Stage 6 Physics: Module 3 — Waves
Submodule 1: Wave Properties (Study Guide)
1. Introduction to Wave Motion
A wave is a disturbance that transfers energy from one point to another through space or a medium without transferring matter.
The Key Principle
Particles of the medium oscillate (vibrate) locally around a fixed equilibrium position.
The wave energy travels continuously forward.
2. Mechanical vs. Electromagnetic Waves
Waves are broadly classified based on whether they require a physical medium to propagate.
| Property | Mechanical Waves | Electromagnetic (EM) Waves |
| Medium Requirement | Must have a physical medium (solid, liquid, or gas). Cannot travel through a vacuum. | Can travel through a vacuum and through transparent media. |
| Mechanism | Relies on the elastic properties and particle interactions of the medium. | Relies on oscillating, mutually perpendicular electric and magnetic fields. |
| Speed | Relatively slow (e.g., sound in air $\approx$ 340 m/s). | Travels at the speed of light ($c \approx 3 \times 10^8 \text{ m/s}$ in a vacuum). |
| Examples | Sound waves, water waves, seismic S and P waves, vibrating strings. | Light, X-rays, Microwaves, Radio waves, UV radiation. |
3. Wave Classification: Transverse vs. Longitudinal
Waves are categorized by how the direction of particle oscillation compares to the direction of wave energy transfer.
.gif){kind=spacer}
A. Transverse Waves
Definition: Waves in which the particles of the medium oscillate perpendicularly (at $180^\circ$) to the direction of energy propagation.
Crest: The point of maximum positive displacement from the equilibrium line.
Trough: The point of maximum negative displacement from the equilibrium line.
Examples: Water surface ripples, secondary seismic waves (S-waves), all electromagnetic waves.
B. Longitudinal Waves
Definition: Waves in which the particles of the medium oscillate parallel to the direction of energy propagation.
Compression: Region where the particles are crowded close together (high pressure/density).
Rarefaction: Region where the particles are spread far apart (low pressure/density).
Examples: Sound waves in air, primary seismic waves (P-waves), a compressed slinky pulse.
4. Graphical Representations of Waves
To extract numeric data from a wave, physicists map it using two distinct graphs. Pay close attention to the horizontal axis label!
Graph 1: Displacement-Distance Graph
Think of this as a snapshot freeze-frame photo of the entire wave at one single moment in time.
What it shows: The position of all particles along the wave's path.
Values you can extract:
Amplitude ($A$): The maximum displacement from equilibrium (measured in meters).
Wavelength ($\lambda$): The distance between two consecutive points in phase (e.g., crest to crest, or compression to compression, measured in meters).
Graph 2: Displacement-Time Graph
Think of this as a video tracking a single particle over a period of time as it bounces up and down or back and forth.
What it shows: The history of motion for one specific coordinate point.
Values you can extract:
Amplitude ($A$): The maximum displacement from equilibrium.
Period ($T$): The total time taken for one single particle to complete one full oscillation cycle (measured in seconds).
5. Quantitative Wave Equations
The Period-Frequency Relationship
Frequency ($f$) is the number of complete wave cycles that pass a given point per second, measured in Hertz (Hz).
The Universal Wave Equation
Because speed is $\text{distance} \div \text{time}$, a wave travels one full wavelength ($\lambda$) in exactly one period ($T$). Substituting $f = \frac{1}{T}$ yields:
Where:
$v = \text{wave velocity } (\text{m/s})$
$f = \text{frequency } (\text{Hz})$
$\lambda = \text{wavelength } (\text{m})$
Crucial Rule: The velocity ($v$) of a wave is determined solely by the properties of the medium (e.g., density, temperature, tension). If frequency increases in a stable medium, wavelength must decrease proportionally to keep $v$ constant.
6. Worked Examples
Example 1: Finding Period and Frequency
A red laser pointer emits light with a wavelength of $6.5 \times 10^{-7} \text{ m}$ through air. Given that the speed of light is $3.0 \times 10^8 \text{ m/s}$, calculate the frequency and period of this light wave.
Step 1: Identify known variables
$v = 3.0 \times 10^8 \text{ m/s}$
$\lambda = 6.5 \times 10^{-7} \text{ m}$
Step 2: Solve for frequency ($f$)
Step 3: Solve for period ($T$)
7. High-Yield Challenge Questions
Try solving these to test your understanding before your assessments:
Conceptual: A sound wave transitions from cool air into a warm air boundary where its speed increases. Explain what happens to its frequency and its wavelength.
Graphical Interpretation: If a wave source completes 240 oscillations in one minute, determine its frequency in Hz and its period in seconds.
Calculation: An ocean swell is observed to have a distance of $15 \text{ meters}$ between successive wave crests. An observer notes that 8 wave crests pass a marker buoy in exactly $20 \text{ seconds}$. Calculate the velocity of the ocean waves.
