What is the relationship between frequency and wavelength

General
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what is the relationship between frequency and wavelength

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What is the relationship between frequency and wavelength?

Answer:
The relationship between frequency and wavelength is a fundamental concept in the study of waves, whether they are sound waves, light waves, or other forms of electromagnetic radiation. This relationship can be described mathematically and conceptually.

1. Understanding Key Terms

  • Frequency (f):

    • Defined as the number of wave cycles that pass a given point per unit of time. It’s measured in hertz (Hz), where one hertz is one cycle per second.

  • Wavelength (λ):

    • The distance between successive crests, troughs, or identical points of a wave. It is usually measured in meters (m).

  • Wave Speed (v):

    • The speed at which the wave propagates through a medium. It is generally in meters per second (m/s).

2. Mathematical Relationship:

The frequency and wavelength of a wave are inversely related to each other through the wave speed. This relationship can be expressed by the formula:

v = f \cdot \lambda

Where:

  • ( v ) is the wave speed.
  • ( f ) is the frequency of the wave.
  • ( \lambda ) is the wavelength of the wave.

3. Inverse Relationship:

Since the product of frequency and wavelength equals the wave speed, if the wave speed is constant:

  • An increase in frequency (( f )) will result in a decrease in wavelength (( \lambda )).
  • A decrease in frequency will result in an increase in wavelength.

This inverse proportionality can be mathematically rearranged as:

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

4. Example Calculations:

Example 1:
If a wave traveling at a speed of 300 meters per second has a frequency of 60 Hz, what is its wavelength?

\lambda = \frac{v}{f} = \frac{300 \text{ m/s}}{60 \text{ Hz}} = 5 \text{ m}

Example 2:
If the wavelength of a wave is 10 meters and it’s traveling at a speed of 200 meters per second, what is its frequency?

f = \frac{v}{\lambda} = \frac{200 \text{ m/s}}{10 \text{ m}} = 20 \text{ Hz}

5. Real-world Applications:

  • In Light and Electromagnetic Waves:

    • Light and other electromagnetic waves follow the same relationship. For visible light, different colors have different wavelengths and frequencies. Blue light has a shorter wavelength and higher frequency compared to red light.

    • The speed of light (( c )) in a vacuum is approximately ( 3 \times 10^8 ) meters per second. This speed remains constant, so variations in wavelength and frequency follow the same inverse relationship:

      c = f \cdot \lambda

  • In Sound Waves:

    • The speed of sound varies depending on the medium (e.g., air, water), but the frequency-wavelength relationship remains consistent. For instance, a musical note of higher pitch (frequency) will have a shorter wavelength than a note of lower pitch.

Final Answer:

In summary, the relationship between frequency and wavelength is given by their product equalling the wave speed (( v = f \cdot \lambda )). This indicates that frequency and wavelength are inversely proportional: as one increases, the other decreases, provided the wave speed is constant. This fundamental concept applies to all types of waves, including sound waves and electromagnetic waves.