An experimenter finds that standing waves on a string fixed...

An experimenter finds that standing waves on a string fixed at both ends occur at $24 \mathrm{Hz}$ and $32 \mathrm{Hz},$ but at no frequencies in between.
a. What is the fundamental frequency?
b. Draw the standing-wave pattern for the string at 32 Hz.

Key Concepts

Nodes and Antinodes

Nodes are points along the string that remain stationary as the wave vibrates, while antinodes are points where the amplitude of vibration is at a maximum. In a string fixed at both ends, the endpoints are nodes due to the boundary conditions, and the number and location of nodes and antinodes determine the harmonic pattern of the standing wave. This concept is crucial for accurately sketching the standing-wave pattern for any given harmonic.

Harmonics and Fundamental Frequency

Harmonics are the integer multiples of the fundamental frequency at which a string fixed at both ends vibrates. The fundamental frequency is the lowest frequency at which the string resonates, and all other natural frequencies (overtones) are integral multiples of it. In this problem, by recognizing that the observed frequencies are harmonics of the fundamental, one can determine the fundamental frequency by finding the greatest common divisor of the given frequencies.

Mode Shapes of Standing Waves

Standing waves on a fixed string occur when waves reflect and interfere, leading to stationary patterns of nodes and antinodes. Each harmonic corresponds to a specific mode where the pattern contains a number of half-wavelength segments equal to its harmonic number. For example, the mode at a higher harmonic will show more nodes and antinodes than the fundamental, and understanding this pattern is key to visualizing and drawing the vibration mode at a specific frequency.

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