Topic 4: Waves
4.1 Oscillations
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)​
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Definition: "Motion in which (the magnitude of) acceleration is proportional and opposite to displacement from a fixed (equilibrium) position (where x = 0)."​​
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Constant quantities: Amplitude, period and frequency. Definitions below for SHM:
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Amplitude (A or xo): Maximum displacement from equilibrium position.
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Period (T): ​Time taken to complete one full oscillation. Unit: s.
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Frequency (f): Number of oscillations completed in one second. Unit: hertz (Hz)
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Examples:​
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Simple pendulum:​​

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Mass-spring system:
Graphical representation (SHM)​
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Acceleration-displacement: Negative gradient and direct proportionality.
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Maximum acceleration at amplitude, zero acceleration at equilibrium position.​
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Energy-displacement:
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Total energy (always constant) = Kinetic energy (EK)​ + Potential energy (PE).
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Displacement, velocity and acceleration versus time:
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Sine or cosine functions of time.
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Phase difference (shift) between graphs:
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Displacement-time and velocity-time: 0.25T​.
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Displacement-time and velocity-time: 0.50T.
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Velocity-time and acceleration-time: 0.25T.
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Think in terms of Calculus!
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Acceleration as the derivative of velocity.
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Velocity as the derivative of velocity.
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When the phase difference is zero or T, the systems are in phase.​
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4.2 Travelling Waves
Wave specifications
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Definition: "A wave is a disturbance that travels in a medium (e.g. air, water etc.)"
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Source: A wave is initiated by a vibrating object and travels away from it.
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Particles of the medium: vibrate about their rest position at the same frequency as the source.
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A wave transfers energy and momentum, but never mass.
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Medium: No large scale movement of the medium as the wave passes through it.
Wave properties
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Wavelength (λ): Shortest distance between two points that are in phase on a wave.
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Two consecutive crests or two consecutive troughs.
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Frequency (f): ​Number of vibrations per second performed by the source of waves.
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Period (T): Time taken for one complete wavelength to pass a fixed point. T = 1/f.
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Displacement (x): Instantaneous distance of the moving object from its mean position (in a specified direction).
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Amplitude (A/xo): Maximum displacement of wave from its rest position.
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Speed (v/c): Depends only on the properties of the medium and not the source.
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v =​ λf or c (speed of light) = λf.
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Graphs:​
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Displacement-distance:​​​
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Displacement-time:
Wave classification
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Mechanical waves: Require a medium to travel through.
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Sound: ​Constant velocity (sqrt(v) proportional to temperature), longitudinal.
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Hearable: 20 Hz to 20000 Hz.​
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Frequency: Pitch.
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Amplitude: Volume (Loudness).
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Electromagnetic waves: May travel in vacuum.
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Speed of light (c): 3 x 10^8 ms^-1.​
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Wavelength: Colour.
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Amplitude: Brightness.
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Electromagnetic spectrum:
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Transverse waves: Displacement of particles is perpendicular to the direction of energy transfer. Both electromagnetic and mechanical waves (e.g in a rope).
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Longitudinal waves: Displacement of particles is parallel to the direction of energy transfer. O​nly mechanical waves, made of compression and rarefaction.

4.3 Wave Characteristics
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)
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Definition: "The intensity of a wave at a point P is the amount of energy arriving at P per unit area per unit time."
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Equation: I = Power/Area = P/4Ï€r^2, where r is the distance.
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Units: J s^-1 m^-2 or W m^-2.
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Inverse-square law: Doubling the distance reduces power received by a quarter.
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Intensity is proportional to amplitude squared.
Superposition
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Definition: "When two or more waves collide, the total displacement is the vector sum of their individual displacements".
Reflection of pulses
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Fixed end: Pulse inverts, due to reaction force (e.g. of the wall).
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Free end: Pulse does not invert.
Polarization
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Definition: "An electromagnetic wave is said to be plane polarized if the electric field oscillates on the same plane".
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Occurrence: ​It only occurs in transverse waves, e.g. light, which is normally unpolarized.
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Polarization of unpolarized wave: Original intensity reduced by half.
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Methods of polarization:
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Passing through a polarizer and an analyzer.​
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Polarizer: Device that produces plane-polarized light from an unpolarized beam.
