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
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.
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This document was uploaded on 03/20/2014 for the course PHYS 215 at Lafayette.
- Fall '09