As we know that, when a wire of length $L$, vibrates its resonant frequency in $n$th mode after stretching its by a tension $T$, if the tuning fork resonates at $L$ it will resonate at $2 L$
So, the sonometer frequency is
$$
v=\frac{n}{2 L} \sqrt{\frac{T}{m}} \quad \quad(n=\text { number of loops })
$$
So, as it vibrates with length $L$, the frequency $\left(v_1\right)$ is :
$$
v_1=\frac{n_1}{2 L} \sqrt{\frac{T_1}{m_1}}
$$
If length is doubled, then frequency $\left(v_2\right)$ is
$$
v_2=\frac{n_2}{2 \times 2 L} \sqrt{\frac{T_2}{m_2}}
$$
Dividing Eq. (i) by Eq. (ii), we get
$$
\frac{v_1}{v_2}=\frac{n_1}{n_2} \times 2 \quad\left(\because m_1=m_2=m ; T_1=T_2=T \text { given }\right)
$$
As tuning fork is same i.e., in both harmonics $n_1$ and $n_2$ frequency of resonance same.
So, $\left(v_1=v_2=v\right)$.
$$
\begin{aligned}
&\left(\frac{\mathrm{v}}{\mathrm{v}}\right)=\frac{n_1}{n_2} \times 2 \Rightarrow 1=\frac{n_1}{n_2} \times 2 \\
&\Rightarrow n_2=2 n_1
\end{aligned}
$$
So, when the length of wire is doubled the number of harmonics loops also get doubled to produce the same resonance frequency of tension. That is it resonates in second harmonic.