Welcome to AS Physics: Waves
Welcome to one of the most fascinating topics in your CCEA AS Physics course! Waves are everywhere: from the light that allows you to read this page, to the sound of your favourite music, and the Wi-Fi signals connecting your devices. In this chapter, we will break down the fundamental physics of waves into clear, step-by-step concepts.
Don't worry if this seems tricky at first! By mastering a few core definitions, learning how to read wave graphs, and practicing the key equations, you will build the confidence needed to tackle any exam question.
1. Progressive Waves & Wave Fundamentals
What is a Progressive Wave?
A progressive wave is an oscillation or disturbance that transfers energy from one place to another through a medium (or vacuum) without transferring matter.
Everyday Analogy: Imagine doing "the Mexican wave" in a sports stadium. The wave travels all the way around the stadium (energy moves), but you stay in your seat (matter does not travel with the wave).
Transverse vs. Longitudinal Waves
Waves are classified based on the direction of their oscillations relative to the direction of energy transfer:
• Transverse Waves: The oscillations are perpendicular (\(90^\circ\)) to the direction of energy transfer.
Examples: All electromagnetic waves (light, radio, X-rays), water ripples, waves on a plucked guitar string.
Features: Consist of crests (peaks) and troughs.
• Longitudinal Waves: The oscillations are parallel to the direction of energy transfer.
Examples: Sound waves, ultrasound, seismic P-waves, a pushed slinky spring.
Features: Consist of compressions (regions of high particle density and pressure) and rarefactions (regions of low particle density and pressure).
Memory Aid:
• Transverse = T-bone / perpendicular (\(\perp\)).
• Longitudinal = Line / along the same direction (\(\parallel\)).
Key Wave Properties and Definitions
You must know these precise definitions for the exam:
• Displacement (\(x\)): The distance and direction of an oscillating particle from its equilibrium (rest) position. Measured in metres (\(\text{m}\)).
• Amplitude (\(A\)): The maximum displacement of an oscillating particle from its equilibrium position. Measured in metres (\(\text{m}\)).
• Wavelength (\(\lambda\)): The minimum distance between two adjacent points oscillating in phase (e.g., peak to peak, or compression to compression). Measured in metres (\(\text{m}\)).
• Period (\(T\)): The time taken for one complete oscillation to occur. Measured in seconds (\(\text{s}\)).
• Frequency (\(f\)): The number of complete oscillations per unit time passing a given point. 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 (\(\text{m s}^{-1}\)).
Essential Equations
The relationship between frequency and time period is:
\(f = \frac{1}{T}\) or \(T = \frac{1}{f}\)
The Wave Equation connects wave speed, frequency, and wavelength:
\(v = f\lambda\)
Phase and Phase Difference (\(\Delta\phi\))
Phase describes the point that an oscillating particle has reached within its cycle. Phase difference compares the relative positions of two points on a wave, or two different waves.
Phase difference is measured in degrees (\(^\circ\)) or radians (\(\text{rad}\)):
• One full cycle = \(360^\circ = 2\pi\text{ rad}\)
• Half a cycle = \(180^\circ = \pi\text{ rad}\)
• Quarter cycle = \(90^\circ = \frac{\pi}{2}\text{ rad}\)
Two Key Relationships:
1. In Phase: Points are at the exact same stage of their cycle (e.g., two crests). Phase difference is \(0\), \(360^\circ\) (\(2\pi\text{ rad}\)), or any whole number multiple of \(360^\circ\) (\(2n\pi\text{ rad}\)). Their separation distance is a whole number of wavelengths (\(n\lambda\)).
2. In Antiphase (Completely out of phase): One point is at a crest while the other is at a trough. Phase difference is \(180^\circ\) (\(\pi\text{ rad}\)), or an odd multiple of \(180^\circ\) (\((2n+1)\pi\text{ rad}\)). Their separation distance is an odd number of half-wavelengths (\((n + \frac{1}{2})\lambda\)).
