Problem

Problem

Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of 440.0 Hz. One of the string is retuned by adjusting its tension. When this is done, 1.5 beats per second are heard when both strings are plucked simultaneously. (a) What are the possible fundamental frequencies of the retuned string? (b) By what fractional amount was the string tension changed if it was (i) increased (ii) decreased?
Intuitive explanation

The key here is understanding that beat frequency arises from two slightly different frequencies interfering with each other. When two sound waves with frequencies f1f_1 and f2f_2 overlap, they create a beat pattern whose frequency is simply ∣f1−f2∣|f_1 - f_2|. Since you know the beat frequency and one string's original frequency, you can find the retuned string's frequency—then use the relationship between tension and frequency (f∝Tf \propto \sqrt{T}) to determine how much the tension changed.

Intuitive explanation

The key here is understanding that beat frequency arises from two slightly different frequencies interfering with each other. When two sound waves with frequencies f1f_1 and f2f_2 overlap, they create a beat pattern whose frequency is simply ∣f1−f2∣|f_1 - f_2|. Since you know the beat frequency and one string's original frequency, you can find the retuned string's frequency—then use the relationship between tension and frequency (f∝Tf \propto \sqrt{T}) to determine how much the tension changed.

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