Oscillations: "Any motion in which the displacement of a particle from a fixed point keeps changing direction and there is a periodicity in the motion, i.e. the motion repeats in some way." (Tsokos, 2014)

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Simple Harmonic Motion (SHM)​

• Definition: "Motion in which (the magnitude of) acceleration is proportional and opposite to displacement from a fixed (equilibrium) position (where x = 0)."​​ 

• Constant quantities: Amplitude, period and frequency. Definitions below for SHM:

  • Amplitude (A or xo): Maximum displacement from equilibrium position.

  • Period (T): ​Time taken to complete one full oscillation. Unit: s.

  • Frequency (f): Number of oscillations completed in one second. Unit: hertz (Hz)

Examples:​

• Simple pendulum:​​

• Mass-spring system:

Graphical representation (SHM)​

• Acceleration-displacement: Negative gradient and direct proportionality.

  • Maximum acceleration at amplitude, zero acceleration at equilibrium position.​

-A

A

• Energy-displacement: 

  • Total energy (always constant) = Kinetic energy (EK)​ + Potential energy (PE). 

• Displacement, velocity and acceleration versus time:

  • Sine or cosine functions of time.

  • Phase difference (shift) between graphs:

    • Displacement-time and velocity-time: 0.25T​.

    • Displacement-time and velocity-time: 0.50T.

    • Velocity-time and acceleration-time: 0.25T.

    • Think in terms of Calculus!

      • Acceleration as the derivative of velocity.

      • Velocity as the derivative of velocity.

  • When the phase difference is zero or T, the systems are in phase.​

Wave specifications

• Definition: "A wave is a disturbance that travels in a medium (e.g. air, water etc.)"

• Source: A wave is initiated by a vibrating object and travels away from it.

• Particles of the medium: vibrate about their rest position at the same frequency as the source.

• A wave transfers energy and momentum, but never mass.

• Medium: No large scale movement of the medium as the wave passes through it.

Wave properties

• Wavelength (λ): Shortest distance between two points that are in phase on a wave.

  • Two consecutive crests or two consecutive troughs.

• Frequency (f): ​Number of vibrations per second performed by the source of waves.

• Period (T): Time taken for one complete wavelength to pass a fixed point. T = 1/f.

• Displacement (x): Instantaneous distance of the moving object from its mean position (in a specified direction).

• Amplitude (A/xo): Maximum displacement of wave from its rest position.

• Speed (v/c): Depends only on the properties of the medium and not the source.

  • v =​ λf or c (speed of light) = λf.

Graphs:​

• Displacement-distance:​​​

• Displacement-time:

Wave classification

• Mechanical waves: Require a medium to travel through.

  • Sound: ​Constant velocity (sqrt(v) proportional to temperature), longitudinal.

    • Hearable: 20 Hz to 20000 Hz.​

    • Frequency: Pitch.

    • Amplitude: Volume (Loudness).

• Electromagnetic waves: May travel in vacuum.

  • Speed of light (c): 3 x 10^8 ms^-1.​

  • Wavelength: Colour.

  • Amplitude: Brightness.

Electromagnetic spectrum:

• Transverse waves: Displacement of particles is perpendicular to the direction of energy transfer. Both electromagnetic and mechanical waves (e.g in a rope). 

• Longitudinal waves: Displacement of particles is parallel to the direction of energy transfer. O​nly mechanical waves, made of compression and rarefaction.

Wavefronts: "Surfaces/lines that join points with the same phase."

Rays: "Lines in the direction of energy transfer." 

Wavefronts and rays are perpendicular to each other.

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Intensity (I)

• Definition: "The intensity of a wave at a point P is the amount of energy arriving at P per unit area per unit time."

• Equation: I = Power/Area = P/4πr^2, where r is the distance. 

• Units: J s^-1 m^-2 or W m^-2.

• Inverse-square law: Doubling the distance reduces power received by a quarter.

• Intensity is proportional to amplitude squared.

Superposition

• Definition: "When two or more waves collide, the total displacement is the vector sum of their individual displacements".

Reflection of pulses

• Fixed end: Pulse inverts, due to reaction force (e.g. of the wall).

• Free end: Pulse does not invert.

Polarization

• Definition: "An electromagnetic wave is said to be plane polarized if the electric field oscillates on the same plane".

  • Occurrence: ​It only occurs in transverse waves, e.g. light, which is normally unpolarized.

• Polarization of unpolarized wave: Original intensity reduced by half.

• Methods of polarization:

  • Passing through a polarizer and an analyzer.​

    • Polarizer: Device that produces plane-polarized light from an unpolarized beam.

    • Analyzer: Polarizer used to detect polarized light.

  • Reflection on non-metallic surfaces (e.g. a lake): Partial polarization into different components, the greatest in the plane parallel to the non-metallic surface.

• Optically active substance (e.g. sugar solutions): Rotates the plane of polarization, normally placed between the polarizer and analyzer.

  • Sugar solution: Length and concentration of solution is proportional to the angle of rotation.​

  • Liquid-crystal displays (LCDs): liquid crystal is sandwiched between two glass electrodes. Rotates the plane of polarization according to pd across it.

