Welcome to the Wave Model!
Hello physicists! Get ready to dive into the world of waves—one of the most fundamental concepts in physics. Waves are how energy gets from A to B, whether it’s the sound of your favourite music, the light from the sun, or the shaking of the ground during an earthquake.
In this chapter (C.2), we are laying the essential foundation: defining what a wave is, classifying the different types, and mastering the key properties that allow us to describe and quantify wave motion. Understanding these basics is crucial, as waves underpin almost all of modern physics, including optics and quantum mechanics. Let’s get started!
1. Defining the Wave: Energy in Motion
What is a Wave?
A wave is a mechanism for transferring energy from one location to another without the net transfer of matter (mass).
Analogy: Think about a stadium wave at a sports game. People stand up and sit down (oscillate), but they don't move around the stadium to transfer the "wave." The energy (the disturbance) moves, but the people (the medium) stay put.
Prerequisite Concept: Oscillation
Waves rely on oscillations (vibrations). An oscillation is simply repetitive back-and-forth motion around a central equilibrium position. The wave is the propagation of this oscillation through space.
Classifying Waves by Medium Requirement
Waves can be grouped based on whether they need material to travel through:
- Mechanical Waves:
- These waves require a medium (solid, liquid, or gas) to propagate.
- They travel due to the restoring forces between the particles of the medium.
- Examples: Sound waves, water waves, seismic waves.
- Electromagnetic (EM) Waves:
- These waves do not require a medium; they can travel through a vacuum (empty space).
- They consist of coupled oscillating electric and magnetic fields.
- Examples: Light, radio waves, X-rays.
Classifying Waves by Direction of Oscillation
This is arguably the most important classification. It depends on the relationship between the direction the particles oscillate and the direction the energy travels.
A. Transverse Waves
In a transverse wave, the oscillation of the particles in the medium is perpendicular (at right angles) to the direction of wave propagation (energy transfer).
- The high points are called crests.
- The low points are called troughs.
- Examples: All electromagnetic waves, water ripples, waves on a string.
Analogy: If you shake a jump rope up and down (perpendicular to the ground), the wave travels horizontally.
B. Longitudinal Waves
In a longitudinal wave, the oscillation of the particles in the medium is parallel (in the same direction) to the direction of wave propagation.
- Regions where the particles are crowded together are called compressions (high pressure/density).
- Regions where the particles are spread apart are called rarefactions (low pressure/density).
- Example: Sound waves, primary seismic waves (P-waves).
Quick Review:
| Wave Type | Oscillation vs. Energy Direction |
| Transverse | Perpendicular (\(90^{\circ}\)) |
| Longitudinal | Parallel (\(0^{\circ}\)) |
Key Takeaway: A wave moves energy, not mass. We classify waves based on whether they need a medium (Mechanical vs. EM) and how the particles vibrate (Transverse vs. Longitudinal).
2. Essential Wave Characteristics and Definitions
To describe any wave mathematically, we need key measurable characteristics. Make sure you know the definition, symbol, and standard unit for each one!
Amplitude (\(A\))
Definition: The maximum displacement of a particle from its equilibrium (rest) position.
Significance: Amplitude is related to the energy carried by the wave. A bigger amplitude means more energy.
Unit: Metres (\(\text{m}\)).
Wavelength (\(\lambda\))
Definition: The shortest distance between two points that are in phase (i.e., the same point on adjacent cycles). Usually measured from crest to crest or trough to trough (for transverse waves), or from compression centre to compression centre (for longitudinal waves).
Symbol: Lambda (\(\lambda\)).
Unit: Metres (\(\text{m}\)).
Period (\(T\))
Definition: The time taken for one complete oscillation or one full wave cycle to pass a fixed point.
Significance: It tells us how long one cycle lasts.
Unit: Seconds (\(\text{s}\)).
Frequency (\(f\))
Definition: The number of complete cycles (waves) that pass a fixed point per unit time.
Significance: It tells us how often cycles occur.
Unit: Hertz (\(\text{Hz}\)), where \(1 \text{ Hz} = 1 \text{ s}^{-1}\) (cycles per second).
Memory Aid: T and f are opposites!
Since the period (\(T\)) is the time for one cycle, and frequency (\(f\)) is the number of cycles per time, they are inversely related:
\[f = \frac{1}{T}\]
or
\[T = \frac{1}{f}\]
Wave Speed (\(v\))
Definition: The distance the wave profile (and the energy) travels per unit time.
Significance: Wave speed is determined entirely by the medium through which the wave travels.
Unit: Metres per second (\(\text{m s}^{-1}\)).
Key Takeaway: Wavelength (\(\lambda\)) and Amplitude (\(A\)) are spatial measurements (distance), while Period (\(T\)) and Frequency (\(f\)) are temporal measurements (time).
3. The Fundamental Wave Equation
We can relate wave speed (\(v\)), frequency (\(f\)), and wavelength (\(\lambda\)) into one powerful equation:
The Relationship between \(v\), \(f\), and \(\lambda\)
Since speed is generally calculated as distance divided by time, and for one full cycle, the distance is the wavelength (\(\lambda\)) and the time is the period (\(T\)):
\[v = \frac{\lambda}{T}\]
And since \(f = 1/T\), we substitute frequency into the equation:
\[v = f\lambda\]
This is the fundamental wave equation. It is used constantly throughout the wave topic!
