A string tied down at both ends has a length of 0.64 m, and its first harmonic has a frequency of 41 Hz. Calculate the first-harmonic frequency for this string if its length is increased to 0.88 m.

25.4 Hz

Explanation:

The fundamental frequency (first harmonic) of a string tied down at both ends is given by

where

L is the length of the string

T is the tension in the string

m is the mass of the string

The equation can be rewritten as

We notice that the term on the right remains constant for this string (we assume the tension remains the same), so we can write

where in this case:

is the fundamental frequency at the beginning

is the initial length of the string

is the final length of the string

Solving for , we find the final fundamental frequency:

wavelength divides in half and speed stays the same.

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