Ncert Exercises & Solutions

Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, the frequencies are in the ratio 1:2:3:4.

Hint: Use the derivation of standing waves.

Step 1: Find the frequency of n-harmonic.

Let, there are n numbers of loops in the string.

Length corresponding each loop is

\frac{λ}{2}

Now, we can write,

L = \frac{nλ}{2} \Rightarrow λ = \frac{2 L}{n} [for n loops]

\Rightarrow \frac{v}{ν} = \frac{2 L}{n} \Rightarrow [∵ v = νλ]

\Rightarrow ν = \frac{n}{2 L} v = \frac{n}{2 L} \sqrt{\frac{T}{μ}} [∵ velocity of transverse wave = \sqrt{T / μ}]

\Rightarrow ν \propto n [∵ length and speed are constants]

Step 2 : Find the ratio of frequencies .

So, ν_{1} : ν_{2} : ν_{3} : ν_{4} = n_{1} : n_{2} : n_{3} : n_{4}

= 1 : 2 : 3 : 4