A 2.0 -m-long string is fixed at both ends and tightened...

A 2.0 -m-long string is fixed at both ends and tightened until the wave speed is $40 \mathrm{m} / \mathrm{s}$. What is the frequency of the standing wave shown in FIGURE EX21.6?
(FIGURE CAN'T COPY)

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Key Concepts

Standing Waves

Standing waves are the result of two waves of the same frequency and amplitude, traveling in opposite directions, interfering to produce a pattern that appears stationary. In a system like a string fixed at both ends, the endpoints remain nodes (points of no movement), and the pattern of nodes and antinodes is determined by the wave's wavelength and the mode of vibration.

Harmonics

Harmonics refer to the specific vibrational modes of a string or other medium that support standing waves. Each mode, indexed by a positive integer n, has a different frequency given by the expression f? = n·v/(2L), where n = 1 corresponds to the fundamental frequency, and higher values of n represent overtones. The number of loops or antinodes increases with n.

Wave Speed

Wave speed is the speed at which a wave propagates through a medium. In problems involving standing waves on a string, the wave speed (v) is combined with the length (L) and the mode number (n) to determine the frequency of the standing wave pattern. It is a crucial parameter in the frequency formula for harmonics, f? = n·v/(2L).

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