To calculate phase difference between two points separated by a distance \(d\):
\(\Delta\phi = \frac{d}{\lambda} \times 360^\circ = \frac{2\pi d}{\lambda}\text{ rad}\)
Interpreting Wave Graphs
Be extremely careful when reading wave graphs in exam questions:
• Displacement–Distance Graph: Represents a "snapshot" photo of the entire wave in space at one instant. The distance between two successive peaks equals the wavelength (\(\lambda\)).
• Displacement–Time Graph: Tracks a single particle's motion at one location over time. The time between two successive peaks equals the time period (\(T\)).
Common Exam Trap: Do not read the wavelength from a displacement-time graph! Always check the label and units on the horizontal axis first.
Key Takeaway
Progressive waves transfer energy without matter. Transverse waves oscillate perpendicular to energy flow, while longitudinal waves oscillate parallel. Use \(v = f\lambda\) and always check your graph axes carefully!
2. The Electromagnetic Spectrum & Polarisation
The Electromagnetic (EM) Spectrum
All electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields. In a vacuum, all EM waves travel at the speed of light, \(c = 3.00 \times 10^8\text{ m s}^{-1}\).
The EM spectrum ordered from longest wavelength / lowest frequency to shortest wavelength / highest frequency:
1. Radio Waves: \(\lambda > 10^{-1}\text{ m}\) (Telecommunications, broadcasting)
2. Microwaves: \(\lambda \approx 10^{-2}\text{ m}\) (Satellite communications, radar, cooking)
3. Infrared (IR): \(\lambda \approx 10^{-5}\text{ m}\) (Thermal imaging, optical fibres, remote controls)
4. Visible Light: \(\lambda \approx 400\text{ nm}\) (violet) to \(700\text{ nm}\) (red)
5. Ultraviolet (UV): \(\lambda \approx 10^{-8}\text{ m}\) (Fluorescence, security marking, tanning)
6. X-rays: \(\lambda \approx 10^{-10}\text{ m}\) (Medical imaging, crystal diffraction)
7. Gamma Rays (\(\gamma\)): \(\lambda < 10^{-12}\text{ m}\) (Cancer treatment, sterilisation)
Mnemonic to remember the order:
Raging Martians Invaded Venus Using X-ray Guns
(Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma)
Polarisation
Unpolarised transverse waves have oscillations occurring in all possible planes perpendicular to the direction of energy propagation.
Plane Polarisation is the process of restricting the oscillations of a transverse wave to a single plane perpendicular to the direction of energy propagation.
Crucial Fact: Only transverse waves can be polarised. Longitudinal waves (like sound) cannot be polarised because their oscillations are already restricted to a single line parallel to the direction of wave travel.
How Polarising Filters (Polaroids) Work:
• When unpolarised light passes through a polarising filter (the polariser), only the component of electric field oscillations aligned with the filter's transmission axis passes through. The transmitted light is plane-polarised, and its intensity drops by half.
• If a second filter (the analyser) is placed behind the first:
– When their transmission axes are parallel (\(0^\circ\)), maximum light passes through.
– When their transmission axes are perpendicular (\(90^\circ\) / crossed), no light passes through.
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 the reflected glare.
• TV and Radio Aerials: Transmitted signals are plane-polarised. The receiving aerial rods must be aligned in the same plane (horizontal or vertical) as the transmitter to receive the strongest signal.
• Stress Analysis: Clear plastics under mechanical stress rotate the plane of polarisation of light, revealing colourful stress patterns when viewed between crossed polaroids.
Key Takeaway
All EM waves travel at \(c = 3.00 \times 10^8\text{ m s}^{-1}\) in a vacuum. Polarisation restricts transverse oscillations to a single plane and provides definitive experimental proof that a wave is transverse.
3. Refraction, Snell's Law & Total Internal Reflection
Refraction & Refractive Index
Refraction is the change in direction of a wave when it crosses a boundary between two different media, caused by a change in wave speed.
