The equations of two sound waves are given by,` y_1 = 3sin ( 100 pi t ) and y_2 =4sin ( 150 pi t ) `, The ratio of intensites of sound produced in the medium is

Options

A. ` 1: 4`
B. ` 2: 3 `
C. ` 3: 4`
D. ` 9: 16`

Answer

To find the ratio of the intensities of the two sound waves given by the equations \( y_1 = 3 \sin(100 \pi t) \) and \( y_2 = 4 \sin(150 \pi t) \), we can follow these steps:

### Step 1: Identify the Amplitudes
The amplitude of a wave is the coefficient in front of the sine function in the wave equation.

- For the first wave \( y_1 = 3 \sin(100 \pi t) \), the amplitude \( A_1 = 3 \).
- For the second wave \( y_2 = 4 \sin(150 \pi t) \), the amplitude \( A_2 = 4 \).

### Step 2: Use the Relationship Between Intensity and Amplitude
The intensity \( I \) of a sound wave is directly proportional to the square of its amplitude. This can be expressed mathematically as:

\[ I \propto A^2 \]

### Step 3: Write the Intensities in Terms of Amplitudes
Let \( I_1 \) be the intensity of the first wave and \( I_2 \) be the intensity of the second wave. Then we have:

\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]

### Step 4: Find the Ratio of Intensities
The ratio of the intensities can be expressed as:

\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]

Substituting the values of the amplitudes:

\[ \frac{I_1}{I_2} = \frac{3^2}{4^2} = \frac{9}{16} \]

### Step 5: Conclusion
Thus, the ratio of the intensities of the two sound waves is:

\[ \frac{I_1}{I_2} = \frac{9}{16} \]

### Final Answer
The ratio of intensities of sound produced in the medium is \( 9:16 \).

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