• Polarimeter: Measures the intensity after the analyzer.​

• Malus' Law (for already polarized light): I = Io cosθ^2 and E = Eo cosθ^2, where θ is the angle between the incident wave and the polarizer or analyzer.

• Brewster's Law: If the reflected ray and the refracted are at right angles to one another, then the reflected ray is totally polarized. Read about the reflection and refraction of waves.

  • The angle of incidence for this condition is known as the polarizing angle.

  • θi + θr = 90º and n = sinθi/sinθr = sinθi/cosθi = tanθi.

• Uses of polarization:

  • Polaroid sunglasses:​ Allows waves with a vertical plane of polarization and absorbs waves with an horizontal plane of polarization.

    • Reduces glare from non-metallic surfaces.​

  • Stress analysis: When white light is passed through stressed plastics, colored lines are observed in regions of maximum stress.

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​Reflection and refraction

• Reflection: Angle of incidence (θ1/θi) = Angle of reflection (θ1'/θr).

• Refraction: Wave travelling from one medium into another.

  • Snell's law: v1/v2 = sinθ1/sinθ2 = n2/n1 = 1n2.

    • Absolute refractive index = n = c/v = speed of light in vacuum/speed if light in medium.

      • High n - optically dense medium.

• A ray will bend towards the normal if entering an optically denser medium.   

• Plane: For reflection and refraction, the rays are always in the same plane.

• Reversibility of light: sinθ1/sinθ2 = 1n2 = 1/2n1 = sinθ2/sinθ1 = 2n1.

• The critical angle (θc​): As the angle of incidence increases, the angle of refraction will approach 90º. At the angle of refraction 90º, the angle of incidence is called critical angle.​

  • sinθc = n2/n1.​

  • If angle of incidence (θ1/θi) > critical angle (θc​), there is total internal reflection.

Diffraction

• Definition: "When a wave passes through a narrow slit, causing spread to bend and creating an interference pattern."

• Occurrence: It takes place when the aperture (slit) ≤ wavelength. It is most evident when the aperture is significantly smaller than the wavelength.

• Quantities that...

  • Remain constant: ​frequency, velocity and wavelength.

  • Change: Amplitude reduces, since the energy is distributed over a larger area.

• Pattern of waves:​

• Uses: CD/DVD or Electron microscope.

Double-source interference

• Definition: "When two similar sources (with the same frequency) and coherent (with a constant phase relationship), emit waves that interfere with each other".

• Path difference: The difference in distance of one specific point from the two sources.

  • Path difference = ∆r =│S1P-S2P│, where S1P is the distance of source 1 to the specific point P and S2P is the distance of source 2 to the specific point P.

  • Constructive interference: When ∆r = nλ, for n = 0, 1, 2, 3,...

  • Destructive interference: When ∆r = (n + 1/2)λ, for n = 0, 1, 2, 3,...

Double-slit interference: Specific double-source interference, in which successive bright fringes are formed, as shown in the diagram below.

• Fringe spacing: s = ​λD/d, where D is the distance between the slits and the screen and d is the distance between the slits.

• Thomas Young experiment:​

Título do Site

• Intensity graph: for negligible slit width.

Definition: "When two travelling waves of equal amplitude and equal frequency travelling with the same speed in opposite directions are superposed, a standing/stationary wave is formed".

Concepts

• Amplitude: Each particle has its own amplitude (A).

• Nodes: Points of destructive interference, i.e. zero amplitude.

• Anti-nodes: Points of constructive interference, i.e. maximum amplitude.

• Phase: Points between consecutive nodes are in phase.

• No energy transfer: A standing wave does not move horizontally, and thus, no energy is transferred and the shape does not change.

Harmonics

• Resonance: Systems, such as pipes and strings, only resonate at very specific frequencies, which are known as the harmonics.

• Harmonics have frequencies that are integral multiples of the first frequency, i.e the fundamental frequency. fn = n f1. They numbered according to n.

Boundary conditions

• Fixed boundary: Always a node, whose reflection causes a 180º phase change.​

  • Example: Walls or edge of a drum-head.​

• Free boundary: Always an anti-node, whose reflection does not cause any phase change.

  • Example: Tuning fork or air.

Cases

• Strings (Length = L): The waves reflect at the fixed ends, generating two identical waves travelling in opposite directions.

  • End condition: node-node: λn = 2L/n or fn = nv/2L.  Walls or edge of a drum-head.​

• Pipes (Length = L): Medium is air - longitudinal waves!

  • Longitudinal waves: Nodes are the centers of compression and rarefaction.

  • End condition: totally closed: node-node. Equal treatment as strings (above).

  • End condition: totally open: anti-node - anti-node. λn = 2L/n or fn = nv/2L 

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  • End condition: partially open (one closed end): node - anti-node. λn = 4L/n or fn = nv/4L, only for odd harmonics, so n = 1, 3, 5...

Comparisson between travelling and standing waves

(Homer & Bowen-Jones, 2014)

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Tip: Instead of trying to memorize the formulas for each specific condition or case in standing waves, try drawing the situation and then reaching the formula!