Question

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?

Answer 1

Answer:

(a). $\dfrac{f_3'}{f_3} =\sqrt{5}.$

(b). The wavelength remains unchanged.

Explanation:

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

$v = \sqrt{\dfrac{T_iL}{m} }$

And if the string is vibrating, its fundamental wavelength is $2L$, and since the frequency $f$ is related to the wave speed and wavelength by

$f = v/\lambda$

the fundamental frequency $f_1$ is

$f_1 =\sqrt{\dfrac{T_iL}{m} }*\dfrac{1}{2L}$

and since the frequency of the third harmonic is

$f_3 = 3f_1$

$f_3 = 3\sqrt{\dfrac{T_iL}{m} }*\dfrac{1}{2L},$

and the wavelength is

$\lambda_3 = \dfrac{2L}{3}.$

(a).

Now, if we increase to the string tension to

$T_f = 5.0T_i$

the third harmonic frequency becomes

$f_3' = 3\sqrt{\dfrac{5T_iL}{m} }*\dfrac{1}{2L},$

The ratio of this new frequency to the old frequency is

$\dfrac{f_3'}{f_3} = \dfrac{3\sqrt{\dfrac{5T_iL}{m} }*\dfrac{1}{2L}}{3\sqrt{\dfrac{T_iL}{m} }*\dfrac{1}{2L}}$

$\boxed{\dfrac{f_3'}{f_3} =\sqrt{5}.}$

(b).

The wavelength of the third harmonic remains unchanged because $\lambda_3 = \dfrac{2L}{3}.$ depends only on the length of the string