The velocities of sound in air and water are 336 m/sec and 1470 m/sec respectively. Two sources producing waves of wavelength 6 m and 7 m sounded together will produce beats

Options

A. than can be heard in both the media
B. that can be heard only in water and not in air
C. than can be heard only in air nd not in water
D. that cannot be heard in either of the media

Answer

To solve the problem of determining the beats produced by two sources of sound with different wavelengths in air and water, we will follow these steps:

### Step 1: Understand the relationship between speed, frequency, and wavelength
The speed of sound (v) in a medium is related to its frequency (f) and wavelength (λ) by the equation:
\[ v = f \cdot \lambda \]
From this, we can express frequency as:
\[ f = \frac{v}{\lambda} \]

### Step 2: Calculate the frequency for the first wavelength in water
Given:
- Speed of sound in water (v) = 1470 m/s
- Wavelength (λ1) = 6 m

Using the formula:
\[ f_1 = \frac{v}{\lambda_1} = \frac{1470 \, \text{m/s}}{6 \, \text{m}} \]
Calculating this gives:
\[ f_1 = 245 \, \text{Hz} \]

### Step 3: Calculate the frequency for the second wavelength in water
Given:
- Wavelength (λ2) = 7 m

Using the formula:
\[ f_2 = \frac{v}{\lambda_2} = \frac{1470 \, \text{m/s}}{7 \, \text{m}} \]
Calculating this gives:
\[ f_2 = 210 \, \text{Hz} \]

### Step 4: Calculate the beat frequency in water
The beat frequency (f_beat) is the difference between the two frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| = |245 \, \text{Hz} - 210 \, \text{Hz}| \]
Calculating this gives:
\[ f_{\text{beat}} = 35 \, \text{Hz} \]

### Step 5: Calculate the frequency for the first wavelength in air
Given:
- Speed of sound in air (v) = 336 m/s
- Wavelength (λ1) = 6 m

Using the formula:
\[ f_1 = \frac{v}{\lambda_1} = \frac{336 \, \text{m/s}}{6 \, \text{m}} \]
Calculating this gives:
\[ f_1 = 56 \, \text{Hz} \]

### Step 6: Calculate the frequency for the second wavelength in air
Given:
- Wavelength (λ2) = 7 m

Using the formula:
\[ f_2 = \frac{v}{\lambda_2} = \frac{336 \, \text{m/s}}{7 \, \text{m}} \]
Calculating this gives:
\[ f_2 = 48 \, \text{Hz} \]

### Step 7: Calculate the beat frequency in air
The beat frequency (f_beat) is the difference between the two frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| = |56 \, \text{Hz} - 48 \, \text{Hz}| \]
Calculating this gives:
\[ f_{\text{beat}} = 8 \, \text{Hz} \]

### Conclusion
In both media (water and air), beats are produced. The beat frequencies are:
- In water: 35 Hz
- In air: 8 Hz

### Final Answer
Beats can be heard in both media.

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