1. Wave Motion and its Applications, Applied Physics -2, (2001), Polytechnic 2nd semester Notes

 

1. Wave Motion and its Applications

1.1 Wave Motion

Wave motion refers to the process by which energy is transferred from one point to another without the physical movement of matter over long distances. Waves can travel through different media such as air, water, and solid materials.

• Types of Waves:

  1. Mechanical Waves: These waves require a medium to travel (e.g., sound waves, water waves, and waves on a string). They cannot travel in a vacuum.
  2. Electromagnetic Waves: These do not require a medium and can travel through a vacuum (e.g., light, radio waves, X-rays).

  Waves are typically described by their wavelength (λ), frequency (f), velocity (v), and amplitude (A).

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1.2 Transverse and Longitudinal Waves with Examples

Wave Classification based on Direction of Particle Motion:

• Transverse Waves:

  • In these waves, the particles of the medium move perpendicular to the direction of wave propagation.
  • Example: Light waves, waves on a string, and water waves.
  • Key Characteristics:
    • Crest: The highest point of the wave.
    • Trough: The lowest point of the wave.
    • Amplitude: The maximum displacement of the particles from the equilibrium position.
    • Wavelength: The distance between two consecutive crests or troughs.
  

• Longitudinal Waves:

  • In these waves, the particles of the medium move parallel to the direction of wave propagation.
  • Example: Sound waves in air, seismic P-waves.
  • Key Characteristics:
    • Compression: The region of high particle density.
    • Rarefaction: The region of low particle density.
    • Wavelength: The distance between two consecutive compressions or rarefactions.
  

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1.3 Definitions of Wave Velocity, Frequency, and Wavelength and Their Relationship

• Wave Velocity (v):

  • It is the speed at which a wave propagates through a medium.
  • Formula: $v = \frac{\lambda}{T}$ or $v = f \times \lambda$, where $T$ is the time period and $f$ is the frequency.

• Frequency (f):

  • The number of oscillations or cycles a wave completes per second. It is measured in Hertz (Hz).
  • Formula: $f = \frac{1}{T}$, where $T$ is the time period.

• Wavelength (λ):

  • The distance between two consecutive crests or troughs of a wave. It is measured in meters (m).

• Relationship:

  • The wave velocity is directly related to the frequency and wavelength by the formula: $$v = f \times \lambda$$
    
  • Where:
    • $v$ = Wave velocity (m/s)
    • $f$ = Frequency (Hz)
    • $\lambda$ = Wavelength (m)

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1.4 Principle of Superposition of Waves

The principle of superposition states that when two or more waves meet at a point, the displacement of the resulting wave is the algebraic sum of the displacements of the individual waves at that point.

• Constructive Interference: When two waves are in phase, their displacements add up, increasing the amplitude.
• Destructive Interference: When two waves are out of phase, their displacements subtract from each other, reducing or canceling out the amplitude.

Example: If two waves with the same frequency and amplitude meet in phase, they create a larger wave. If they meet out of phase, they may cancel each other.

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1.5 Stationary Waves and Resonance Tube

• Stationary Waves:

  • Stationary waves are formed when two waves of the same frequency, amplitude, and wavelength travel in opposite directions and interfere with each other.
  • Key Features:
    • Nodes: Points where there is no displacement (destructive interference).
    • Antinodes: Points where the displacement is maximum (constructive interference).
  • Example: A vibrating string fixed at both ends forms stationary waves.
  

  

• Resonance Tube:

  • A resonance tube is used to study sound waves and resonance phenomena. It is a tube where standing waves are formed at specific frequencies.
  • Application: A resonance tube helps determine the speed of sound in air. When the frequency of the sound wave matches the natural frequency of the tube, resonance occurs, and the amplitude becomes very large.

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1.6 Simple Harmonic Motion (SHM)

1.6.1 Definition

Simple Harmonic Motion (SHM) is the motion of an object where the restoring force is proportional to the displacement from the equilibrium position, directed toward the center.

