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


Welcome to the world of oscillations and waves, a fascinating area of classical mechanics that underlies a vast range of natural phenomena. The aim of this lesson is to clarify the concepts associated with oscillations and waves, explaining them in simple terms and with various examples to ensure clarity and comprehensive understanding.

Understanding oscillations

Oscillation means a repetitive back-and-forth movement around an equilibrium position. You can think of a child swinging on a swing, a vibrating guitar string, or the pendulum in a grandfather clock. These are all oscillating systems.

Simple harmonic motion (SHM)

One of the most basic types of oscillation is simple harmonic motion (SHM). It is characterized by the fact that the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.

F = -kx

Here, F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.

The motion of an object in SHM can be described using the following equations:

x(t) = a cos(ωt + φ)
v(t) = -Aωsin(ωt + φ)
a(t) = -Aω² cos(ωt + φ)

Where:

  • A is the dimension
  • ω is the angular frequency
  • t is the time
  • φ is the phase constant
Mass

Examples of oscillation

Anchor

Imagine a simple pendulum, such as the one found in a clock. It consists of a mass called a bob, attached to a string or rod. When it is released from its equilibrium position, it will oscillate back and forth.

Mass-spring system

Consider a mass attached to a spring. When you pull and release it, the mass oscillates around the balance point. This system provides a classic example of SHM.

Properties of oscillation

Dimensions

The amplitude is the maximum displacement from the equilibrium position. In the pendulum example, it is the maximum distance the bob moves to one side.

Duration and frequency

Period is the time taken for one complete cycle of oscillation. Frequency is the number of complete cycles per unit time.

T = 2π√(m/k)
F = 1/T

Energy in oscillations

In an oscillating system the energy transfers between potential energy and kinetic energy. In a frictionless system the total mechanical energy remains constant.

e = (1/2) k a² = (1/2) mv² + (1/2) kx²

Understanding waves

Now let's learn about waves. A wave is a disturbance that transfers energy from one point to another without moving matter. Waves can occur in different mediums, such as air, water or even solids.

Types of waves

Transverse waves

In a transverse wave the displacement of the medium is perpendicular to the direction of wave propagation. Think of waves on a string.

Longitudinal waves

In a longitudinal wave the displacement of the medium is parallel to the direction of wave propagation. Sound waves in air are a typical example of this.

Wave characteristics

Some of the basic characteristics of waves are as follows:

Wavelength

Wavelength is the distance between successive crests (or troughs) in a wave.

Frequency and duration

The frequency of a wave is the number of wavelengths passing a given point per unit time. The period is the time taken for one complete cycle of the wave.

V = fL

where v is the speed of the wave, f is the frequency, and λ is the wavelength.

Wave motion

The speed of a wave is determined by the medium through which it travels. It can be calculated as follows:

v = d/t

Types of mechanical waves

Surface waves

These waves travel on the surface of the medium. The movement of waves in the ocean or ripples on the surface of water are common examples.

Standing waves

Standing waves are produced when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other.

Wave interference

When two or more waves overlap each other, they interfere with each other. According to the principle of superposition the resultant wave is the sum of the individual waves.

Constructive interference

This occurs when the waves are in the same phase, causing an increase in amplitude.

Destructive interference

This occurs when the waves are out of phase, causing a decrease in amplitude.

Applications of waves and oscillations

Oscillations and waves have a profound impact on various aspects of our world. Some of the major applications include:

  • Sound waves: used in communication and musical instruments.
  • Seismic waves: Help in understanding earthquakes.
  • Electromagnetic waves: This includes light, radio waves, and X-rays.
  • Medical applications: Ultrasound uses sound waves for imaging the body.

Conclusion

By understanding vibrations and waves, we gain insight into a wide range of phenomena, from the music we love to the technology we depend on. These concepts are not just theoretical; they also have practical applications in science and engineering. Through this exploration of vibrations and waves, we have discovered fundamental principles that govern a significant portion of the physical world.


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Oscillations and waves

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