(1) A violin string vibrates at 441 Hz when unfingered. At what frequency will it vibrate if it

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(1) A violin string vibrates at 441 Hz when unfingered. At...

(1) A violin string vibrates at 441 $\mathrm{Hz}$ when unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.)

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Key Concepts

Standing Waves

Standing waves are patterns of vibration that occur when waves traveling in opposite directions superpose and create fixed nodes and antinodes along a medium. In a string, these patterns form due to reflections at the endpoints where the displacement is forced to be zero, leading to discrete resonant frequencies.

Fundamental Frequency

The fundamental frequency is the lowest resonant frequency at which a system naturally vibrates. For a string, it is determined by the wave speed and the effective vibrating length, and serves as the basis for the harmonic series, with higher harmonics being integer multiples of this fundamental frequency.

Frequency-Length Relationship

The frequency at which a string vibrates is inversely related to the length of the vibrating portion of the string. This means that if the effective length of the string is reduced, the wavelength of the fundamental mode decreases, causing the frequency to increase accordingly.

Effective Length

Effective length refers to the segment of the string that is free to vibrate between its fixed ends. Changing the effective length, such as by fingering the string at a particular point, directly influences the possible standing wave patterns and hence alters the resonant frequency.

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