The equation of displacement of two waves are given as
` y_(1) = 10 sin( 3 pi t + (pi)/(3)) , y_(2) = 5 [ sin 3 pi t + sqrt(3) cos 3 pi t]`
Then what is the ratio of their amplitudes

Options

A. ` 1: 2`
B. `2 : 1`
C. `1 :1`
D. None of these

Explanation

To find the ratio of the amplitudes of the two waves given by the equations:

1. **Identify the amplitudes from the equations:**
- The first wave is given by \( y_1 = 10 \sin(3\pi t + \frac{\pi}{3}) \).
- The amplitude \( A_1 \) of the first wave is \( 10 \).
- The second wave is given by \( y_2 = 5 \left[ \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \right] \).
- We need to find the amplitude \( A_2 \) of the second wave.

2. **Rewrite the second wave's equation:**
- The term \( \sqrt{3} \cos(3\pi t) \) can be rewritten using the sine function:
\[ \cos(3\pi t) = \sin\left(3\pi t + \frac{\pi}{2}\right) \]
- Therefore, we can express \( y_2 \) as:
\[ y_2 = 5 \sin(3\pi t) + 5\sqrt{3} \cos(3\pi t) = 5 \sin(3\pi t) + 5\sqrt{3} \sin\left(3\pi t + \frac{\pi}{2}\right) \]

3. **Use the concept of vector addition to find the resultant amplitude:**
- The two components \( 5 \sin(3\pi t) \) and \( 5\sqrt{3} \cos(3\pi t) \) can be treated as vectors:
- \( A_x = 5 \) (along the x-axis)
- \( A_y = 5\sqrt{3} \) (along the y-axis)
- The resultant amplitude \( A_R \) can be calculated using the Pythagorean theorem:
\[ A_R = \sqrt{A_x^2 + A_y^2} = \sqrt{5^2 + (5\sqrt{3})^2} \]
\[ A_R = \sqrt{25 + 75} = \sqrt{100} = 10 \]

4. **Calculate the ratio of the amplitudes:**
- Now that we have both amplitudes:
- \( A_1 = 10 \)
- \( A_2 = 10 \)
- The ratio of the amplitudes is:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{10}{10} = 1 \]

5. **Final answer:**
- The ratio of the amplitudes of the two waves is \( 1:1 \).