A guitar player can change the frequency of a string by...

A guitar player can change the frequency of a string by "bending" it-pushing it along a fret that is perpendicular to its length. This stretches the string. increasing its tension and its frequency. The G string on a guitar is $64 \mathrm{cm}$ long and has a tension of $74 \mathrm{N}$. The guitarist pushes this string down against a fret located at the center of the string, which gives it a frequency of 392 Hz. He then bends the string, pushing with a force of $4.0 \mathrm{N}$ so that it moves $8.0 \mathrm{mm}$ along the fret. What is the new frequency?

Key Concepts

Wave Propagation in Strings

This concept deals with the way waves travel along a stretched string, where the speed of the wave is determined by the tension in the string divided by its mass per unit length. The formula v = ?(T/?) connects these quantities, showing that any change in tension directly affects the wave speed.

Standing Waves and Harmonics

When a string is fixed at both ends, it can support standing waves that form due to the interference of waves reflecting off the boundaries. The frequencies at which these standing waves occur (harmonics) depend on the wave speed and the length of the string, with the fundamental frequency given by f = v/(2L).

Effect of Tension on Frequency

Because the frequency of a vibrating string is directly related to the wave speed (which in turn depends on the tension), an increase in tension leads to an increase in frequency. This is why applying additional force to a string, causing it to stretch, can raise its pitch.

Geometric Effects of String Bending

Bending a string involves displacing it from its straight position, which alters its geometry and increases its effective length. This deformation results in additional stretching and consequently an increase in tension. Understanding this geometric relationship is crucial to analyzing how manipulations like bending change the vibrational characteristics of the string.

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