Submodule 2: Wave Behaviour
Inquiry Question: How do waves behave when they encounter barriers, changes in medium, or other waves?
This submodule covers the core physical phenomena that govern all wave types.
Key Content Focus Areas:
Reflection: Analyze how pulses bounce off fixed boundaries (phase inversion) versus free boundaries (no inversion), and apply the Law of Reflection.
Refraction: Explain the bending of waves using Snell's Law and changes in wave velocity when moving between media of different refractive indices:
$$\frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2} = \frac{\sin(\theta_1)}{\sin(\theta_2)}$$Diffraction: Investigate the spreading of waves through gaps or around obstacles, noting why the effect is most pronounced when the gap size matches the wavelength ($\lambda \approx w$).
Superposition & Interference: Apply the principle of superposition to construct resultant wave diagrams for constructive and destructive interference.
Stage 6 Physics: Module 3 — Waves
Submodule 2: Wave Behaviour (Study Guide)
1. Reflection: Interacting with Boundaries
When a wave pulse encounters a boundary or the end of a medium, it doesn't just disappear—part or all of it bounces back into the original medium. The way it reflects depends entirely on whether the boundary is fixed or free.
A. Fixed Boundary Reflection (Phase Inversion)
The Scenario: The end of the medium is securely anchored to an unyielding wall or a much denser medium.
The Behaviour: When a crest hits a fixed boundary, it exerts an upward force on the wall. According to Newton’s Third Law, the wall exerts an equal and opposite downward force on the medium.
The Result: The wave reflects with a $180^\circ$ phase inversion (a crest reflects back as a trough).
B. Free Boundary Reflection (No Inversion)
The Scenario: The end of the medium is free to move up and down (e.g., a string tied to a light ring sliding frictionlessly on a vertical pole).
The Behaviour: As the crest reaches the free end, the medium suffers no restraining force and actually overshoots its equilibrium position.
The Result: The wave reflects in phase (a crest reflects back as a crest).
C. The Law of Reflection
For two-dimensional or three-dimensional wavefronts (like light reflecting off a mirror or water waves hitting a sea wall), the path can be modeled using rays.
Where:
$\theta_i = \text{Angle of incidence}$ (measured between the incoming ray and the perpendicular normal line)
$\theta_r = \text{Angle of reflection}$ (measured between the reflected ray and the normal line)
2. Refraction: Changing Media
Refraction is the bending of a wave path as it crosses a boundary from one medium to another. This bending occurs because the wave changes its speed.
The Mechanics of Refraction
Frequency ($f$) stays CONSTANT: The source determining how many wave cycles are produced per second does not change. Therefore, $f_1 = f_2$.
Speed ($v$) changes: The physical properties of the new medium alter the propagation speed.
Wavelength ($\lambda$) adjusts: Because $v = f\lambda$, if the wave slows down ($v \downarrow$), its wavelength must shorten ($\lambda \downarrow$) to match.
Snell's Law and Mathematical Modeling
The geometric relationship governing this change is expressed through Snell's Law:
Directional Trends to Memorize:
Fast Medium $\rightarrow$ Slow Medium: Wave bends towards the normal ($\theta_2 < \theta_1$). Wavelength compresses.
Slow Medium $\rightarrow$ Fast Medium: Wave bends away from the normal ($\theta_2 > \theta_1$). Wavelength stretches.
3. Diffraction: Bending Around Obstacles
Diffraction is the spreading or bending of wavefronts as they pass through a narrow gap (aperture) or edge around an obstacle. It is a defining characteristic of wave behavior that particles cannot replicate.
The Crucial Condition ($\lambda \approx w$)
The extent of diffraction depends heavily on the relationship between the wavelength ($\lambda$) of the wave and the width ($w$) of the gap:
When $w \gg \lambda$ (Wide Gap): The wave passes straight through with minimal bending at the very edges.
When $w \approx \lambda$ or $w < \lambda$ (Narrow Gap): Significant diffraction occurs. The gap begins acting like a completely new point source, sending out highly curved, circular wavefronts.
Real-World Application: You can hear someone talking around a corner even if you cannot see them. This happens because sound waves have long wavelengths ($0.1\text{m}$ to $10\text{m}$) that easily diffract around standard architectural doorways and corners ($w \approx 1\text{m}$). Light waves have microscopic wavelengths ($\approx 10^{-7}\text{m}$), meaning they pass straight through without noticeable macroscopic diffraction.
4. Superposition and Interference
Unlike solid matter, two or more waves can occupy the exact same point in space at the exact same time.
