What is the wavelength of a sound wave whose frequency is 820 Hz and speed is 420 m/s in a given medium?
0.51 m
Understanding Sound Wave Wavelength Calculation
This question asks us to find the wavelength of a sound wave given its frequency and speed in a specific medium. Sound waves, like all waves, have fundamental properties that are related to each other. These properties include wavelength, frequency, and speed.
Key Concepts: Wavelength, Frequency, and Wave Speed
- Wavelength (\(\lambda\)): This is the spatial period of the wave—the distance over which the wave's shape repeats. For a sound wave, it's the distance between two consecutive points that are in the same phase, such as two compressions or two rarefactions. It is typically measured in meters (m).
- Frequency (\(f\)): This is the number of wave cycles that pass a given point per unit of time. For a sound wave, it relates to the pitch we hear. It is measured in Hertz (Hz), which is equal to cycles per second (s<sup>-1</sup>).
- Speed (\(v\)): This is how fast the wave propagates through the medium. For a sound wave, the speed depends on the properties of the medium (like temperature, density, and elasticity). It is measured in meters per second (m/s).
Formula for Wave Calculation
The relationship between wave speed, frequency, and wavelength is a fundamental equation in wave physics:
\(v = f \lambda\)
Where:
- \(v\) is the speed of the wave.
- \(f\) is the frequency of the wave.
- \(\lambda\) is the wavelength of the wave.
To find the wavelength (\(\lambda\)), we can rearrange this formula:
\(\lambda = \frac{v}{f}\)
Applying the Formula to Find Wavelength
We are given the following values:
- Frequency (\(f\)) = 820 Hz
- Speed (\(v\)) = 420 m/s
Now, we can substitute these values into the rearranged formula to calculate the wavelength:
\(\lambda = \frac{420 \text{ m/s}}{820 \text{ Hz}}\)
Performing the division:
\(\lambda \approx 0.512195 \text{ m}\)
Rounding this value to two decimal places, we get 0.51 m.
Let's check this result against the given options:
| Option | Wavelength Value |
|---|---|
| 1 | 3.52 m |
| 2 | 2.52 m |
| 3 | 1.52 m |
| 4 | 0.51 m |
Our calculated wavelength, approximately 0.51 m, matches Option 4.
Revision Table: Sound Wave Properties Summary
| Property | Symbol | Unit (SI) | Definition | Relationship to v & f |
|---|---|---|---|---|
| Speed | \(v\) | m/s | Distance wave travels per unit time | \(v = f \lambda\) |
| Frequency | \(f\) | Hz (s<sup>-1</sup>) | Number of cycles per unit time | \(f = v / \lambda\) |
| Wavelength | \(\lambda\) | m | Spatial period of the wave | \(\lambda = v / f\) |
Additional Information: Types of Waves
Waves can be broadly classified into different types based on how they propagate and what they require to travel:
- Mechanical Waves: These waves require a medium (like solid, liquid, or gas) to travel. Sound waves are mechanical waves because they need a material medium to propagate. Other examples include water waves and seismic waves.
- Electromagnetic Waves: These waves do not require a medium to travel and can propagate through a vacuum. Light waves, radio waves, microwaves, X-rays, and gamma rays are examples of electromagnetic waves. They travel at the speed of light in a vacuum.
- Transverse Waves: In these waves, the particles of the medium move perpendicular to the direction of wave propagation. Light waves are examples of transverse waves. Shaking a rope up and down to create waves is also a demonstration of transverse waves.
- Longitudinal Waves: In these waves, the particles of the medium move parallel to the direction of wave propagation. Sound waves in air or fluids are examples of longitudinal waves. They cause compressions and rarefactions in the medium.
The formula \(v = f \lambda\) is applicable to both mechanical and electromagnetic waves, though the speed \(v\) depends on the specific type of wave and the medium it is traveling through.
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