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 their displacement equations, we can follow these steps:

### Step 1: Identify the amplitude of the first wave
The first wave is given by the equation:
\[ y_1 = 10 \sin(3\pi t + \frac{\pi}{3}) \]
From this equation, we can directly see that the amplitude \( A_1 \) of the first wave is:
\[ A_1 = 10 \]

### Step 2: Simplify the second wave equation
The second wave is given by:
\[ y_2 = 5 \left[ \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \right] \]
We can factor out the 5:
\[ y_2 = 5 \left[ \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \right] = 5 \left[ \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \right] \]

### Step 3: Convert the sine and cosine terms into a single sine function
We can express the combination of sine and cosine as a single sine function using the sine addition formula:
\[ R \sin(3\pi t + \phi) \]
where \( R = \sqrt{A^2 + B^2} \) and \( A = 1 \) (coefficient of sine), \( B = \sqrt{3} \) (coefficient of cosine).

Calculating \( R \):
\[ R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 4: Find the amplitude of the second wave
Thus, the amplitude \( A_2 \) of the second wave is:
\[ A_2 = 5 \cdot R = 5 \cdot 2 = 10 \]

### Step 5: Calculate the ratio of the amplitudes
Now that we have both amplitudes, we can find the ratio of their amplitudes:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{10}{10} = 1 \]

### Final Answer
The ratio of the amplitudes of the two waves is:
\[ \text{Ratio} = 1 \]

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