Waves: Step-by-Step Worked Examples (6)

Question

A string of length $L = 1$ m and linear mass density $\mu = 0.01$ kg/m is fixed at both ends and stretched with tension $T = 100$ N. Find the frequencies of the first three harmonics. Then find the position of the node closest to the midpoint in the third harmonic.

Solution — Step by Step

Why This Works

A string fixed at both ends supports only standing waves where both ends are nodes. That forces the length $L$ to be an integer number of half-wavelengths: $L = n\lambda/2$. Each value of $n$ is a harmonic.

Wave speed depends only on the medium ($T$ and $\mu$), not the frequency. So once we know $v$, every harmonic frequency is just $v$ divided by the corresponding wavelength.

Alternative Method

Use the direct formula for a string fixed at both ends:

$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = \frac{n}{2(1)}\sqrt{\frac{100}{0.01}} = 50n \text{ Hz}$

Plug $n = 1, 2, 3$ to get $50, 100, 150$ Hz directly.

Final answer: $f_1 = 50$ Hz, $f_2 = 100$ Hz, $f_3 = 150$ Hz. Closest node to midpoint in 3rd harmonic is at $x = L/3$.