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)


  • Simple pendulum:​

  • Mass-spring system:

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.


  • 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.

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.



  • 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.


  • 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.

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.

  • 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.


  • 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.

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".


  • 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.


  • 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.



  • 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 

    • 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)

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! and are student initiatives to provide free material to help international students prepare themselves for the IB exams.

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