Important Rule to Remember (SL & HL):
The speed of a wave (\(v\)) is dictated by the medium. If the medium doesn't change, the speed \(v\) doesn't change.
If a wave passes from one medium to another (e.g., air to water), the speed \(v\) changes, but the frequency (\(f\)) remains constant. This means any change in speed must result in a compensating change in wavelength (\(\lambda\)).
Common Mistake to Avoid:
Do not assume increasing the frequency of a source (like a speaker) makes the waves travel faster. It doesn't! Increasing \(f\) simply decreases the wavelength \(\lambda\), keeping \(v\) constant, as long as the medium is unchanged.
Key Takeaway: \(v = f\lambda\). Speed depends on the medium. Frequency depends on the source and is constant between media.
4. Visualizing Waves: Graphs
Waves can be confusing because they involve both space and time. We use two main types of graphs to analyze them.
Graph Type 1: Displacement vs. Position (The Snapshot Graph)
This graph shows the shape of the wave at a single moment in time (a "snapshot").
- The vertical axis is Displacement (\(y\)), measured in metres (\(\text{m}\)). The maximum displacement is the Amplitude (\(A\)).
- The horizontal axis is Position (\(x\)), measured in metres (\(\text{m}\)).
- The distance between two consecutive peaks (or any two points in phase) on this graph gives the Wavelength (\(\lambda\)).
Graph Type 2: Displacement vs. Time (The Point-in-Motion Graph)
This graph shows how the displacement of a single particle in the medium changes over time as the wave passes.
- The vertical axis is Displacement (\(y\)), measured in metres (\(\text{m}\)). The maximum displacement is still the Amplitude (\(A\)).
- The horizontal axis is Time (\(t\)), measured in seconds (\(\text{s}\)).
- The time taken for the particle to complete one full oscillation (one cycle) on this graph gives the Period (\(T\)).
Step-by-Step Interpretation Trick:
Look at the horizontal axis first!
If the axis is Distance (\(x\)), you find \(\lambda\).
If the axis is Time (\(t\)), you find \(T\) (and thus \(f\)).
Key Takeaway: Position graphs give spatial information (\(\lambda\)). Time graphs give temporal information (\(T\)). Both graphs show the Amplitude (\(A\)).
5. Wave Intensity and Energy Transfer
The intensity of a wave tells us how much energy is being delivered per unit area. This is why sound gets quieter the further away you are from the source.
Intensity (\(I\)) Definition
Definition: Intensity is defined as the Power (\(P\)) transported by the wave per unit area (\(A\)) in a direction perpendicular to the area.
\[I = \frac{P}{A}\]
Unit: Watts per square metre (\(\text{W m}^{-2}\)).
Intensity and Amplitude Relationship
The energy carried by a wave is proportional to the square of its amplitude (\(A^2\)). Since intensity is proportional to energy flow (power):
\[I \propto A^2\]
This is crucial! Doubling the amplitude of a wave (e.g., a sound wave) increases its intensity by a factor of four (\(2^2 = 4\)).
The Inverse Square Law (For Spherical Waves)
For waves propagating freely in three dimensions from a point source (like light from a lamp or sound from a speaker), the wave energy spreads out over a constantly increasing spherical surface area.
The surface area of a sphere is \(A = 4\pi r^2\). Therefore, intensity \(I\) is inversely proportional to the square of the distance (\(r\)) from the source:
\[I \propto \frac{1}{r^2}\]
Practical Example: If you stand 2 metres away from a speaker, and then walk back to 4 metres away (doubling the distance \(r\)), the intensity of the sound you hear will drop to one quarter (\(1/2^2\)) of the original intensity.
Did you know? This inverse square relationship applies to almost all forces and fields that spread out equally in all directions, including gravitational fields, electric fields, light, and sound!
Key Takeaway: Intensity is power per area (\(P/A\)). Intensity is linked to the energy of the wave, meaning \(I\) is proportional to \(A^2\). As the wave spreads, intensity drops by the inverse square of the distance.
Chapter C.2 Summary Checklist
You have successfully completed the foundation of the Wave Model! Make sure you can confidently:
- Define a wave as an energy transfer mechanism without net mass transfer.
- Distinguish between mechanical and electromagnetic waves.
- Draw and label transverse (crests/troughs) and longitudinal (compressions/rarefactions) waves.
- Define and apply the terms: Amplitude (\(A\)), Wavelength (\(\lambda\)), Period (\(T\)), Frequency (\(f\)), and Speed (\(v\)).
- Use the fundamental wave equation: \(v = f\lambda\).
- Interpret wave characteristics (\(\lambda\) and \(T\)) from Displacement-Position and Displacement-Time graphs.
- Understand the relationships \(I \propto A^2\) and \(I \propto 1/r^2\).
You got this! This understanding sets the stage for analyzing wave behaviour like reflection, refraction, and interference in the next chapters.