The Principle of Superposition
When two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements of each wave.
$$y_{\text{total}} = y_1 + y_2 + y_3 + \dots$$
Types of Interference
| Type | Condition | Visual Outcome |
| Constructive Interference | Waves meet completely in-phase (Crest meets Crest / Trough meets Trough). | The displacements reinforce each other, creating a larger resultant amplitude. |
| Destructive Interference | Waves meet out-of-phase (Crest meets Trough). | The displacements counteract each other, reducing or completely canceling out the amplitude. |
5. Worked Examples
Example 1: Calculating Refraction
A deep-water ocean wavefront approaches a shallow sandbar at an angle of $35^\circ$ relative to the normal. In deep water, the waves travel at $4.5 \text{ m/s}$ with a wavelength of $12 \text{ meters}$. Upon entering the shallow water over the sandbar, the wave speed drops to $2.8 \text{ m/s}$.
Calculate the new angle of refraction ($\theta_2$) and the new wavelength ($\lambda_2$).
Step 1: Calculate the new wavelength ($\lambda_2$)
Step 2: Calculate the angle of refraction ($\theta_2$)
The wave slowed down and bent closer to the normal line ($20.9^\circ < 35^\circ$), verifying our conceptual rules.
6. High-Yield Challenge Questions
Try working through these practice problems to consolidate your understanding:
Conceptual Challenge: A pulse traveling along a thin string reaches a junction where it connects to a significantly thicker, heavier rope. Describe the nature of the reflected pulse and the transmitted pulse at this boundary interface.
Graphical Assignment: Sketch a step-by-step vector diagram showing two square wave pulses of equal amplitude ($A = 2\text{ cm}$) traveling toward each other from opposite directions at the exact instant they perfectly overlap if:
Both are positive crest pulses.
One is a positive crest pulse and the other is a negative trough pulse.
Diffraction Reasoning: AM radio stations use frequencies around $1000\text{ kHz}$ ($\lambda \approx 300\text{ m}$), while FM stations use frequencies around $100\text{ MHz}$ ($\lambda \approx 3\text{ m}$). Use your knowledge of diffraction to explain why AM radio signals can generally be received behind hills and deep in valleys where FM signals fail.
Stage 6 Physics: Module 3 — Waves
Submodule 3: Sound Waves
Inquiry Question: Why are sound waves useful?
Sound waves are longitudinal mechanical waves. They require a physical medium to travel through and propagate via a series of alternating compressions (high-pressure regions) and rarefactions (low-pressure regions).
1. The Nature of Sound & Human Perception
The objective, measurable physical properties of a sound wave directly dictate how it is subjectively perceived by the human ear.
Frequency $\rightarrow$ Pitch: * Frequency ($f$) is the number of wave cycles passing a point per second (Hz).
The higher the frequency, the higher the perceived pitch of the sound. The normal human hearing range is roughly 20 Hz to 20,000 Hz.
Amplitude $\rightarrow$ Loudness: * Amplitude ($A$) corresponds to the maximum pressure variation from equilibrium.
A larger amplitude carries more energy, which the human auditory system perceives as increased loudness (measured on a logarithmic decibel scale).
2. The Doppler Effect
The Doppler Effect is the apparent change in the frequency of a wave due to the relative motion between the wave source and an observer.
Moving Towards Each Other: Wavefronts are bunched closer together. The observer detects a shorter apparent wavelength, resulting in a higher apparent frequency ($f' > f$).
Moving Away From Each Other: Wavefronts are stretched further apart. The observer detects a longer apparent wavelength, resulting in a lower apparent frequency ($f' < f$).
The general quantitative relationship is modeled by:
Sign Convention Rule: > * Numerator ($v_{\text{observer}}$): Use $+$ if the observer moves towards the source; use $-$ if moving away.
Denominator ($v_{\text{source}}$): Use $-$ if the source moves towards the observer; use $+$ if moving away.
3. Standing Waves & Resonance
When a sound wave is confined within a boundary (like a pipe or a stretched string), it reflects back on itself. If the incoming and reflected waves have the same frequency and amplitude, they superpose to form a standing wave.
Nodes ($N$): Points of zero particle displacement due to absolute destructive interference.
Antinodes ($A$): Points of maximum particle displacement due to absolute constructive interference.
Acoustic Systems and Harmonics
| Boundary Condition | Standing Wave Boundary States | First Harmonic (Fundamental) | Harmonic Series Allowed |
| Stretched Strings | Fixed ends must be Nodes. | $\lambda_1 = 2L \implies f_1 = \frac{v}{2L}$ | All integers ($n = 1, 2, 3, 4\dots$) |
| Open Air Columns (Open both ends) | Open ends must be Antinodes. | $\lambda_1 = 2L \implies f_1 = \frac{v}{2L}$ | All integers ($n = 1, 2, 3, 4\dots$) |
| Closed Air Columns (Closed at one end) | Closed end = Node Open end = Antinode | $\lambda_1 = 4L \implies f_1 = \frac{v}{4L}$ | Odd integers only ($n = 1, 3, 5\dots$) |
4. Acoustic Measurements: The Inverse-Square Law
Sound waves spread out spherically from a point source. As the wave front expands, the total power ($P$) emitted by the source is distributed over a larger surface area ($4\pi r^2$).
