1. Overview
Progressive waves are oscillations that travel through a medium (or a vacuum, in the case of electromagnetic waves), transferring energy from one position to another without the net transfer of matter. The particles of the medium oscillate about fixed equilibrium positions, but they do not travel with the wave.
Waves are classified into two fundamental categories—transverse and longitudinal—based on the geometric relationship between the direction of the oscillations and the direction of energy transfer (propagation). Understanding this distinction is essential for analyzing wave phenomena such as polarization, interference, and stationary waves.
Key Definitions
To earn full marks in Cambridge 9702 exams, definitions must be precise. Pay close attention to the bolded keywords.
- Progressive Wave: A wave that carries energy from one place to another as a result of oscillations without the transfer of the medium itself.
- Transverse Wave: A wave in which the oscillations of the particles (or fields) are perpendicular (at $90^\circ$) to the direction of energy transfer.
- Longitudinal Wave: A wave in which the oscillations of the particles are parallel to the direction of energy transfer.
- Displacement ($x$ or $y$): The distance and direction of a point on the wave from its equilibrium position. It is a vector quantity.
- Amplitude ($A$): The maximum displacement of a particle from its equilibrium position.
- Wavelength ($\lambda$): The minimum distance between two points on a wave that are oscillating in phase (e.g., the distance from one peak to the next adjacent peak).
- Period ($T$): The time taken for one complete oscillation of a point in the wave.
- Frequency ($f$): The number of oscillations per unit time at a point in the wave. Measured in Hertz ($Hz$), where $1 \text{ Hz} = 1 \text{ s}^{-1}$.
- Phase Difference ($\phi$): The difference in the relative positions of the crests or troughs of two waves, or the difference in the stage of cycle between two points on the same wave. It is measured in degrees ($^\circ$) or radians ($rad$).
- Compression: A region in a longitudinal wave where the particles are closer together than their equilibrium positions, resulting in high pressure.
- Rarefaction: A region in a longitudinal wave where the particles are further apart than their equilibrium positions, resulting in low pressure.
Content
3.1 Comparison of Transverse and Longitudinal Waves
The primary distinction lies in the orientation of the oscillation.
| Feature | Transverse Waves | Longitudinal Waves |
|---|---|---|
| Direction of Oscillation | Perpendicular to the direction of energy transfer. | Parallel to the direction of energy transfer. |
| Key Components | Consists of peaks (crests) and troughs. | Consists of compressions and rarefactions. |
| Polarisation | Can be polarised because oscillations occur in a plane perpendicular to propagation. | Cannot be polarised because oscillations are restricted to the direction of propagation. |
| Medium Requirements | Can travel through solids and on the surface of liquids. Cannot travel through the bulk of a liquid or gas (except for EM waves). | Can travel through solids, liquids, and gases (any medium with elasticity). |
| Vacuum Propagation | Electromagnetic waves can travel through a vacuum. | Cannot travel through a vacuum (requires a medium). |
| Examples | Light, Radio, X-rays (all EM waves), waves on a plucked string, S-waves (seismic). | Sound waves, ultrasound, pressure waves, P-waves (seismic). |
Visualizing the Motion:
- Transverse: Imagine a rope tied to a wall. If you move your hand up and down, the wave moves toward the wall (horizontally), but the rope particles move up and down (vertically).
- Longitudinal: Imagine a slinky spring. If you push and pull the end of the spring toward and away from the wall, the "pulse" moves toward the wall, and the coils of the spring vibrate back and forth in that same horizontal line.
3.2 Graphical Representations of Waves
In the 9702 syllabus, you must be able to extract data from two distinct types of graphs. Always check the x-axis units first.
1. Displacement–Distance Graphs ($x$ vs $d$) This graph represents a "snapshot" in time. It shows the position of all particles in the medium at one specific instant ($t = \text{constant}$).
- Amplitude ($A$): The height from the center line to a peak.
- Wavelength ($\lambda$): The horizontal distance between two adjacent peaks or two adjacent troughs.
- Phase Difference: You can determine how far "out of step" two points are along the distance of the wave.
2. Displacement–Time Graphs ($x$ vs $t$) This graph tracks the history of a single particle at one specific location ($d = \text{constant}$). It shows how that one particle moves up and down (or back and forth) as time passes.
