Introduction: Welcome to the World of Waves!
Have you ever wondered how music travels from your headphones into your ears, how sunlight reaches our planet through the vacuum of space, or how your smartphone downloads a video using Wi-Fi? The answer to all of these questions is waves!
In this chapter of your CCEA AS 2 Physics course, we will explore the fundamental properties of waves. Don't worry if physics sometimes feels intimidating—we will break down every concept into small, manageable steps with clear everyday examples. Let's dive in!
1. What is a Progressive Wave?
A progressive wave is an oscillation or vibration that transfers energy from one place to another through a medium (or vacuum) without transferring any matter.
Everyday Analogy: Imagine doing the "Mexican Wave" in a sports stadium. You stand up and sit down in your seat. You haven't moved to the other side of the stadium, but the wave itself travelled all the way around the arena! The people (matter) stay in their places, while the ripple (energy) moves.
Types of Waves: Transverse vs. Longitudinal
All progressive waves can be classified into one of two categories based on the direction of their oscillations relative to the direction of energy transfer:
1. Transverse Waves
• In a transverse wave, the oscillations of the particles are perpendicular (\(90^\circ\)) to the direction of energy propagation.
• Features: Peaks (crests) and troughs.
• Examples: All electromagnetic waves (light, radio, X-rays), water waves, S-waves in earthquakes, and waves on a plucked guitar string.
2. Longitudinal Waves
• In a longitudinal wave, the oscillations of the particles are parallel to the direction of energy propagation.
• Features: Compressions (regions where particles are squashed together, high pressure) and Rarefactions (regions where particles are spread apart, low pressure).
• Examples: Sound waves, ultrasound, and P-waves in earthquakes.
Common Mistake to Avoid: Never say "longitudinal waves move back and forth". You must use the word oscillate or vibrate, and state clearly that the oscillations are parallel to the direction of energy transfer.
Key Takeaway
Progressive waves transfer energy, not matter. In transverse waves, oscillations are perpendicular to wave travel. In longitudinal waves, oscillations are parallel to wave travel.
2. Describing Wave Characteristics
To analyse waves mathematically, we need a precise vocabulary. Here are the core definitions you must learn for your exams:
• Displacement (\(x\) or \(y\)): The distance and direction of an oscillating particle from its equilibrium (rest) position. Measured in metres (\(\text{m}\)).
• Amplitude (\(A\)): The maximum displacement of a particle from its equilibrium position. Measured in metres (\(\text{m}\)). Amplitude is a measure of the wave's energy.
• Wavelength (\(\lambda\)): The minimum distance between two adjacent points on a wave that are in phase (e.g., from peak to peak, or compression to compression). Measured in metres (\(\text{m}\)).
• Period (\(T\)): The time taken for one complete wave oscillation to pass a given point. Measured in seconds (\(\text{s}\)).
• Frequency (\(f\)): The number of complete wave cycles passing a fixed point per unit time (per second). Measured in hertz (\(\text{Hz}\)), where \(1\text{ Hz} = 1\text{ s}^{-1}\).
• Wave Speed (\(v\)): The speed at which energy is transmitted through the medium. Measured in metres per second (\(\text{m s}^{-1}\)).
Key Equations
Frequency and period are inversely related:
\(f = \frac{1}{T}\) \(\implies\) \(T = \frac{1}{f}\)
The Wave Equation relates speed, frequency, and wavelength:
\(v = f\lambda\)
Step-by-Step Example:
A sound wave has a frequency of \(440\text{ Hz}\) and travels through air at a speed of \(330\text{ m s}^{-1}\). Calculate its wavelength.
1. State the formula: \(v = f\lambda\)
2. Rearrange for wavelength: \(\lambda = \frac{v}{f}\)
3. Substitute the values: \(\lambda = \frac{330}{440} = 0.75\text{ m}\)
Understanding Wave Graphs: Look at the Axes!
In physics exam papers, you will encounter two types of wave graphs. Pay close attention to the x-axis:
• Displacement-Distance Graph: Represents a "snapshot" of the whole wave in space at a single instant in time. The distance between two consecutive peaks gives the wavelength (\(\lambda\)).
