Question

A composite string is made up by joining two strings of different mass per unit length $r$ and $4r$. The composite string is under the same tension. A transverse wave pulse is sent along the lighter string towards the joint. The ratio of wave number of incident wave to the transmitted wave is

• A. $2:1$
• B. $1:2$
• C. $3:2$
• D. $2:3$

Answer 1

The correct option is B $1:2$

We know that frequency $f$ will remain same for incident as well as transmitted wave as it depends on source.
Also, $f=\frac{v}{\lambda }$ where $v$ is wave velocity and $\lambda$ is wavelength.

So, $f=\frac{v_i}{{\lambda }_i}=\frac{v_t}{{\lambda }_t}$
$\Rightarrow \frac{{\lambda }_t}{{\lambda }_i}=\frac{v_t}{v_i}=\frac{\sqrt{\frac{T}{4r}}}{\sqrt{\frac{T}{r}}}$
$\Rightarrow \frac{{\lambda }_t}{{\lambda }_i}=\frac{1}{2}$ .........(1)
Now, we know wave number is given by,
$k=\frac{2\pi }{\lambda }$
So,
$\frac{k_i}{k_t}=\frac{{\lambda }_t}{{\lambda }_i}$
$\Rightarrow \frac{k_i}{k_t}=\frac{1}{2}=1:2$ [ from (1) ]