Ultrasonic waves of frequency 3 × 10 5Hz are passed through a medium where speed of sound is 10 times that in air (Speed of sound is 300 m/s). The wavelength of this wave in the medium will be of the order of
1 cm
Understanding Ultrasonic Wave Properties in a Medium
This question asks us to determine the wavelength of ultrasonic waves when they travel through a specific medium. We are given the frequency of the waves and the speed of sound in both air and the given medium. To solve this, we will use the fundamental relationship between wave speed, frequency, and wavelength.
Given Information
- Frequency of ultrasonic waves, $f = 3 \times 10^5$ Hz
- Speed of sound in air = 300 m/s
- Speed of sound in the medium is 10 times that in air
Calculating the Speed of Sound in the Medium
The speed of sound in the medium is given as 10 times the speed of sound in air.
Speed of sound in medium, $v_{\text{medium}} = 10 \times \text{Speed of sound in air}$
$v_{\text{medium}} = 10 \times 300 \text{ m/s}$
$v_{\text{medium}} = 3000 \text{ m/s}$
Relationship Between Speed, Frequency, and Wavelength
The speed ($v$) of a wave is related to its frequency ($f$) and wavelength ($\lambda$) by the formula:
\(v = f\lambda\)
To find the wavelength ($\lambda$), we can rearrange the formula:
\(\lambda = \frac{v}{f}\)
Calculating the Wavelength in the Medium
Using the calculated speed in the medium and the given frequency:
\(\lambda_{\text{medium}} = \frac{v_{\text{medium}}}{f}\)
\(\lambda_{\text{medium}} = \frac{3000 \text{ m/s}}{3 \times 10^5 \text{ Hz}}\)
\(\lambda_{\text{medium}} = \frac{3000}{300000} \text{ m}\)
\(\lambda_{\text{medium}} = \frac{3}{300} \text{ m}\)
\(\lambda_{\text{medium}} = \frac{1}{100} \text{ m}\)
\(\lambda_{\text{medium}} = 0.01 \text{ m}\)
Converting Wavelength to Centimeters
The options are given in centimeters. We need to convert the wavelength from meters to centimeters. We know that 1 meter = 100 centimeters.
\(\lambda_{\text{medium}} \text{ (in cm)} = \lambda_{\text{medium}} \text{ (in m)} \times 100\)
\(\lambda_{\text{medium}} \text{ (in cm)} = 0.01 \text{ m} \times 100\)
\(\lambda_{\text{medium}} \text{ (in cm)} = 1 \text{ cm}\)
Determining the Order of Magnitude
The calculated wavelength is exactly 1 cm. This value is of the order of 1 cm.
Comparing with Options
Let's compare our calculated wavelength with the given options:
- 1 cm
- 10 cm
- 100 cm
- 0.1 cm
Our calculated wavelength of 1 cm matches the first option.
Conclusion
The wavelength of the ultrasonic waves in the given medium is 1 cm.
Revision Table: Wave Properties
| Property | Symbol | Units | Description |
|---|---|---|---|
| Speed of Wave | $v$ | m/s | How fast the wave disturbance travels through the medium. |
| Frequency | $f$ | Hz (or s-1) | Number of complete oscillations or cycles per second. |
| Wavelength | $\lambda$ | m | The spatial period of the wave; the distance over which the wave's shape repeats. |
Additional Information: Ultrasonic Waves and Speed of Sound
Ultrasonic Waves: These are sound waves with frequencies higher than the upper audible limit of human hearing, typically above 20 kHz ($20 \times 10^3$ Hz). The frequency in the question ($3 \times 10^5$ Hz) is well within the ultrasonic range.
Speed of Sound in Different Media: The speed of sound depends on the properties of the medium through which it travels, such as density, temperature, and stiffness. Generally, sound travels faster in solids and liquids than in gases. The speed of sound in a given medium is typically constant for a specific temperature and pressure, regardless of the frequency or wavelength of the sound wave.
Wave Equation: The relationship $v = f\lambda$ is a fundamental equation that applies to all types of waves, including sound waves, light waves, water waves, etc.
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