A string is tied down at both ends. Some of the standing...

A string is tied down at both ends. Some of the standing waves on this string have the following frequencies: $100 \mathrm{Hz}$ $200 \mathrm{Hz}, 250 \mathrm{Hz},$ and $300 \mathrm{Hz} .$ It is also known that there are no standing waves with frequencies between $250 \mathrm{Hz}$ and $300 \mathrm{Hz}$. (a) What is the fundamental frequency of this string? (b) What is the frequency of the third harmonic?

A string is tied down at both ends. Some of the standing waves on this string have the following frequencies: $100 \mathrm{Hz}$ $200 \mathrm{Hz}, 250 \mathrm{Hz},$ and $300 \mathrm{Hz} .$ It is also known that there are no standing waves with frequencies between $250 \mathrm{Hz}$ and $300 \mathrm{Hz}$.
(a) What is the fundamental frequency of this string?
(b) What is the frequency of the third harmonic?

Key Concepts

Standing Waves on a String

A standing wave on a string is created when waves reflected from the fixed ends interfere with the incoming waves, producing nodes (points of zero amplitude) and antinodes (points of maximum amplitude) that remain stationary in time. This phenomenon occurs only at specific frequencies that satisfy the boundary conditions, leading to a discrete set of possible vibration modes.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a standing wave can be sustained on a string fixed at both ends. It corresponds to the simplest mode of vibration, having only one half-wavelength along the length of the string. All higher vibration frequencies (harmonics) are integer multiples of this fundamental frequency.

Harmonics

Harmonics are the higher modes of vibration on a string, each corresponding to an integer multiple of the fundamental frequency. Each harmonic represents a standing wave pattern with a specific number of nodes and antinodes. The spacing between successive harmonics is constant, ensuring that no additional standing wave frequencies exist between one harmonic and the next in an ideal system.

Harmonic Series and Frequency Gaps

The harmonic series for a string fixed at both ends is a set of frequencies that are integer multiples of the fundamental frequency. The fact that there are no standing waves between two successive harmonic frequencies (as in the gap between 250 Hz and 300 Hz) confirms that the frequencies must follow a strict integer multiple relationship. This discrete set of frequencies is a direct consequence of the boundary conditions imposed on the string.

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