Sound Intensity ($I$) is defined as power per unit area:
Because power stays constant, intensity is inversely proportional to the square of the distance ($r$) from the source:
If you double your distance from a sound source ($2r$), the perceived sound intensity drops to one-quarter ($\frac{1}{4}$) of its original value.
Submodule 4: Ray Model of Light
Inquiry Question: What evidence supports the ray model of light?
The ray model of light treats light as traveling in straight lines called "rays." This geometric approach simplifies the analysis of how light behaves when reflecting off surfaces or crossing optical boundaries.
1. Refractive Index & Snell’s Law
When light transitions from one transparent medium to another, its speed shifts depending on the optical density of the material. The refractive index ($n$) of a medium is a dimensionless ratio comparing the speed of light in a vacuum ($c \approx 3 \times 10^8 \text{ m/s}$) to its velocity in that medium ($v$):
Snell's Law
Snell’s Law calculates the exact geometric bending angle of a light ray as it strikes a boundary line between two media with differing refractive indices:
Into a denser medium ($n_2 > n_1$): Light slows down and bends towards the normal ($\theta_2 < \theta_1$).
Into a less dense medium ($n_2 < n_1$): Light speeds up and bends away from the normal ($\theta_2 > \theta_1$).
2. Total Internal Reflection (TIR)
When light travels from an optically denser medium to a less dense medium ($n_1 > n_2$), the refracted ray bends away from the normal. As the angle of incidence ($\theta_1$) increases, the angle of refraction ($\theta_2$) eventually reaches $90^\circ$, skimming perfectly along the boundary surface.
The angle of incidence that causes this configuration is called the critical angle ($\theta_c$):
If the angle of incidence exceeds the critical angle ($\theta_1 > \theta_c$), no refraction can occur. Instead, 100% of the light energy is reflected back into the initial medium. This phenomenon forms the operational basis of fiber-optic communication cables.
3. Mirrors and Lenses (Geometric Optics)
Light rays can be systematically traced to locate and define the images produced by curved optical surfaces.
Essential Terminology
Real Image: Formed when light rays physically intersect at a point. Can be projected onto a screen.
Virtual Image: Formed when light rays diverge and only appear to intersect when traced backward. Cannot be projected onto a screen.
Focal Length ($f$): The distance from the center of the lens or mirror to the focal point where parallel rays converge (or appear to diverge).
The Mirror and Thin Lens Equations
The positions of the object ($d_o$) and the resulting image ($d_i$) are mathematically related to the focal length ($f$) by:
Linear Magnification ($M$)
Sign Conventions Table:
Variable Positive (+) Condition Negative (−) Condition Focal Length ($f$) Concave Mirror / Converging Lens Convex Mirror / Diverging Lens Image Distance ($d_i$) Real Image (In front of mirror / behind lens) Virtual Image (Behind mirror / in front of lens) Magnification ($M$) Upright image (Virtual) Inverted image (Real)
High-Yield Submodule 3 & 4 Exemplar Problems
Problem 1: Sound Columns
A closed organ pipe has a physical length of $0.85\text{ m}$. If the speed of sound in air is $340\text{ m/s}$, calculate the frequency of its third harmonic.
Solution:
For a closed pipe, the harmonic frequencies follow the odd-integer sequence ($n = 1, 3, 5\dots$). The third harmonic corresponds to $n = 3$.
Write down the structural equation for a closed pipe harmonic:
$$f_n = \frac{nv}{4L}$$Substitute the values:
$$f_3 = \frac{3 \times 340}{4 \times 0.85} = \frac{1020}{3.4} = 300\text{ Hz}$$
Problem 2: Optical Fiber Limits
A light ray travels inside the core of a flint-glass fiber-optic cable ($n_{\text{core}} = 1.66$) surrounded by a cladding material ($n_{\text{cladding}} = 1.42$). Calculate the critical angle required to maintain total internal reflection within the core.
Solution:
Set up the critical angle equation:
$$\sin(\theta_c) = \frac{n_2}{n_1} = \frac{n_{\text{cladding}}}{n_{\text{core}}}$$Substitute the indices:
$$\sin(\theta_c) = \frac{1.42}{1.66} \approx 0.8554$$Take the inverse sine:
$$\theta_c = \sin^{-1}(0.8554) \approx 58.8^\circ$$Any light ray striking the cladding interface at an angle greater than $58.8^\circ$ will remain perfectly trapped within the core.