Standing waves on a 1.0- m-long string that is fixed at both...

Standing waves on a $1.0-$ m-long string that is fixed at both ends are seen at successive frequencies of $24 \mathrm{Hz}$ and $36 \mathrm{Hz}$
a. What are the fundamental frequency and the wave speed?
b. Draw the standing-wave pattern when the string oscillates at
$36 \mathrm{Hz}$

Key Concepts

Wave Speed

Wave speed refers to the speed at which waves travel along a medium. On a stretched string, the wave speed is influenced by the tension in the string and its linear density, and it is related to the frequency and wavelength by the equation v = f?.

Standing Waves

Standing waves are formed when two waves of identical frequency and amplitude travel in opposite directions and interfere, resulting in a pattern that appears to be stationary. In strings fixed at both ends, the interference between incident and reflected waves produces nodes (points of no motion) and antinodes (points of maximum motion).

Harmonics

Harmonics are the series of standing wave patterns that occur at discrete frequencies, each corresponding to an integer multiple of the fundamental mode. In a string fixed at both ends, the harmonics are determined by the condition that the string’s length must be an integer multiple of half-wavelengths.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a standing wave can be sustained, corresponding to the first harmonic. It is determined by the speed of the wave on the string and the string's length, with the relation that the wavelength of the fundamental mode is twice the length of the string.

Nodes and Antinodes

In the context of standing waves on a string, nodes are fixed points where the medium does not move, and antinodes are points where the displacement reaches a maximum. The locations of nodes and antinodes are determined by the harmonic mode of vibration, and drawing the pattern helps visualize how the string is partitioned into segments vibrating in phase.

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