A uniform rope of length ' \(L\) ' and mass ' \(\mathrm{m}_1\) ' hangs vertically from a rigid support. A block of mass ' \(\mathrm{m}_2\) ' is attached to the free end of the rope. A transverse wave of wavelength ' \(\lambda_1\) ' is produced at the lower end of the rope. The wavelength of the wave when it reaches the top of the rope is ' \(\lambda_2\) '. The ratio \(\frac{\lambda_1}{\lambda_2}\) is

Let velocity of pulse at lower end be \(v_1\) and at top be \(\mathrm{v}_2\)
\(\therefore \quad \frac{\lambda_2}{\lambda_1}=\frac{\mathrm{v}_2}{\mathrm{v}_1} \quad\left(\because \lambda=\frac{\mathrm{v}}{\mathrm{n}} \text { and } \mathrm{n}=\text { constant }\right)\)
Velocity of transverse wave on a string is
\(\mathrm{v}=\sqrt{\frac{\mathrm{T}}{\mathrm{m}}}\)
where, \(m\) is linear density.
In this case, \(\mathrm{v} \propto \sqrt{\mathrm{T}}\)
\(\therefore \quad \frac{\lambda_2}{\lambda_1}=\frac{\mathrm{v}_2}{\mathrm{v}_1}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}}=\sqrt{\frac{\left(\mathrm{m}_2+\mathrm{m}_1\right)}{\mathrm{m}_2}}\)
Where, \(T_2\) is tension at upper end of rope and \(T_1\) is tension at lower end of rope.
\(\Rightarrow \frac{\lambda_1}{\lambda_2}=\sqrt{\frac{m_2}{m_2+m_1}}\)