Question

In the figure shown below, mass per unit length of string$2$ is nine times that of string$1$. Tension in both the strings is same. A transverse wave is incident at the boundary as shown in the figure. A part of the wave is reflected in medium$1$ and a part is transmitted in medium$2$. Then, which of the following option(s) is/are correct?

• A. Ratio of amplitudes of reflected and transmitted waves, $\frac{A_1}{A_2}=1$
• B. Ratio of amplitudes of reflected and transmitted waves, $\frac{A_1}{A_2}=3$
• C. Ratio of velocities, $\frac{v_1}{v_2}=3$
• D. Ratio of velocities,$\frac{v_1}{v_2}=\frac{1}{3}$

Answer 1

Answer: (A), (C)

Ratio of velocities, $\frac{v_1}{v_2}=3$

Let mass per unit length of string$1$ be ${\mu }_1$ and mass per unit length of string $2$ be ${\mu }_2=9{\mu }_1$.

Wave speed on string$-1$ , $v_1=\sqrt{\frac{T}{{\mu }_1}}$

Wave speed on string$-2$, $v_2=\sqrt{\frac{T}{9{\mu }_1}}$
$\Rightarrow v_1=3v_2$

Amplitude of the reflected wave is
$A_1=|\frac{v_2-v_1}{v_1+v_2}|A_i....\left(1\right)$

Amplitude of the transmitted wave is.
$A_2=\left(\frac{2v_2}{v_1+v_2}\right)A_i.....\left(2\right)$

From $\left(1\right)$ and $2)$ we get,
$\frac{A_1}{A_2}=\left|\frac{\left(\frac{v_2-v_1}{v_1+v_2}\right)}{\left(\frac{2v_2}{v_1+v_2}\right)}\right|$

$\Rightarrow \frac{A_1}{A_2}=\left|\frac{v_2-v_1}{2v_2}\right|=\left|\frac{1-3}{2}\right|=1$

Thus, options (a) and (c) are the correct answers.