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Unformatted text preview: Standing Waves A standing wave is a wave that appears to remain in a constant position. It usually results in a stationary medium (such as air or on a string) as a result of interference between two waves traveling in opposite directions. Such a situation can be carefully arranged based on the length of the medium, the speed within the medium and the frequency. e wave must propagate with just such as combination so that the areas of destructive interference and constructive interference always occur at the same locations. Boundary Conditions A key in creating a standing wave is that the wave must reflect from the end of the medium so that the reflected wave interferes with the next wave to create the standing wave pattern. ere are two ways that a wave may reflect from the end of the medium depending on the boundary conditions . In one case, the boundary is closed . In the case of a string, the end of the string would be fixed and not allowed to move or in the case of an air column, the end of a tube would be blocked off. Under such conditions, the wave reflects from the boundary upside down as illustrated: e second kind of boundary condition is the open case. For a string, we could imagine that the end of the string has a loop and is placed over a post. e loop is free to move up and down on the post. In such a case, the wave is reflected from the end right side up. For an air column, we could imagine a tube with an open end. e open boundary condition is illustrated below: Summer 2011 1 Lectures The Physics of Light & Sound W EEK 3 L ECTURES GSCI 121 S UMMER 2011 Standing Waves on a String We will consider the case where a string is fixed at both ends. Only certain frequencies will produce standing wave patterns which requires that the waves traveling along the string always constructively and destructively interfere in the same locations along the string despite the fact that the wave pulses are constantly moving. Consider the minimum wave pattern which is that the midpoint of the wave be at the endpoints of the string with a single wave crest in the center of the string. To produce this pattern, each subsequent wave crest has to be in step with the one already on the string. us the crests are timed to be spaced exactly equal to the round-trip time of a single wave on the string. A whole number wave fits into one round trip or a half of a wave is stretched across the string. us the wavelength is 2L. e frequency can then be determined from the relation f = c. is is the so-called fundamental frequency or first harmonic . e situation is illustrated below: We must always start and finish at a point where the sine wave is zero. e next point in the cycle of a sine wave that returns to zero is one full cycle. is would give us the second harmonic : And for the third harmonic, we would have to advance one more half cycle to end on the next time the since function yielded a value of zero: A pattern has now emerged that allows us to find a general expression for the frequency of the...
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- Spring '08