- Amplitude ($A$): The maximum displacement of that specific particle.
- Period ($T$): The time interval between two adjacent peaks on the graph.
- Frequency ($f$): Calculated using $f = \frac{1}{T}$.
3.3 Interpreting Longitudinal Waves on Graphs
This is a frequent source of confusion. Because longitudinal waves oscillate parallel to the wave direction, we use a sign convention to plot them on a standard $y$-axis:
- Positive Displacement (+): Particle has moved to the right (or in the direction of wave travel).
- Negative Displacement (-): Particle has moved to the left (or opposite to the direction of wave travel).
Identifying Compressions and Rarefactions on a Displacement-Distance Graph: Assume a wave is traveling to the right.
- Compression: Look for the point where displacement is zero and the gradient is negative.
- Logic: To the left of this point, displacement is positive (particles moved right toward the point). To the right of this point, displacement is negative (particles moved left toward the point). Particles are "piling up."
- Rarefaction: Look for the point where displacement is zero and the gradient is positive.
- Logic: To the left, particles moved left (negative displacement). To the right, particles moved right (positive displacement). Particles are "spreading out."
3.4 Phase Difference ($\phi$)
Phase difference describes the relative positions of two points on a wave or two different waves. It is expressed as an angle.
- In Phase: Points are at the same stage of their cycle (e.g., both at peaks). $\phi = 0^\circ, 360^\circ$ or $0, 2\pi \text{ rad}$.
- Antiphase: Points are doing the exact opposite (e.g., one at a peak, one at a trough). $\phi = 180^\circ$ or $\pi \text{ rad}$.
Calculations:
- By Distance: If two points are separated by distance $x$: $$\phi = \frac{x}{\lambda} \times 360^\circ \quad \text{or} \quad \phi = \frac{x}{\lambda} \times 2\pi \text{ rad}$$
- By Time: If one wave lags behind another by time $t$: $$\phi = \frac{t}{T} \times 360^\circ \quad \text{or} \quad \phi = \frac{t}{T} \times 2\pi \text{ rad}$$
3.5 The Wave Equation
The speed of a wave ($v$) is the distance it travels per unit time. In one period ($T$), the wave travels exactly one wavelength ($\lambda$).
$$v = f\lambda$$
- $v$: Wave speed ($m s^{-1}$)
- $f$: Frequency ($Hz$)
- $\lambda$: Wavelength ($m$)
Note: Wave speed depends solely on the properties of the medium (e.g., tension in a string, density of air), not on the frequency or amplitude.
Worked Example 1 — Analyzing Particle Motion
A transverse wave travels from left to right. The diagram shows a displacement-distance graph at time $t = 0$. A particle P is located at a distance of $\frac{1}{4}\lambda$ from the origin, currently at maximum positive displacement.
Question: Determine the direction of motion of particle P at $t = 0$.
Step-by-Step Solution:
- Visualize the "Next Moment": To find the direction of a particle's velocity, imagine the wave has moved a tiny distance to the right (the direction of propagation).
- Shift the Wave: Draw a dotted line representing the wave a fraction of a second later. The peak that was at P has now moved to the right.
- Observe P's Position: The particle P can only move vertically (it is a transverse wave). On the new dotted line, the displacement at the horizontal position of P is lower than it was at $t = 0$.
- Conclusion: Since the displacement is decreasing from the maximum, the particle P must be moving downwards toward the equilibrium position.
Worked Example 2 — Frequency and Wavelength Calculation
A stationary microphone is connected to an oscilloscope. It detects a longitudinal sound wave traveling at $330 \text{ m s}^{-1}$. The oscilloscope trace shows that the time between the first compression and the fifth compression passing the microphone is $10.0 \text{ ms}$.
Question: Calculate the wavelength of the sound wave.
