Question

PLEASE SHOW ALL STEPS and equations used.

Two identical strings are fixed on both ends. The first string is vibrating in its fundamental mode and it is observed that the other string begins to vibrate at its third harmonic, driven by the first string. What is the ratio of the tension of the second string to that of the first string.

This was answered else where but I don't understand the answers. If possible please draw a picture.

Answer 1

By "identical", this tells us that the strings have the same linear density, and the same length. Therefore, it would only be tension driving them to vibrate differently.

The third harmonic, has three times the frequency, as the fundamental, for any given case of tension/linear density/length. This means the fundamental frequency of the second string, is 1/3 of its third harmonic's frequency.

Because the first string is driving the vibration of the other string, this means that the other string's driven harmonic is at the same frequency as the first strings fundamental.

Thus, fundamental frequencies of each string relate as such:
f2 = f1/3

For any given string, the speed relates to string length and fundamental frequency as:
v = 2*L*f

because only half of a fundamental wavelength, vibrates in the length of the string.

Construct for version 1 and 2:
v1 = 2*L*f1
v2 = 2*L*f2

And formulas for speed of the string wave:
v = sqrt(T/mu)

sqrt(T1/mu) = 2*L*f1
sqrt(T2/mu) = 2*L*f2
sqrt(T2/mu) = 2*L*(f1/3)

Solve for f1, in terms of T1:
sqrt(T1/mu) = 2*L*f1
f1 = sqrt(T1/mu)/(2*L)

Plug in:
sqrt(T2/mu) = 2*L*(f1/3)
sqrt(T2/mu) = 2*L*((sqrt(T1/mu)/(2*L))/3)

Cancel the common terms:
sqrt(T2) = sqrt(T1)/3

Square both sides, and here is your answer:
T2 = T1/sqrt(3)

The ratio is 1/sqrt(3) to 1. Also said as 0.577 to 1. The second string only tightened to 57.7% of the tension of the first string.