Mechanical Waves

Introduction

Mechanical waves are disturbances that propagate through a medium, transferring energy from one point to another without the actual transfer of matter. These waves can be classified into two main types: transverse waves and longitudinal waves. Understanding the properties and behavior of mechanical waves is crucial in the study of physics.

Wave Properties

1. Amplitude

• Definition: The maximum displacement of a particle from its equilibrium position.
• Example: If the amplitude of a wave is $5$ cm, and the wave is represented by the equation $y = 5\sin(2\pi t)$, calculate the displacement of a particle at $t = 0.25$ s.
  • Solution:
    • Substitute $t = 0.25$ into the equation:
    • $y = 5\sin(2\pi \times 0.25) = 5\sin(\pi/2) = 5$
    • The displacement of the particle is $5$ cm.

2. Wavelength

• Definition: The distance between two consecutive points in a wave that are in phase.
• Example: If the velocity of a wave is $300$ m/s and its frequency is $50$ Hz, calculate the wavelength.
  • Solution:
    • Use the formula: $v = f\lambda$
    • $\lambda = \frac{v}{f} = \frac{300}{50} = 6$ m
    • The wavelength of the wave is $6$ m.

3. Frequency

• Definition: The number of complete oscillations or cycles of a wave that occur in one second.
• Example: A wave has a frequency of $20$ Hz. Calculate the period of the wave.
  • Solution:
    • Use the formula: $f = \frac{1}{T}$
    • $T = \frac{1}{f} = \frac{1}{20} = 0.05$ s
    • The period of the wave is $0.05$ s.

4. Wave Speed

• Definition: The speed at which a wave propagates through a medium.
• Example: A wave with a frequency of $100$ Hz and a wavelength of $2$ m travels through a medium. Calculate the wave speed.
  • Solution:
    • Use the formula: $v = f\lambda$
    • $v = 100 \times 2 = 200$ m/s
    • The wave speed is $200$ m/s.

Common Mistakes

• Confusing amplitude with wavelength: Remember, amplitude is the maximum displacement from equilibrium, while wavelength is the distance between two consecutive points in phase.
• Incorrectly calculating wave speed: Ensure you use the correct formula $v = f\lambda$ and substitute the values correctly.

Key Points

• Mechanical waves transfer energy through a medium.
• Transverse waves have oscillations perpendicular to the direction of wave propagation, while longitudinal waves have oscillations parallel to the direction of wave propagation.
• The speed of a wave is determined by the frequency and wavelength.
• Amplitude, wavelength, frequency, and wave speed are key properties of waves.

Practice Questions

1. A transverse wave has an amplitude of $2$ cm and a wavelength of $8$ cm. Calculate the speed of the wave if its frequency is $40$ Hz.
2. A longitudinal wave travels through a medium with a speed of $350$ m/s. If the frequency of the wave is $25$ Hz, calculate its wavelength.
3. A wave has a period of $0.02$ s and a wavelength of $4$ m. Determine the frequency of the wave.
4. Explain the difference between transverse and longitudinal waves.
5. Calculate the amplitude of a wave if the maximum displacement of a particle from its equilibrium position is $10$ mm.

Answers

1. Given: Amplitude ($A$) = $2$ cm, Wavelength ($\lambda$) = $8$ cm, Frequency ($f$) = $40$ Hz. Wave speed ($v$) = $f\lambda = 40 \times 8 = 320$ cm/s.
2. Given: Wave speed ($v$) = $350$ m/s, Frequency ($f$) = $25$ Hz. Wavelength ($\lambda$) = $\frac{v}{f} = \frac{350}{25} = 14$ m.
3. Given: Period ($T$) = $0.02$ s, Wavelength ($\lambda$) = $4$ m. Frequency ($f$) = $\frac{1}{T} = \frac{1}{0.02} = 50$ Hz.
4. Transverse waves have oscillations perpendicular to the direction of wave propagation, while longitudinal waves have oscillations parallel to the direction of wave propagation.
5. Amplitude ($A$) = $10$ mm.