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Analyzer: Polarizer used to detect polarized light.
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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.
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Optically active substance (e.g. sugar solutions): Rotates the plane of polarization, normally placed between the polarizer and analyzer.
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Sugar solution: Length and concentration of solution is proportional to the angle of rotation.​
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Liquid-crystal displays (LCDs): liquid crystal is sandwiched between two glass electrodes. Rotates the plane of polarization according to pd across it.
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Polarimeter: Measures the intensity after the analyzer.​
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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.
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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.
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The angle of incidence for this condition is known as the polarizing angle.
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θi + θr = 90º and n = sinθi/sinθr = sinθi/cosθi = tanθi.
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Uses of polarization:
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Polaroid sunglasses:​ Allows waves with a vertical plane of polarization and absorbs waves with an horizontal plane of polarization.
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Reduces glare from non-metallic surfaces.​
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Stress analysis: When white light is passed through stressed plastics, colored lines are observed in regions of maximum stress.
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4.4 Wave Behavior
​Reflection and refraction
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Reflection: Angle of incidence (θ1/θi) = Angle of reflection (θ1'/θr).
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Refraction: Wave travelling from one medium into another.
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Snell's law: v1/v2 = sinθ1/sinθ2 = n2/n1 = 1n2.
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Absolute refractive index = n = c/v = speed of light in vacuum/speed if light in medium.
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High n - optically dense medium.
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It's important to remember that, whenever refraction takes place, so does reflection!
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A ray will bend towards the normal if entering an optically denser medium.
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Plane: For reflection and refraction, the rays are always in the same plane.
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Reversibility of light: sinθ1/sinθ2 = 1n2 = 1/2n1 = sinθ2/sinθ1 = 2n1.
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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.​
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sinθc = n2/n1.​
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If angle of incidence (θ1 or θi) > critical angle (θc​), there is total internal reflection.
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Diffraction
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Definition: "When a wave passes through a narrow slit, causing spread to bend and creating an interference pattern."
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Occurrence: It takes place when the aperture (slit) ≤ wavelength. It is most evident when the aperture is significantly smaller than the wavelength.
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Quantities that...
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Remain constant: ​frequency, velocity and wavelength.
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Change: Amplitude reduces, since the energy is distributed over a larger area.
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Pattern of waves:​
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Uses: CD/DVD or Electron microscope.
Double-source interference
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Definition: "When two similar sources (with the same frequency) and coherent (with a constant phase relationship), emit waves that interfere with each other".
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Path difference: The difference in distance of one specific point from the two sources.
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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.
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Constructive interference: When ∆r = nλ, for n = 0, 1, 2, 3,...
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Destructive interference: When ∆r = (n + 1/2)λ, for n = 0, 1, 2, 3,...
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Double-slit interference: Specific double-source interference, in which successive bright fringes are formed, as shown in the diagram below.
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Fringe spacing: s = ​λD/d, where D is the distance between the slits and the screen and d is the distance between the slits.
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Thomas Young experiment:​
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Intensity graph: for negligible slit width.
4.5 Standing Waves
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
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Amplitude: Each particle has its own amplitude (A).
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Nodes: Points of destructive interference, i.e. zero amplitude.
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Anti-nodes: Points of constructive interference, i.e. maximum amplitude.
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Phase: Points between consecutive nodes are in phase.
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No energy transfer: A standing wave does not move horizontally, and thus, no energy is transferred and the shape does not change.
Harmonics
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Resonance: Systems, such as pipes and strings, only resonate at very specific frequencies, which are known as the harmonics.
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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
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Fixed boundary: Always a node, whose reflection causes a 180º phase change.​
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Example: Walls or edge of a drum-head.​
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Free boundary: Always an anti-node, whose reflection does not cause any phase change.
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Example: Tuning fork or air.
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Cases
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Strings (Length = L): The waves reflect at the fixed ends, generating two identical waves travelling in opposite directions.
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End condition: node-node: λn = 2L/n or fn = nv/2L. Walls or edge of a drum-head.​
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Pipes (Length = L): Medium is air - longitudinal waves!
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Longitudinal waves: Nodes are the centers of compression and rarefaction.
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End condition: totally closed: node-node. Equal treatment as strings (above).
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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...
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Comparison between travelling and standing waves
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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!