Module 112. Waves and Oscillations

Waves and oscillations are fundamental phenomena in physics that describe periodic motion and the transfer of energy through various media. They play a central role in understanding sound, light, mechanical vibrations, and even quantum mechanics. While oscillations refer to the repetitive motion of a system about an equilibrium position, waves represent the propagation of these oscillations through space and time. Together, they form the basis for a wide range of natural and technological processes, from musical tones to electromagnetic communication.

Concept of Oscillations

An oscillation is a repeated back-and-forth or up-and-down motion about a mean or equilibrium position. It occurs when a system experiences a restoring force proportional to its displacement, as described by Hooke’s law.
Mathematically, the simplest form of oscillatory motion is Simple Harmonic Motion (SHM), represented by:
x=Asin⁡(ωt+ϕ)x = A \sin(\omega t + \phi)x=Asin(ωt+ϕ)
where:

• xxx = displacement from equilibrium,
• AAA = amplitude (maximum displacement),
• ω\omegaω = angular frequency (2πf2\pi f2πf),
• ttt = time,
• ϕ\phiϕ = phase constant.

Key characteristics of oscillations include:

• Amplitude (A): Maximum displacement from equilibrium.
• Frequency (f): Number of oscillations per second (measured in hertz, Hz).
• Period (T): Time taken for one complete oscillation (T=1/fT = 1/fT=1/f).
• Phase: Describes the state of motion of the oscillating body at a given instant.

Examples of oscillatory systems include a pendulum, a vibrating tuning fork, or the alternating current in an electrical circuit.

Types of Oscillations

1. Free Oscillations: Occur when a system oscillates naturally without any external periodic force after being displaced from its equilibrium position, e.g., a simple pendulum swinging in air.
2. Damped Oscillations: The amplitude of oscillation gradually decreases over time due to energy loss (usually from friction or resistance). For example, a tuning fork’s vibration diminishes as sound energy dissipates.
3. Forced Oscillations: Occur when an external periodic force continuously drives the system, e.g., a building shaking during an earthquake or a loudspeaker diaphragm vibrating due to an electric signal.
4. Resonance: A special case of forced oscillation where the driving frequency matches the natural frequency of the system, resulting in maximum amplitude. Examples include the vibration of bridges due to wind and the shattering of glass by a specific sound frequency.

Wave Motion

A wave is a disturbance that travels through a medium (or space) transferring energy without transferring matter. Waves occur as a consequence of oscillations propagated through space.
The general wave equation is expressed as:
y=Asin⁡(kx−ωt)y = A \sin(kx – \omega t)y=Asin(kx−ωt)
where k=2π/λk = 2\pi/\lambdak=2π/λ (wave number) and λ\lambdaλ is the wavelength.
The wave speed (v) is given by:
v=fλv = f\lambdav=fλ
where fff is the frequency and λ\lambdaλ is the wavelength.

Types of Waves

Waves can be categorised based on the direction of particle motion and the medium through which they travel.
1. Mechanical Waves: Require a material medium (such as air, water, or solids) for propagation.

• Transverse Waves: Particles of the medium vibrate perpendicular to the direction of wave propagation. Examples include waves on a string and electromagnetic waves.
• Longitudinal Waves: Particles vibrate parallel to the direction of wave travel, creating compressions and rarefactions. Sound waves in air are typical examples.
• Surface Waves: Exhibit characteristics of both transverse and longitudinal motion, such as water waves.

2. Electromagnetic Waves: Do not require a medium; they can propagate through a vacuum. They consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. Examples include light, radio waves, X-rays, and microwaves.
3. Matter Waves: Associated with moving particles, as described by quantum mechanics (de Broglie hypothesis). These waves demonstrate the wave–particle duality of matter.

Characteristics of Waves

• Amplitude (A): Indicates the energy carried by the wave; greater amplitude means more energy.
• Wavelength (λ): The distance between two consecutive crests or compressions.
• Frequency (f): Number of complete waves passing a point per second.
• Velocity (v): Speed at which the wave propagates through the medium.
• Phase and Phase Difference: Represent the relative position of oscillating particles.

Wave Phenomena

Waves exhibit several characteristic behaviours that illustrate their energy transfer and interaction properties:

• Reflection: When a wave bounces back upon striking a barrier, following the law of reflection (angle of incidence=angle of reflection\text{angle of incidence} = \text{angle of reflection}angle of incidence=angle of reflection).
• Refraction: Change in wave direction when moving between media with different densities, due to variation in speed.
• Diffraction: Bending of waves around obstacles or through small openings.
• Interference: Superposition of two or more waves resulting in regions of constructive and destructive interference.
• Polarisation: Restriction of wave vibrations to a single plane; occurs only in transverse waves.

Sound Waves and Their Properties

Sound waves are longitudinal mechanical waves produced by vibrating sources and propagated through air, liquids, or solids. The human ear perceives sound frequencies between 20 Hz and 20,000 Hz.
Key properties include:

• Pitch: Determined by frequency.
• Loudness: Depends on amplitude.
• Quality (Timbre): Determined by the waveform and harmonics, allowing distinction between different sounds of the same pitch and loudness.

Sound phenomena such as Doppler effect (change in frequency due to relative motion between source and observer) and resonance in musical instruments are direct applications of wave theory.

Energy in Oscillations and Waves

In oscillating systems, energy continually shifts between potential and kinetic forms. For simple harmonic motion, total energy remains constant:
E=12kA2E = \frac{1}{2}kA^2E=21​kA2
where kkk is the force constant.
In wave motion, energy is transmitted across space. The energy carried by a wave is proportional to the square of its amplitude (E∝A2E \propto A^2E∝A2).

Applications of Waves and Oscillations

Waves and oscillations are foundational to numerous scientific and technological applications:

• Communication systems: Radio, television, and mobile networks rely on electromagnetic waves.
• Medicine: Ultrasonic waves are used in diagnostic imaging (ultrasound) and therapy.
• Engineering: Understanding vibration and resonance is crucial for designing stable structures and machinery.
• Navigation: SONAR (Sound Navigation and Ranging) uses sound waves to detect underwater objects.
• Astronomy: Electromagnetic waves provide information about distant celestial bodies through spectroscopy.
• Quantum physics: Wave–particle duality explains the behaviour of electrons and photons.

Wave Superposition and Standing Waves

When two waves of the same frequency and amplitude travel in opposite directions, they interfere to form standing waves, characterised by nodes (points of zero displacement) and antinodes (points of maximum displacement). Standing waves are important in acoustics, as they determine resonance conditions in musical instruments and air columns.
The principle of superposition states that when two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements. This principle explains interference patterns, such as those observed in Young’s double-slit experiment, confirming the wave nature of light.

Damped and Driven Oscillatory Systems

In real-world systems, oscillations are affected by resistive forces like friction or air resistance, leading to damped oscillations. The rate of damping determines whether the system is under-damped, critically damped, or over-damped.
When an external periodic force drives the system, forced oscillations occur. At resonance, the amplitude reaches its maximum, which can be beneficial (in radio tuning) or destructive (as in structural failures due to vibrations).