Question

Given a wave of amplitude $A$, is travelling on a string. At some point, the linear mass density of string is increased by four times. Choose the correct option

• A. The amplitude of the transmitted wave is $\frac{2A}{3}$.
• B. Amplitude of reflected wave is $\frac{A}{3}$.
• C. Amplitude of reflected wave is $\frac{2A}{3}$.
• D. The velocity of wave is not same in both the strings.

Answer 1

Answer: (A), (B), (D)

The velocity of wave is not same in both the strings.

When a wave travels in a string with different density, it gets partly reflected and partly transmitted.

The formula for amplitude of a reflected wave while the wave is travelling from one medium to the other is given by,

$A_r=\frac{\sqrt{{\mu }_r}-\sqrt{{\mu }_i}}{\sqrt{{\mu }_i}+\sqrt{{\mu }_r}}\times A$

${\mu }_i:$ Linear mass density of $1^{st}$ string
${\mu }_r:$ Linear mass density of second string
$A_r:$ Amplitude of reflected wave.

Let ${\mu }_i=\mu$,

Given that, ${\mu }_r=4\mu$

$\Rightarrow A_r=\frac{\sqrt{4\mu }-\sqrt{\mu }}{\sqrt{4\mu }+\sqrt{\mu }}\times A$

$\Rightarrow A_r=\frac{A}{3}$

The amplitude of the transmitted wave is given by,

$A_t=\frac{2\sqrt{{\mu }_i}}{\sqrt{{\mu }_i}+\sqrt{{\mu }_t}}\times A$

$\Rightarrow A_t=\frac{2\sqrt{\mu }}{\sqrt{\mu }+2\sqrt{\mu }}A=\frac{2A}{3}$

The velocity of the wave in a string is given by,

$V_w=\sqrt{\frac{T}{\mu }}$

As, the tension in both the string is same, and $\mu$ is more for second string, the velocity of the wave are unequal in both the strings.

Hence, $A,B\text{and}D$ are correct.