• Displacement-Time Graph: Shows how a single particle moves up and down over time. The time between two consecutive peaks gives the time period (\(T\)).
Phase and Phase Difference (\(\phi\))
Phase describes the point that an oscillating particle has reached within its cycle.
• Two points are in phase when they are at the exact same point in their cycle and moving in the same direction. Their phase difference is \(0\) or a whole number multiple of \(360^\circ\) (\(2\pi\text{ rad}\)).
• Two points are in antiphase (completely out of phase) when one is at a maximum positive displacement while the other is at a maximum negative displacement. Their phase difference is \(180^\circ\) (\(\pi\text{ rad}\)).
To calculate the phase difference \(\Delta \phi\) between two points separated by a distance \(x\):
\(\Delta \phi = \frac{x}{\lambda} \times 360^\circ\) (in degrees) or \(\Delta \phi = \frac{x}{\lambda} \times 2\pi\) (in radians)
Key Takeaway
Master the wave equation \(v = f\lambda\). Always check the x-axis of a graph: distance gives \(\lambda\), while time gives \(T\).
3. The Electromagnetic (EM) Spectrum
All electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields at right angles to each other. They do not require a medium and all travel at the same speed in a vacuum: the speed of light, \(c = 3.00 \times 10^8\text{ m s}^{-1}\).
The EM Spectrum in Order
From longest wavelength / lowest frequency to shortest wavelength / highest frequency:
1. Radio Waves: Wavelength \(\gt 10^{-1}\text{ m}\) (telecommunications, broadcasting)
2. Microwaves: Wavelength \(\approx 10^{-2}\text{ m}\) (cooking, satellite communication, mobile phones)
3. Infrared (IR): Wavelength \(\approx 10^{-5}\text{ m}\) (thermal imaging, remote controls, heating)
4. Visible Light: Wavelength \(\approx 400\text{ nm}\) (violet) to \(700\text{ nm}\) (red)
5. Ultraviolet (UV): Wavelength \(\approx 10^{-8}\text{ m}\) (tanning beds, detecting counterfeit notes, sterilisation)
6. X-Rays: Wavelength \(\approx 10^{-10}\text{ m}\) (medical bone imaging, airport security)
7. Gamma Rays (\(\gamma\)): Wavelength \(\lt 10^{-12}\text{ m}\) (cancer radiotherapy, sterilising surgical instruments)
Helpful Mnemonic to Remember the Order:
Raging Martians Invaded Venus Using X-ray Guns
(Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma)
Key Takeaway
All EM waves travel at \(c = 3.00 \times 10^8\text{ m s}^{-1}\) in a vacuum. As wavelength decreases across the spectrum, frequency and photon energy increase.
4. Polarisation
Polarisation is a property exclusive to transverse waves. It provides definitive evidence that light is a transverse wave and not a longitudinal wave.
What is Plane Polarisation?
• In unpolarised light, oscillations occur in many planes perpendicular to the direction of wave travel.
• In plane-polarised light, the oscillations are restricted to a single plane perpendicular to the direction of energy propagation.
The Fence Analogy: Imagine shaking a rope through the vertical slats of a wooden fence. If you shake the rope up and down (vertical oscillations), the wave passes straight through the vertical slats. If you shake the rope side to side (horizontal oscillations), the slats block the wave completely!
Polaroid Filters
• When unpolarised light passes through a polarising filter (Polaroid), only the component of light vibrating parallel to the filter's transmission axis is transmitted.
• If two polarising filters are placed one after the other with their transmission axes crossed at \(90^\circ\), no light passes through the second filter (complete extinction).
Applications of Polarisation
• Polaroid Sunglasses: Light reflected from surfaces like water or roads is partially horizontally polarised (glare). Polaroid lenses have vertical transmission axes, blocking this glare and improving visibility.
• Liquid Crystal Displays (LCDs): Use polarisation of light to switch pixels on and off in screens and digital watches.
• Stress Analysis: Transparent plastics placed between crossed polarising filters reveal colourful patterns showing areas of mechanical stress.
Did You Know? Longitudinal waves (like sound) cannot be polarised because their oscillations are already along a single line—parallel to the direction of wave travel.