When light enters a medium that is optically denser (slower speed):
• Speed \(v\) decreases
• Wavelength \(\lambda\) decreases
• Frequency \(f\) remains constant (frequency depends only on the source)
• The ray bends towards the normal
The absolute refractive index (\(n\)) of a material is defined as:
\(n = \frac{c}{v}\)
where \(c\) is the speed of light in a vacuum (\(3.00 \times 10^8\text{ m s}^{-1}\)) and \(v\) is the speed of light in the material. For air or vacuum, \(n \approx 1.00\).
Snell's Law
Snell's Law relates the angles of incidence and refraction to the refractive indices of the media:
\(n_1 \sin\theta_1 = n_2 \sin\theta_2\)
where:
• \(n_1\) = refractive index of medium 1
• \(\theta_1\) = angle of incidence (measured to the normal)
• \(n_2\) = refractive index of medium 2
• \(\theta_2\) = angle of refraction (measured to the normal)
Common Mistake: Always measure angles from the normal line (the line perpendicular to the surface at \(90^\circ\)), never from the surface interface itself!
Total Internal Reflection (TIR) and the Critical Angle
When light travels from an optically denser medium (higher \(n_1\)) towards a less dense medium (lower \(n_2\)), the ray bends away from the normal.
• 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 (\(\theta_c\) or \(c\)).
• When the angle of incidence \(\theta_1 > \theta_c\), no light refracts; all the light is reflected back into the denser medium. This is Total Internal Reflection (TIR).
Two Necessary Conditions for TIR:
1. Light must travel from a medium of higher refractive index to lower refractive index (\(n_1 > n_2\)).
2. The angle of incidence must be greater than the critical angle (\(\theta_1 > \theta_c\)).
Critical Angle Equation:
Setting \(\theta_1 = \theta_c\) and \(\theta_2 = 90^\circ\) (so \(\sin 90^\circ = 1\)) in Snell's Law:
\(\sin\theta_c = \frac{n_2}{n_1}\)
If medium 2 is air (\(n_2 = 1\)):
\(\sin\theta_c = \frac{1}{n}\)
Step-Index Optical Fibres
Optical fibres transmit data over long distances using total internal reflection.
A step-index optical fibre consists of:
• Core: Narrow central cylinder of high refractive index glass (\(n_{\text{core}}\)).
• Cladding: Outer protective layer of glass with a lower refractive index (\(n_{\text{cladding}} < n_{\text{core}}\)).
Functions of the Cladding:
1. Provides a lower refractive index boundary so that TIR occurs at the core-cladding boundary.
2. Prevents scratches and damage to the core surface.
3. Prevents signal leakage ("cross-talk") between adjacent fibres.
Modal (Multipath) Dispersion:
Light rays entering at different angles travel along paths of different lengths. Rays travelling straight down the centre arrive faster than rays bouncing repeatedly via TIR. This causes pulse broadening (spreading out of pulses), which can lead to overlapping data signals.
Solution: Use a very narrow, single-mode core (diameter \(\approx 8\text{ }\mu\text{m}\)) so light only travels along one direct path.
Key Takeaway
Light slows down in denser media (\(n = c/v\)). TIR only occurs when going from dense to less dense media at angles greater than \(\theta_c\), where \(\sin\theta_c = \frac{n_2}{n_1}\).
4. Superposition, Interference & Diffraction
The Principle of Superposition
When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements of the waves at that point.
• Constructive Interference: Two waves meet in phase (crest meets crest or trough meets trough). Displacements add up to produce a wave of maximum amplitude (\(A_{\text{resultant}} = A_1 + A_2\)).
• Destructive Interference: Two waves meet in antiphase (crest meets trough). Displacements subtract to produce a wave of minimum or zero amplitude (\(A_{\text{resultant}} = |A_1 - A_2|\)).
Coherence
Two wave sources are coherent if they maintain a constant phase difference and have the same frequency (and wavelength).
Did you know? You cannot observe stable interference patterns from two separate light bulbs because they emit random, independent bursts of light photons with continuously changing phase differences (they are incoherent).
Path Difference
Path difference is the difference in distance travelled by two waves from their respective sources to a given meeting point.
