/2. This condition 20waves-String.html tells us that if the left-going and right-going traveling waves are to set up this pattern by their interference, they must have the wavelength λ=2L. A second simple pattern meeting the requirement of nodes at the fixed ends is shown as the second wave in Figure 3. This pattern has three nodes and two antinodes and is said to be a two-loop pattern. For the left-going and right-going waves to set it up, they must have wavelength L=λ. A third pattern is the third wave in Figure 3. It has four nodes, three antinodes, and three loops. Its wavelength is λ= (2/3) L. We would continue this progression by drawing increasingly more complicated patterns. In each step of the progression, the pattern would have one more node and one more antinode than the preceding step. Thus, a standing wave can be set up on a string of length L by a wave with a wavelength equal to one of these values. 2 for 0, 1, 2, 3, … (22) In other words, a standing wave can exist on the string ONLY if its wavelength is one of the values given by equation (22). The nth possible wavelength of equation (22) is just the right size so that its nth node is located at the end of the string (x=L). NOTE: Other wavelengths, which would be perfectly acceptable wavelengths for a traveling wave, cannot exist as a standing wave of length L because they cannot meet the boundary conditions (x=L) requiring a node at each end of the string. If stand...
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This document was uploaded on 03/20/2014 for the course PHYS 215 at Lafayette.

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