The Wave Equation in Physics

Introduction

The wave equation is fundamental in physics, relating wave speed, frequency, and wavelength.

Key Formulas

1. Wave Speed Equation:

   • [ v = f \times \lambda ]
   • Where:
     • ( v ) is wave speed in meters per second (m/s)
     • ( f ) is frequency in hertz (Hz)
     • ( \lambda ) is wavelength in meters (m)

2. Frequency and Time Period Relationship:

   • [ f = \frac{1}{T} ]
   • Where:
     • ( f ) is frequency in hertz (Hz)
     • ( T ) is time period in seconds (s)

Formula Triangle

• Used to rearrange the wave speed equation easily.

Worked Examples

1. Visible Light Cycle Calculation:

   • Given:
     • Frequency ( f = 6 \times 10^{14} \text{Hz} )
   • Calculate time period ( T ):
     • [ T = \frac{1}{f} = \frac{1}{6 \times 10^{14}} = 1.67 \times 10^{-15} \text{s} ]

2. Sound Wave Calculation:

   • Given:
     • Wave speed ( v = 330 \text{m/s} )
     • Time period ( T = 0.0001 \text{s} )
   • Calculate the frequency ( f ):
     • [ f = \frac{1}{T} = \frac{1}{0.0001} = 10,000 \text{Hz} ]
   • Calculate the wavelength ( \lambda ):
     • [ \lambda = \frac{v}{f} = \frac{330}{10,000} = 0.033 \text{m} ]

3. Radio Wave Calculation:

   • Given:
     • Frequency ( f = 200 \text{kHz} = 200,000 \text{Hz} )
     • Wavelength ( \lambda = 1500 \text{m} )
   • Calculate wave speed ( v ):
     • [ v = f \times \lambda = 200,000 \times 1500 = 3 \times 10^8 \text{m/s} ]

Tips and Tricks

• Ensure correct units: meters for wavelength, m/s for speed, and Hz for frequency.
• Use formula triangles to simplify rearranging formulas.
• Beware of units like kHz to Hz conversions.

Applications

• The wave equation is applicable to all wave types, including sound and electromagnetic waves.

Conclusion

Understanding and applying the wave equation is crucial for solving problems about wave properties in various physical contexts such as sound and light waves.