Two waves of wavelengths 99 cm and 100 cm produce 4 beats per second. Velocity of sound in the medium is

Options

A. 100 m/s
B. 90 m/s
C. 196 m/s
D. 396 m/s

Answer

To solve the problem, we need to find the velocity of sound in the medium given two waves of different wavelengths that produce beats. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Wavelength of the first wave, \( \lambda_1 = 99 \, \text{cm} = 0.99 \, \text{m} \)
- Wavelength of the second wave, \( \lambda_2 = 100 \, \text{cm} = 1.00 \, \text{m} \)
- Beat frequency, \( f_b = 4 \, \text{Hz} \)

### Step 2: Write the formula for frequency
The frequency \( f \) of a wave is given by the formula:
\[ f = \frac{V}{\lambda} \]
where \( V \) is the velocity of sound in the medium, and \( \lambda \) is the wavelength.

### Step 3: Calculate the frequencies of the two waves
- Frequency of the first wave:
\[ f_1 = \frac{V}{\lambda_1} = \frac{V}{0.99} \]
- Frequency of the second wave:
\[ f_2 = \frac{V}{\lambda_2} = \frac{V}{1.00} \]

### Step 4: Write the expression for beat frequency
The beat frequency \( f_b \) is given by the absolute difference of the two frequencies:
\[ f_b = |f_1 - f_2| \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ 4 = \left| \frac{V}{0.99} - \frac{V}{1.00} \right| \]

### Step 5: Simplify the equation
Taking \( V \) common:
\[ 4 = V \left( \frac{1}{0.99} - \frac{1}{1.00} \right) \]
Calculating the difference:
\[ \frac{1}{0.99} - \frac{1}{1.00} = \frac{1.00 - 0.99}{0.99 \times 1.00} = \frac{0.01}{0.99} \]
Thus, we have:
\[ 4 = V \left( \frac{0.01}{0.99} \right) \]

### Step 6: Solve for \( V \)
Rearranging the equation to find \( V \):
\[ V = 4 \times \frac{0.99}{0.01} \]
Calculating:
\[ V = 4 \times 99 = 396 \, \text{m/s} \]

### Final Answer
The velocity of sound in the medium is \( V = 396 \, \text{m/s} \).

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