Key Takeaway
Polarisation restricts oscillations to a single plane. Because only transverse waves have oscillations perpendicular to wave travel, only transverse waves can be polarised.
5. Reflection, Refraction, and Total Internal Reflection
Reflection
When any wave hits a barrier, it reflects.
• Law of Reflection: The angle of incidence (\(i\)) is equal to the angle of reflection (\(r\)).
• Angles are always measured between the ray and the normal (an imaginary line drawn perpendicular to the surface at \(90^\circ\)).
Refraction & Snell's Law
Refraction is the change in direction of a wave as it passes across a boundary from one medium into another, caused by a change in wave speed.
• When light enters a denser medium (e.g., air to glass), it slows down and bends towards the normal (\(r \lt i\)).
• When light enters a less dense medium (e.g., glass to air), it speeds up and bends away from the normal (\(r \gt i\)).
• During refraction, the frequency remains constant. Therefore, because \(v = f\lambda\), if speed \(v\) decreases, wavelength \(\lambda\) must also decrease.
The refractive index (\(n\)) of a material is a measure of how much it slows down light:
\(n = \frac{c}{v}\)
where \(c\) is the speed of light in a vacuum and \(v\) is the speed of light in the material.
Snell's Law:
\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)
For light passing from air (\(n_1 \approx 1\)) into a medium of refractive index \(n\) (\(n_2 = n\)):
\(n = \frac{\sin i}{\sin r}\)
Total Internal Reflection (TIR) and the Critical Angle
When light travels from an optically denser medium to a less dense medium (e.g., glass to air):
1. At small angles of incidence, light refracts away from the normal with a weak reflected ray.
2. As the angle of incidence increases, the angle of refraction reaches \(90^\circ\). The angle of incidence at which this happens is called the critical angle (\(c\) or \(\theta_c\)).
3. If the angle of incidence is increased beyond the critical angle (\(i \gt \theta_c\)), no light refracts out; all light is reflected back into the denser medium. This is Total Internal Reflection (TIR).
Formula for Critical Angle:
Using Snell's law with \(\theta_1 = \theta_c\) and \(\theta_2 = 90^\circ\) (\(\sin 90^\circ = 1\)):
\(\sin \theta_c = \frac{n_2}{n_1}\)
For a boundary with air (\(n_2 = 1\)):
\(\sin \theta_c = \frac{1}{n}\)
Two Essential Conditions for TIR to Occur:
1. The light must be travelling from an optically denser medium to an optically less dense medium (\(n_1 \gt n_2\)).
2. The angle of incidence must be greater than the critical angle (\(i \gt \theta_c\)).
Step-by-Step Optical Fibre Example
Optical fibres use TIR to transmit light signals over vast distances.
• Structure: A central glass core surrounded by a glass cladding of slightly lower refractive index (\(n_{\text{core}} \gt n_{\text{cladding}}\)).
• Function of Cladding: Protects the core from scratches, prevents signal leakage (crosstalk) between adjacent fibres, and ensures the boundary conditions for TIR are maintained.
Key Takeaway
Refraction occurs because waves change speed at boundaries. Total internal reflection happens only when light goes from high to low refractive index at an angle greater than the critical angle (\(\sin \theta_c = \frac{1}{n}\)).
6. The Principle of Superposition & Interference
The Principle of Superposition
When two or more waves meet at a point in space, the resultant displacement at that point is equal to the vector (algebraic) sum of the individual displacements of the waves.
Interference
When two continuous waves superpose, they produce an interference pattern:
• Constructive Interference: Occurs when two waves arrive in phase (peaks meet peaks, troughs meet troughs). The amplitudes add together to produce a wave of maximum displacement (\(A_{\text{total}} = A_1 + A_2\)).
• Destructive Interference: Occurs when two waves arrive in antiphase (a peak meets a trough). The displacements cancel each other out (\(A_{\text{total}} = |A_1 - A_2|\)). If the amplitudes are equal, the resultant displacement is zero!
Path Difference
The path difference is the difference in distance travelled by two waves from their respective coherent sources to the point where they meet.
