Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Oscillations and waves ↓


Wave interference and superposition


Wave interference and superposition are fundamental concepts in the study of waves and oscillations within classical mechanics. Understanding these principles is important in fields such as physics, engineering, and various applied sciences. Before we delve deeper into the details, it is necessary to establish some fundamental concepts about waves.

A wave is a disturbance or oscillation that travels through space, accompanied by a transfer of energy. Waves can be mechanical, such as sound waves traveling through air, or electromagnetic, such as light waves traveling through a vacuum. The basic properties of waves include amplitude, wavelength, frequency, and wave speed.

The principle of superposition states that when two or more waves overlap in space, the resulting wave is the sum of the individual waves. This leads us to wave interference, which occurs when two or more waves meet while traveling in the same medium. Interference can be classified as constructive or destructive.

Constructive and destructive interference

When two waves meet, their interaction depends on their phases – that is, whether their crests (crests) and troughs are aligned. Here's an overview:

  • Constructive interference: This occurs when the crest of one wave aligns with the crest of another wave, resulting in a wave with a greater amplitude. In simple terms, the waves combine to form a stronger wave.
  • Destructive interference: This occurs when the peak of one wave aligns with the trough of another wave. In this case, the waves begin to cancel each other out, making the resultant amplitude low or zero.

Visual example of constructive and destructive interference

In the SVG view above:

  • The first three overlapping waves (in blue and red) produce a resultant wave (black) in the upper part for constructive interference. Note that this wave is more powerful than any of the individual waves.
  • The next three overlapping waves exhibit destructive interference, where the amplitude of the resultant wave is reduced as the waves cancel each other out.

Mathematical representation

When waves interfere, their mathematical representation is guided by the superposition principle. For simplicity let's consider two sinusoidal waves:

Wave 1: y₁(x, t) = A₁ sin(k₁x - ω₁t + φ₁)
Wave 2: y₂(x, t) = A₂ sin(k₂x - ω₂t + φ₂)

Here:

  • A is the dimension.
  • k is the wavenumber, which is related to the wavelength (λ) by ( k = frac{2pi}{lambda} ).
  • ω is the angular frequency, which is related to the frequency (f) by ( omega = 2pi f ).
  • φ is the phase constant.

The resultant wave due to their interference ( y(x, t) ) can be expressed as:

y(x, t) = y₁(x, t) + y₂(x, t)

This is the essence of the superposition principle; the total wave is the simple sum of the individual waves.

Applications of wave interference

Wave interference has many practical applications in daily life and advanced technologies:

  • Noise-canceling headphones: These headphones use destructive interference. They detect ambient noise and generate sound waves of opposite phase to reduce unwanted sounds.
  • Radio transmission: Constructive interference improves signal strength when multiple antennas transmit waves in the same phase.
  • Musical instruments: Interference patterns create complex sound textures and beats when many notes are played simultaneously.

Example: beats in sound waves

Beats are caused by the interference of two sound waves of slightly different frequencies. This effect is often noticeable when tuning musical instruments.

Let the frequency of wave 1 = f₁
Let the frequency of wave 2 = f₂

The beat frequency, or "pulsation" frequency, is given by:
f_beat = | f₁ - f₂ |

Further explorations in superposition

While interference is the visual or audible result of superposition, the mathematical concept applies more broadly. Superposition can be applied to various forms of waves, including solutions of wave equations in various contexts such as string vibrations and electromagnetic fields.

Consider individual waves on a string. The total displacement of the string at any point is the sum of the displacements made by each individual wave. When analyzing complex systems, superposition allows decomposing them into simpler parts, solving for the individual components, and reconstructing a general solution.

Conclusion

Understanding wave interference and superposition is crucial for analyzing and predicting the behavior of various wave phenomena in nature and engineered systems. These principles explain why and how waves interact, reinforcing the beautiful complexities and simple truths that underlie the physical world around us.


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Wave interference and superposition

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