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)

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:​

pendulum.gif
  • Mass-spring system:

Spring.gif

Graphical representation (SHM)

  • Acceleration-displacement: Negative gradient and direct proportionality.

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

  • 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.​

4.2 Travelling Waves

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:

DistancexDisplacement.jpg
  • Displacement-time:

TimexDisplacement.jpg

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.

electromagneticwave.png
EM-Wave_Final.gif

Electromagnetic spectrum:

spectrum.png
  • 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.

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.

 

 

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".

wavefronts.jpg
interference.jpg

Reflection of pulses

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

  • Free end: Pulse does not invert.

reflectionfixedend.gif
interferenceExtra.jpg
freeend1.png
freeend2.png

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.

Polarization.png
  • 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.

MalusLaw.png
  • 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.

4.4 Wave Behavior

 

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 = = c/v = speed of light in vacuum/speed if light in medium.

        • High n - optically dense medium.

ReflectionRefraction.gif

It's important to remember that, whenever refraction takes place, so does reflection!

airxglass.jpg
ReflectionRefractionAngle.png
  • 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 or 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."

TotalInternalReflection.png
diffraction.png
Diffraction.jpg
  • 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,...

SingleSlitDiffractionGraph.png
SingleSlitDiffraction2.png
Doubleslit3D.gif
maxminintensity.jpg

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

DoubleSlitExtra.jpg
  • Intensity graph: for negligible slit width.

DoubleSlitGraph.jpg
ConsDestInterferenceExtra.jpg

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".

standingwave.gif

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.​

StandingWaveNodeAntinode.jpg
1st2nd3rdHarmonic.jpg
  • 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 

    • 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...

AirTube.jpg

Comparison between travelling and standing waves

TravellingxStanding.png

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!

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