A string consists of two parts attached at x=0 , The right part of the string `(xlt 0)` has mass per unit length `mu_r` and the left part of the string ` (xlt 0)` has mass per unit length ` mu _l` The tension in the string is T. If a wave of units amplitude travels along the left part of the string,what is the amplitude of the wave that is transmitted to the right part of the string?

Options

A. ` 1`
B. ` (2)/( 1+sqrt (mu _l //mu _y))`
C. ` (2)/( 1+sqrt (mu _l //mu _y))`
D. ` (sqrt( mu_l //mu_y)-1)/( sqrt(mu_l //mu_y )+ 1)`

Answer

To solve the problem, we need to determine the amplitude of the wave transmitted to the right part of the string when a wave of unit amplitude travels along the left part of the string.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let \( \mu_l \) be the mass per unit length of the left part of the string.
- Let \( \mu_r \) be the mass per unit length of the right part of the string.
- Let \( T \) be the tension in the string.
- The amplitude of the incident wave (on the left part) is \( A_i = 1 \) (unit amplitude).

2. **Calculate the Wave Velocities:**
- The velocity of the wave in the left part of the string \( v_l \) is given by:
\[ v_l = \sqrt{\frac{T}{\mu_l}} \]
- The velocity of the wave in the right part of the string \( v_r \) is given by:
\[ v_r = \sqrt{\frac{T}{\mu_r}} \]

3. **Use the Amplitude Transmission Formula:**
- When a wave travels from one medium to another, the amplitude of the transmitted wave \( A_t \) can be calculated using the formula:
\[ A_t = \frac{2 v_r}{v_l + v_r} A_i \]
- Substituting \( A_i = 1 \):
\[ A_t = \frac{2 v_r}{v_l + v_r} \]

4. **Substitute the Values of \( v_l \) and \( v_r \):**
- Substitute \( v_l \) and \( v_r \) into the amplitude formula:
\[ A_t = \frac{2 \sqrt{\frac{T}{\mu_r}}}{\sqrt{\frac{T}{\mu_l}} + \sqrt{\frac{T}{\mu_r}}} \]

5. **Simplify the Expression:**
- Factor out \( \sqrt{T} \):
\[ A_t = \frac{2 \sqrt{\frac{T}{\mu_r}}}{\sqrt{T} \left( \frac{1}{\sqrt{\mu_l}} + \frac{1}{\sqrt{\mu_r}} \right)} \]
- This simplifies to:
\[ A_t = \frac{2 \sqrt{\mu_l}}{\sqrt{\mu_r} \left( \sqrt{\mu_l} + \sqrt{\mu_r} \right)} \]

### Final Result:
The amplitude of the wave transmitted to the right part of the string is:
\[ A_t = \frac{2 \sqrt{\mu_l}}{\sqrt{\mu_r} \left( \sqrt{\mu_l} + \sqrt{\mu_r} \right)} \]