Two sound waves are represented by `y_(1)=sinomegat+cosomegat` and `y_(2)=(sqrt(3))/(2)sinomegat+(1)/(2)cosomegat`. The ratio of their amplitude is

Options

A. `1 : 1`
B. `sqrt(3) : 2`
C. `2 : sqrt(3)`
D. `sqrt(2) : 1`

Explanation

To find the ratio of the amplitudes of the two sound waves represented by the equations \( y_1 = \sin(\omega t) + \cos(\omega t) \) and \( y_2 = \frac{\sqrt{3}}{2} \sin(\omega t) + \frac{1}{2} \cos(\omega t) \), we will convert each equation into the standard form of a wave equation, which is \( y = A \sin(\omega t + \phi) \), where \( A \) is the amplitude.

### Step-by-Step Solution:

1. **Convert \( y_1 \) to standard form**:
\[ y_1 = \sin(\omega t) + \cos(\omega t) \]
We can express this as:
\[ y_1 = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin(\omega t) + \frac{1}{\sqrt{2}} \cos(\omega t) \right) \]
Recognizing that \( \frac{1}{\sqrt{2}} = \sin(45^\circ) \) and \( \frac{1}{\sqrt{2}} = \cos(45^\circ) \), we can rewrite it as:
\[ y_1 = \sqrt{2} \left( \sin(\omega t) \cos(45^\circ) + \cos(\omega t) \sin(45^\circ) \right) \]
This simplifies to:
\[ y_1 = \sqrt{2} \sin(\omega t + 45^\circ) \]
Thus, the amplitude \( A_1 \) of wave 1 is:
\[ A_1 = \sqrt{2} \]

2. **Convert \( y_2 \) to standard form**:
\[ y_2 = \frac{\sqrt{3}}{2} \sin(\omega t) + \frac{1}{2} \cos(\omega t) \]
We can express this as:
\[ y_2 = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} \left( \frac{\frac{\sqrt{3}}{2}}{\sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2}} \sin(\omega t) + \frac{\frac{1}{2}}{\sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2}} \cos(\omega t) \right) \]
First, calculate the amplitude:
\[ A_2 = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]
Now, we can express \( y_2 \) as:
\[ y_2 = 1 \left( \sin(\omega t \cos(30^\circ) + \cos(\omega t) \sin(30^\circ) \right) \]
This simplifies to:
\[ y_2 = \sin(\omega t + 30^\circ) \]
Thus, the amplitude \( A_2 \) of wave 2 is:
\[ A_2 = 1 \]

3. **Calculate the ratio of the amplitudes**:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{\sqrt{2}}{1} = \sqrt{2} \]

### Final Answer:
The ratio of the amplitudes \( A_1 : A_2 = \sqrt{2} : 1 \).

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