A string under tension τi oscillates in the third harmonic at frequency f3, and the waves on the string have wavelength λ3. If the tension is increased to τf = 5.0τi and the string is again made to oscillate in the third harmonic, what then are (a) the ratio of frequency of oscillation to f3 and (b) the ratio of the wavelength of the waves to λ3?

(a).

(b). The wavelength remains unchanged.

Explanation:

The speed of the waves on the string with tension is given by

And if the string is vibrating, its fundamental wavelength is , and since the frequency is related to the wave speed and wavelength by

the fundamental frequency is

and since the frequency of the third harmonic is

and the wavelength is

(a).

Now, if we increase to the string tension to

the third harmonic frequency becomes

The ratio of this new frequency to the old frequency is

(b).

The wavelength of the third harmonic remains unchanged because depends only on the length of the string

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