For two coherent sources emitting in phase:
• Constructive Interference (Bright fringe / Loud sound):
\(\text{Path Difference} = n\lambda\) where \(n = 0, 1, 2, 3, \dots\) (whole number of wavelengths)
• Destructive Interference (Dark fringe / Quiet sound):
\(\text{Path Difference} = (n + \frac{1}{2})\lambda\) where \(n = 0, 1, 2, 3, \dots\) (odd number of half wavelengths)
Young's Double-Slit Experiment
Thomas Young proved the wave nature of light using double-slit interference. A monochromatic coherent light source shines through two narrow slits separated by a distance \(a\), creating an interference pattern of equally spaced bright and dark fringes on a screen at distance \(D\).
The Double-Slit Equation:
\(\lambda = \frac{a y}{D}\) or \(y = \frac{\lambda D}{a}\)
where:
• \(\lambda\) = wavelength of light (\(\text{m}\))
• \(a\) = separation between the centres of the two slits (\(\text{m}\))
• \(y\) = fringe separation (distance between centres of two consecutive bright or dark fringes) (\(\text{m}\))
• \(D\) = distance from the slits to the screen (\(\text{m}\))
Safety Note: When using laser light in the laboratory, never look directly along the beam or point it at others, and use diffuse screens to avoid specular reflections that could damage the retina.
Diffraction & The Diffraction Grating
Diffraction is the spreading of waves as they pass through an aperture (gap) or around an obstacle. Maximum diffraction occurs when the gap width is approximately equal to the wavelength (\(w \approx \lambda\)).
A diffraction grating is a glass or plastic slide with thousands of closely spaced parallel lines etched onto it per millimetre. When monochromatic light passes through, each slit acts as a coherent source, producing very sharp, bright, widely spaced spectral lines.
The Diffraction Grating Equation:
\(d \sin\theta = n\lambda\)
where:
• \(d\) = grating spacing (distance between adjacent slit centres) in metres (\(\text{m}\))
• \(\theta\) = angle of diffraction for the \(n\text{th}\) order maximum from the central line (\(^\circ\))
• \(n\) = order number of the maximum (\(n = 0\) is the central beam, \(n = 1\) is the first order, etc.)
• \(\lambda\) = wavelength of light (\(\text{m}\))
Calculating Slit Spacing \(d\):
If a grating has \(N\) lines per millimetre, first convert to lines per metre (\(N \times 10^3\text{ lines m}^{-1}\)), then:
\(d = \frac{1}{\text{number of lines per metre}}\)
Example: A grating with \(500\text{ lines mm}^{-1}\) has \(d = \frac{1}{500 \times 10^3} = 2.00 \times 10^{-6}\text{ m}\).
Finding the Maximum Number of Orders:
Since \(\sin\theta\) cannot exceed \(1\) (\(\theta \le 90^\circ\)):
\(n_{\text{max}} \le \frac{d}{\lambda}\)
Always round down to the nearest integer for \(n_{\text{max}}\). The total number of visible bright spots on a screen is \(2n_{\text{max}} + 1\) (accounting for both sides and the central \(n=0\) order).
Key Takeaway
Interference requires coherent sources. Use \(\lambda = \frac{ay}{D}\) for double slits and \(d\sin\theta = n\lambda\) for diffraction gratings. Remember to round \(n_{\text{max}}\) down to the nearest integer!
5. Stationary (Standing) Waves
How Stationary Waves Form
A stationary wave (or standing wave) is formed by the superposition of two progressive waves of the same frequency and amplitude, travelling in opposite directions along the same line.
Nodes and Antinodes
Unlike progressive waves, stationary waves do not transfer net energy through the medium:
• Nodes (N): Points of zero amplitude and zero energy transfer. Destructive interference occurs here permanently.
• Antinodes (A): Points of maximum amplitude. Constructive interference occurs here permanently.