• Key Characteristics:

  • The motion is periodic (repeats over time).
  • The displacement follows a sinusoidal pattern.
  • The restoring force is given by Hooke's law: $F = -kx$, where:
    • $F$ is the restoring force.
    • $k$ is the spring constant (or force constant).
    • $x$ is the displacement from equilibrium.

• Mathematical Form: The equation of SHM is:

  $$x(t) = A \cos(\omega t + \phi)$$
  

  where:

  • $A$ = amplitude of motion.
  • $\omega$ = angular frequency.
  • $\phi$ = phase constant.
  • $t$ = time.

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1.6.2 Simple Harmonic Progressive Wave and Energy Transfer

• Simple Harmonic Progressive Wave:
  • A progressive wave in SHM travels through a medium, transferring energy from one point to another without moving the medium’s particles over large distances.
• Energy in SHM:
  • Total Energy (E) in SHM is the sum of kinetic energy ($K$) and potential energy ($U$) at any point in time.
  • Formula for Total Energy: $$E = \frac{1}{2} m \omega^2 A^2$$where:
    • $m$ = mass.
    • $\omega$ = angular frequency.
    • $A$ = amplitude.
  • Kinetic Energy: $$K = \frac{1}{2} m v^2$$
    
  • Potential Energy: $$U = \frac{1}{2} k x^2$$
    

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Practice Questions

Long Answer Questions:

1. Explain the principle of superposition of waves and describe its applications in real life.

   • Answer: The principle of superposition states that when two or more waves overlap, the resultant displacement is the sum of the individual displacements at that point. In the case of constructive interference, the waves add up, leading to a wave with a larger amplitude. For destructive interference, the waves cancel each other out, reducing the amplitude. This principle is fundamental to understanding interference patterns in sound waves (e.g., noise-canceling headphones), light waves, and even water waves.

2. Describe Simple Harmonic Motion (SHM) and derive its equation of motion.

   • Answer: SHM is defined as a motion where the restoring force is proportional to the displacement. The motion follows a sinusoidal pattern, and the force is directed towards the equilibrium position. The equation of SHM is: $$F = -kx$$The second law of motion gives: $$F = ma = m \frac{d^2x}{dt^2}$$ Equating the forces: $$m \frac{d^2x}{dt^2} = -kx$$Rearranging: $$\frac{d^2x}{dt^2} + \frac{k}{m} x = 0$$This is the equation of motion for SHM, where $\frac{k}{m}={\omega }^{\mathrm{2,\; and }}$$\omega$ is the angular frequency.

Numerical Problems:

1. A wave has a frequency of 50 Hz and a wavelength of 2 meters. Find the wave velocity.

   • Solution: Given:
     • $f = 50$ Hz
     • $\lambda = 2$ m Using the formula $v = f \times \lambda$:
     $$v = 50 \times 2 = 100 \text{ m/s}$$So, the wave velocity is 100 m/s.

2. A simple pendulum oscillates with an amplitude of 10 cm and a frequency of 2 Hz. Find the maximum speed of the pendulum.

   • Solution: Given:
     • $A = 10$ cm = 0.1 m
     • $f = 2$ Hz
     • The angular frequency $\omega =2\pi f=2\pi \times 2=4\mathrm{\pi \; rad/s\; The\; maximum\; speed\; (}$$v_{\text{max}}$) occurs at equilibrium, and is given by:
     $$v_{\text{max}} = A \omega = 0.1 \times 4\pi = 1.26 \text{ m/s}$$
     

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Important MCQs:

1. What is the relationship between frequency and time period of a wave?
   a) $f = T$
   b) $f = \frac{1}{T}$
   c) $f = T^2$
   d) $f = \frac{1}{2T}$
   Answer: b) $f = \frac{1}{T}$

2. In SHM, the restoring force is proportional to the:
   a) Velocity
   b) Displacement
   c) Kinetic energy
   d) Amplitude
   Answer: b) Displacement