Answer:

When a sonometer vibrates, the length of the wire is doubled. The sonometer will resonate at 2L if the tuning fork resonates at L. The frequency of the sonometer is expressed using the equation below.

\[f=\frac{n}{2L}\sqrt{\frac{T}{\mu }}=\frac{nv}{2L}\]

The velocity of the sonometer’s wave is constant. Even if the length of the wire is changed, the resonance between the tuning fork and the wire will remain.

Then,

n/L = constant

Which implies that,

n₁/L₁ = n₂/L₂

n₁/L₁ = n₂/2L₂

n₂ = 2n₁

As a result, when the wire length is doubled, the resonance doubles as well.