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 step by step, we will follow the concepts of wave frequency, wavelength, and beat frequency.

### Step 1: Identify the given values
We are given:
- Wavelength of the first wave, \( \lambda_1 = 99 \, \text{cm} \)
- Wavelength of the second wave, \( \lambda_2 = 100 \, \text{cm} \)
- Beat frequency, \( f_{\text{beat}} = 4 \, \text{Hz} \)

### Step 2: Use the relationship between frequency and wavelength
The frequency \( f \) of a wave is related to its speed \( v \) and wavelength \( \lambda \) by the formula:
\[ f = \frac{v}{\lambda} \]
Thus, we can express the frequencies of the two waves as:
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{99 \, \text{cm}} \]
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{100 \, \text{cm}} \]

### Step 3: Set up the equation for beat frequency
The beat frequency is given by the absolute difference between the two frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| = \left| \frac{v}{99} - \frac{v}{100} \right| \]
Since \( \lambda_1 < \lambda_2 \), we can simplify this to:
\[ f_{\text{beat}} = \frac{v}{99} - \frac{v}{100} \]

### Step 4: Simplify the equation
To combine the fractions, we find a common denominator:
\[ f_{\text{beat}} = \frac{100v - 99v}{99 \times 100} = \frac{v}{9900} \]
Setting this equal to the given beat frequency:
\[ \frac{v}{9900} = 4 \]

### Step 5: Solve for the velocity \( v \)
Now, we can solve for \( v \):
\[ v = 4 \times 9900 = 39600 \, \text{cm/s} \]

### Step 6: Convert to meters per second
To convert from centimeters per second to meters per second, we divide by 100:
\[ v = \frac{39600}{100} = 396 \, \text{m/s} \]

### Final Answer
The velocity of sound in the medium is:
\[ \boxed{396 \, \text{m/s}} \]

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