Two mechanical waves, `y_(1) = 2 sin 2 pi (5- t - 2x)` & `y_(2) = 4 sin 2 pi(fax + 100 t)` propagate in a medium with same speed.

Options

A. The ratio of their intensities is `1 : 16`
B. The ratio of their intensities is `1 : 4`
C. The value of 'a' is `4` units
D. The value of 'a' is `2` units.

Explanation

To solve the problem, we need to analyze the two wave equations provided and find their properties, particularly the wave speed and the intensity ratio.

### Step-by-Step Solution:

1. **Identify the wave equations:**
The given wave equations are:
\[ y_1 = 2 \sin(2\pi(5 - t - 2x)) \]
\[ y_2 = 4 \sin(2\pi(fax + 100t)) \]

2. **Rewrite the equations in standard form:**
The standard form of a wave equation is:
\[ y = A \sin(\omega t + kx) \quad \text{or} \quad y = A \sin(\omega t - kx) \]
For \(y_1\):
\[ y_1 = 2 \sin(2\pi(5 - t - 2x)) = 2 \sin(2\pi(5) - 2\pi t - 4\pi x) \]
This can be simplified to:
\[ y_1 = 2 \sin(-2\pi t - 4\pi x + 10\pi) \]
Here, \(\omega_1 = 2\pi\) and \(k_1 = 4\pi\).

For \(y_2\):
\[ y_2 = 4 \sin(2\pi(fax + 100t)) = 4 \sin(2\pi(100t + fax)) \]
Here, \(\omega_2 = 200\pi\) and \(k_2 = 2\pi f\).

3. **Calculate the wave speed for both waves:**
The wave speed \(v\) is given by:
\[ v = \frac{\omega}{k} \]
For \(y_1\):
\[ v_1 = \frac{2\pi}{4\pi} = \frac{1}{2} \text{ units} \]

For \(y_2\):
\[ v_2 = \frac{200\pi}{2\pi f} = \frac{100}{f} \text{ units} \]

4. **Set the wave speeds equal to each other:**
Since both waves propagate at the same speed:
\[ \frac{1}{2} = \frac{100}{f} \]
Solving for \(f\):
\[ f = 200 \text{ Hz} \]

5. **Calculate the intensity ratio:**
The intensity \(I\) of a wave is proportional to the square of the amplitude and the square of the angular frequency:
\[ I \propto A^2 \omega^2 \]
For \(y_1\):
\[ I_1 \propto (2^2)(2\pi)^2 = 4 \times 4\pi^2 = 16\pi^2 \]

For \(y_2\):
\[ I_2 \propto (4^2)(200\pi)^2 = 16 \times 40000\pi^2 = 640000\pi^2 \]

6. **Find the ratio of intensities:**
\[ \frac{I_1}{I_2} = \frac{16\pi^2}{640000\pi^2} = \frac{16}{640000} = \frac{1}{40000} \]

### Final Results:
- The frequency \(f\) is \(200 \text{ Hz}\).
- The intensity ratio is \(1:40000\).