A string with a mass density = 2.40×10 -3 kg / m is under a tension of F = 131 N

Question

A string with a mass density = 2.40×10-3kg/m is under a tension of F = 131 N and is fixed at both ends. One of its resonance frequencies is 646.0 Hz. The next higher resonance frequency is 861.3 Hz. What is the fundamental frequency of this string?
215.3 Hz

Which harmonic does the resonance frequency at 646.0 Hz correspond to? (i.e. what is n at this frequency?)
3

What is the length of the string?
5.42×10-1 m

Now, suppose the same string is detached at one end and connected by a ring to a frictionless post, so that it can move freely. Find the wavelength of the first (fundamental) harmonic.

What is the frequency of the third (n = 3) harmonic in this case?

Explanation / Answer

from one end of the open string

f = nv/4L

wavelength = 4L

for n = 1

4L = 1*V/f

4L = 343/(215.3)

4L = 1.6 m

for n =2 ,

wavelength 4L = 2*1.6 = 3.2 m

wavelength 4L = 3*1.6 = 4.8 m

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