A bat emits ultrasonic sound of frequency 100 kHz in air. If this sound meets a water surface, the wavelengths of the reflected and transmitted sound are (Speed of sound in air = 340 m `s^(-1)` and in water = 1500 m `s^(-1)` )

Options

A. 3.4 mm, 30 mm
B. 6.8 mm, 15 mm
C. 3.4 mm, 15 mm
D. 6.8 mm, 30 mm

Explanation

To solve the problem, we need to find the wavelengths of the reflected and transmitted ultrasonic sound when a bat emits a sound of frequency 100 kHz in air, and this sound meets a water surface. We will use the formulas for wavelength, which is given by:

\[ \lambda = \frac{v}{f} \]

where \( \lambda \) is the wavelength, \( v \) is the speed of sound in the medium, and \( f \) is the frequency of the sound.

### Step 1: Convert the frequency from kilohertz to hertz
The frequency of the ultrasonic sound emitted by the bat is given as 100 kHz. We need to convert this to hertz:

\[ f = 100 \, \text{kHz} = 100 \times 10^3 \, \text{Hz} = 10^5 \, \text{Hz} \]

### Step 2: Calculate the wavelength of the reflected sound in air
The speed of sound in air is given as 340 m/s. We can now calculate the wavelength of the reflected sound using the formula:

\[ \lambda_a = \frac{v_a}{f} \]

Substituting the values:

\[ \lambda_a = \frac{340 \, \text{m/s}}{10^5 \, \text{Hz}} = \frac{340}{100000} = 3.4 \times 10^{-3} \, \text{m} = 3.4 \, \text{mm} \]

### Step 3: Calculate the wavelength of the transmitted sound in water
The speed of sound in water is given as 1500 m/s. We can calculate the wavelength of the transmitted sound using the same formula:

\[ \lambda_w = \frac{v_w}{f} \]

Substituting the values:

\[ \lambda_w = \frac{1500 \, \text{m/s}}{10^5 \, \text{Hz}} = \frac{1500}{100000} = 15 \times 10^{-3} \, \text{m} = 15 \, \text{mm} \]

### Final Results
- Wavelength of the reflected sound in air: \( \lambda_a = 3.4 \, \text{mm} \)
- Wavelength of the transmitted sound in water: \( \lambda_w = 15 \, \text{mm} \)

### Conclusion
Thus, the wavelengths of the reflected and transmitted sounds are \( 3.4 \, \text{mm} \) and \( 15 \, \text{mm} \) respectively.

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