Four standing wave segments, or loops, are observed on a string fixed at both ends as it vibrates at a frequency of 240 Hz. What is the fundamental frequency of the string?

Given f1 = 60 Hz

Let L be the length of the string vibrating in four loops. so each will represent half wavelength and four loops will be 2 wavelength long.

λn = 2 L / n where n=1,2,3,4 ...

For n=4

λ4 = L/2

To fundamental frequency it should vibrate in one loop so

For n=1

λ1 = 2 L / 1

we know that f1 * λ1 = f2 * λ2 = f3 * λ3 = f4 * λ4

⇒f1 * λ1 = f4 * λ4

f1 = (f4 * λ4) / λ1

f1 = (240 Hz * L/2) / 2L

f1 = 60 Hz

Fundamental frequency = 60 Hz

Explanation:

The fundamental frequency can be found by

fo = (f_4*λ_4) / λ_o

Where

λ_o = 2L/1 = 2L

λ_4 = 2L/4 = L/2

fo = (240*L/2) / 2L

L cancels out and equation becomes

fo = (240) / 4

fo = 60

Therefore, the fundamental frequency of the string is 60 Hz.