• For Constructive Interference (bright fringe / loud sound):
\(\text{Path Difference} = n\lambda\) (where \(n = 0, 1, 2, 3, \dots\))
• For Destructive Interference (dark fringe / quiet sound):
\(\text{Path Difference} = (n + \frac{1}{2})\lambda\) (where \(n = 0, 1, 2, 3, \dots\))
What are Coherent Sources?
Two wave sources are said to be coherent if they have a constant phase difference (which requires them to have the same frequency/wavelength).
Key Takeaway
Superposition means adding displacements. Whole-wavelength path differences (\(n\lambda\)) give constructive interference; half-wavelength path differences (\((n + \frac{1}{2})\lambda\)) give destructive interference.
7. Stationary (Standing) Waves
How is a Stationary Wave Formed?
A stationary wave is formed by the superposition of two progressive waves that have:
• The same frequency (and wavelength)
• The same amplitude (or similar amplitude)
• Travelling in opposite directions along the same line.
This typically happens when a progressive wave travels along a medium (like a string or air column) and reflects back on itself from a fixed boundary.
Nodes and Antinodes
Unlike progressive waves, stationary waves do not transmit energy through the medium. Instead, they trap energy in a fixed pattern consisting of:
• Nodes (N): Points along the wave where the amplitude of vibration is always zero (caused by complete destructive interference).
• Antinodes (A): Points along the wave where the amplitude of vibration is at its maximum (caused by constructive interference).
Important Spatial Relationships:
• Distance between two adjacent nodes: \(\frac{\lambda}{2}\)
• Distance between two adjacent antinodes: \(\frac{\lambda}{2}\)
• Distance between a node and the next adjacent antinode: \(\frac{\lambda}{4}\)
Key Differences: Progressive vs. Stationary Waves
This is a classic CCEA exam question. Learn this comparison carefully:
• Energy Transfer: Progressive waves transfer energy from place to place. Stationary waves store/trap energy; there is no net transfer of energy.
• Amplitude: In progressive waves, all particles vibrate with the same amplitude. In stationary waves, amplitude varies from zero at nodes to maximum at antinodes.
• Phase: In progressive waves, phase varies continuously along the wave. In stationary waves, all particles between two adjacent nodes vibrate in phase with each other, and in antiphase with particles in the adjacent inter-node section.
Harmonics on Stretched Strings
When a string fixed at both ends is plucked, standing waves form with a node at each fixed end:
• Fundamental Mode (1st Harmonic):
Pattern: Node - Antinode - Node (N-A-N)
Length of string: \(L = \frac{\lambda_1}{2}\) \(\implies\) \(\lambda_1 = 2L\)
Fundamental frequency: \(f_1 = \frac{v}{2L}\)
• Second Harmonic:
Pattern: N-A-N-A-N (two complete loops)
Length of string: \(L = \lambda_2\) \(\implies\) \(\lambda_2 = L\)
Frequency: \(f_2 = 2f_1 = \frac{v}{L}\)
• Third Harmonic:
Pattern: N-A-N-A-N-A-N (three loops)
Length of string: \(L = \frac{3\lambda_3}{2}\) \(\implies\) \(\lambda_3 = \frac{2L}{3}\)
Frequency: \(f_3 = 3f_1 = \frac{3v}{2L}\)
Key Takeaway
Stationary waves form when two identical waves travelling in opposite directions superpose. They have nodes (zero displacement) and antinodes (maximum displacement) and transfer no net energy.
Quick Summary Checklist for Revision
Before sitting your exam, make sure you can confidently:
1. State the difference between transverse and longitudinal waves with examples.
2. Calculate wave speed, frequency, wavelength, and period using \(v = f\lambda\) and \(f = \frac{1}{T}\).
3. List the 7 regions of the EM spectrum in order of frequency and wavelength.
4. Explain polarisation and state why it proves light is a transverse wave.
5. Apply Snell's law and calculate the critical angle for total internal reflection (\(\sin \theta_c = \frac{1}{n}\)).
6. Explain path difference and conditions for constructive (\(n\lambda\)) and destructive (\((n + \frac{1}{2})\lambda\)) interference.
7. Describe how stationary waves form and list three key differences between stationary and progressive waves.