Key Spacing Rules:
• Distance between two adjacent nodes = \(\frac{\lambda}{2}\)
• Distance between two adjacent antinodes = \(\frac{\lambda}{2}\)
• Distance between an adjacent node and antinode = \(\frac{\lambda}{4}\)
Progressive vs. Stationary Waves: Quick Comparison
• Energy Transfer: Progressive waves transfer energy in the direction of travel; stationary waves store energy without net transfer.
• Amplitude: In progressive waves, all particles oscillate with the same amplitude; in stationary waves, amplitude varies from zero at nodes to maximum at antinodes.
• Phase: In progressive waves, phase varies continuously across one wavelength; in stationary waves, all particles between two adjacent nodes are in phase, while particles on opposite sides of a node are in antiphase (\(180^\circ\) / \(\pi\text{ rad}\)).
Stationary Waves on Stretched Strings (Fixed at Both Ends)
Because the string is clamped at both ends, each fixed end must be a Node.
For a string of length \(L\):
• Fundamental / 1st Harmonic (\(f_1\)):
Pattern: Node–Antinode–Node (1 loop)
\(L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L\)
Frequency: \(f_1 = \frac{v}{2L}\)
• 2nd Harmonic / 1st Overtone (\(f_2 = 2f_1\)):
Pattern: N–A–N–A–N (2 loops)
\(L = \lambda_2 \implies \lambda_2 = L\)
Frequency: \(f_2 = \frac{v}{L} = 2f_1\)
• 3rd Harmonic / 2nd Overtone (\(f_3 = 3f_1\)):
Pattern: N–A–N–A–N–A–N (3 loops)
\(L = \frac{3\lambda_3}{2} \implies \lambda_3 = \frac{2L}{3}\)
Frequency: \(f_3 = \frac{3v}{2L} = 3f_1\)
In general for strings fixed at both ends: \(\lambda_n = \frac{2L}{n}\) and \(f_n = n f_1\), where \(n = 1, 2, 3, \dots\)
Stationary Waves in Air Columns (Pipes)
Boundary rules for sound waves in pipes:
• Closed End: Air molecules cannot move \(\implies\) Displacement Node (N).
• Open End: Air molecules are free to move with maximum displacement \(\implies\) Displacement Antinode (A).
1. Pipe Closed at One End (Open at the other):
• Fundamental (1st Harmonic): Node at closed end, Antinode at open end.
\(L = \frac{\lambda_1}{4} \implies \lambda_1 = 4L \implies f_1 = \frac{v}{4L}\)
• Next Harmonic (3rd Harmonic): \(L = \frac{3\lambda_3}{4} \implies \lambda_3 = \frac{4L}{3} \implies f_3 = 3f_1\)
Important: Closed pipes produce only odd harmonics (\(f_1, 3f_1, 5f_1, \dots\)).
2. Pipe Open at Both Ends:
• Fundamental (1st Harmonic): Antinodes at both open ends, Node in the centre.
\(L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L \implies f_1 = \frac{v}{2L}\)
• Produces all harmonics (\(f_1, 2f_1, 3f_1, 4f_1, \dots\)).
Key Takeaway
Stationary waves store energy between nodes (zero amplitude) and antinodes (maximum amplitude). The distance between consecutive nodes is \(\frac{\lambda}{2}\). Strings and open pipes produce all integer harmonics, while closed-end pipes produce only odd harmonics.
Chapter Summary & Quick Formula Sheet
Keep these fundamental formulas at your fingertips for your CCEA AS exam:
• Frequency & Period: \(f = \frac{1}{T}\)
• Wave Equation: \(v = f\lambda\)
• Refractive Index: \(n = \frac{c}{v}\)
• Snell's Law: \(n_1 \sin\theta_1 = n_2 \sin\theta_2\)
• Critical Angle: \(\sin\theta_c = \frac{n_2}{n_1}\) (\(\sin\theta_c = \frac{1}{n}\) for air)
• Young's Double Slit: \(\lambda = \frac{a y}{D}\)
• Diffraction Grating: \(d\sin\theta = n\lambda\)
• Node-to-Node Distance: \(\text{Separation} = \frac{\lambda}{2}\)