String-1 is connected with string-2 The mass per unit length in string-1 is ` mu _1` and mass per unit length in string-2 is `4mu _1.` The tension in the strings is T .A travelling wave is coming from the left .What fraction of the energy in the incident wave goes into string-2?

Options

A. `(1)/(8)`
B. `(4)/(9)`
C. ` (2)/(3)`
D. ` (8)/(9)`

Answer

To solve the problem of finding the fraction of energy in the incident wave that goes into string-2, we will follow these steps:

### Step 1: Understand the Energy of the Wave
The energy of a wave traveling in a string can be expressed as:
\[ E = \frac{1}{2} \mu \omega^2 A^2 v \]
where:
- \( \mu \) = mass per unit length of the string,
- \( \omega \) = angular frequency of the wave,
- \( A \) = amplitude of the wave,
- \( v \) = velocity of the wave in the string.

### Step 2: Calculate the Energy in String-1
For string-1, with mass per unit length \( \mu_1 \):
\[ E_1 = \frac{1}{2} \mu_1 \omega^2 A^2 v_1 \]

### Step 3: Determine the Velocity in String-1
The velocity of the wave in string-1 is given by:
\[ v_1 = \sqrt{\frac{T}{\mu_1}} \]
where \( T \) is the tension in the string.

### Step 4: Calculate the Energy in String-2
For string-2, with mass per unit length \( \mu_2 = 4\mu_1 \):
\[ E_2 = \frac{1}{2} \mu_2 \omega^2 A_t^2 v_2 \]
where \( A_t \) is the transmitted amplitude and \( v_2 \) is the velocity in string-2.

### Step 5: Calculate the Velocity in String-2
The velocity of the wave in string-2 is:
\[ v_2 = \sqrt{\frac{T}{\mu_2}} = \sqrt{\frac{T}{4\mu_1}} = \frac{1}{2} \sqrt{\frac{T}{\mu_1}} = \frac{v_1}{2} \]

### Step 6: Calculate the Transmitted Amplitude
The transmitted amplitude \( A_t \) can be calculated using the formula:
\[ A_t = A \cdot \frac{2 \sqrt{\mu_1}}{\sqrt{\mu_1} + \sqrt{4\mu_1}} = A \cdot \frac{2 \sqrt{\mu_1}}{\sqrt{\mu_1} + 2\sqrt{\mu_1}} = A \cdot \frac{2}{3} \]

### Step 7: Substitute Values into Energy Equation for String-2
Now substitute \( A_t \) and \( v_2 \) into the energy equation for string-2:
\[ E_2 = \frac{1}{2} (4\mu_1) \omega^2 \left(\frac{2A}{3}\right)^2 \left(\frac{v_1}{2}\right) \]
\[ E_2 = \frac{1}{2} (4\mu_1) \omega^2 \left(\frac{4A^2}{9}\right) \left(\frac{v_1}{2}\right) \]
\[ E_2 = \frac{4\mu_1 \omega^2 A^2 v_1}{9} \]

### Step 8: Calculate the Fraction of Energy Transmitted
Now we need to find the fraction of energy transmitted:
\[ \text{Fraction} = \frac{E_2}{E_1} \]
Substituting \( E_1 \) and \( E_2 \):
\[ \text{Fraction} = \frac{\frac{4\mu_1 \omega^2 A^2 v_1}{9}}{\frac{1}{2} \mu_1 \omega^2 A^2 v_1} \]
\[ \text{Fraction} = \frac{4/9}{1/2} = \frac{4}{9} \cdot 2 = \frac{8}{9} \]

### Final Answer
The fraction of the energy in the incident wave that goes into string-2 is:
\[ \frac{8}{9} \]