Step 1: Identify the number of cycles. The time between the 1st and 5th compression represents 4 complete oscillations (cycles). $4T = 10.0 \text{ ms} = 10.0 \times 10^{-3} \text{ s}$
Step 2: Calculate the Period ($T$). $T = \frac{10.0 \times 10^{-3}}{4} = 2.5 \times 10^{-3} \text{ s}$
Step 3: Calculate the Frequency ($f$). $f = \frac{1}{T} = \frac{1}{2.5 \times 10^{-3}} = 400 \text{ Hz}$
Step 4: Calculate the Wavelength ($\lambda$) using the wave equation. $v = f\lambda \implies \lambda = \frac{v}{f}$ $\lambda = \frac{330}{400} = 0.825 \text{ m}$
Answer: $0.83 \text{ m}$ (to 2 significant figures).
Worked Example 3 — Phase Difference between Two Points
A wave has a frequency of $50 \text{ Hz}$ and a speed of $20 \text{ m s}^{-1}$. Calculate the phase difference between two points on the wave that are $15 \text{ cm}$ apart. Give your answer in radians.
Step 1: Calculate the wavelength ($\lambda$). $\lambda = \frac{v}{f} = \frac{20}{50} = 0.40 \text{ m}$
Step 2: Convert the distance between points to SI units. $x = 15 \text{ cm} = 0.15 \text{ m}$
Step 3: Apply the phase difference formula. $\phi = \frac{x}{\lambda} \times 2\pi$ $\phi = \frac{0.15}{0.40} \times 2\pi$ $\phi = 0.375 \times 2\pi = 0.75\pi \text{ rad}$
Step 4: Final calculation. $\phi \approx 2.36 \text{ rad}$
Answer: $2.4 \text{ rad}$ (to 2 significant figures).
Key Equations
| Equation | Description | Data Sheet? |
|---|---|---|
| $v = f\lambda$ | Wave speed equation | Yes |
| $f = \frac{1}{T}$ | Relationship between frequency and period | Yes |
| $\phi = \frac{x}{\lambda} \times 360^\circ$ | Phase difference (using distance) | No |
| $\phi = \frac{t}{T} \times 360^\circ$ | Phase difference (using time lag) | No |
| $\text{rad} = \text{deg} \times \frac{\pi}{180}$ | Converting degrees to radians | No |
Common Mistakes to Avoid
- ❌ Confusing $x-d$ and $x-t$ graphs: Students often read the wavelength from a displacement-time graph.
- ✓ Right: Always check the x-axis. If the axis is Time, you are looking at the Period ($T$). If the axis is Distance/Position, you are looking at the Wavelength ($\lambda$).
- ❌ Incorrect Wavelength Identification: Identifying the distance from a peak to the next trough as $\lambda$.
- ✓ Right: Peak-to-trough is half a wavelength ($0.5\lambda$). Wavelength is peak-to-peak.
- ❌ Polarization Errors: Stating that sound waves can be polarized.
- ✓ Right: Sound is longitudinal. Only transverse waves (like light or waves on a string) can be polarized.
- ❌ Unit Neglect: Using frequency in $kHz$ or wavelength in $cm$ directly in $v = f\lambda$.
- ✓ Right: Convert all values to SI base units ($Hz, m, s$) before calculating to avoid power-of-ten errors.
- ❌ Phase Difference Units: Giving a phase difference without units or confusing degrees and radians.
- ✓ Right: Check if the question specifies radians or degrees. If not specified, radians are standard in A-Level Physics, but degrees are acceptable if clearly labeled.
Exam Tips
- The 2-Mark Comparison: If asked to compare transverse and longitudinal waves, your first mark is almost always for the direction of oscillation relative to the direction of energy transfer. Use the words "perpendicular" and "parallel" explicitly.
- Significant Figures: Cambridge 9702 is strict. If the data in the question is given to 2 s.f. (e.g., $330 \text{ m s}^{-1}$ and $10 \text{ ms}$), your final answer should be given to 2 s.f. (or 3 s.f. as a safe margin). Never provide 1 s.f.
- Longitudinal Particle Motion: Remember that particles in a longitudinal wave move back and forth along the line of the wave. If the wave is horizontal, the particles move horizontally. They never move up and down.
- Phase Difference Logic:
- Distance of $\lambda \implies 360^\circ$ ($2\pi$ rad)
- Distance of $\lambda/2 \implies 180^\circ$ ($\pi$ rad)
- Distance of $\lambda/4 \implies 90^\circ$ ($\pi/2$ rad)
- Sketching Graphs: If asked to sketch a graph of a wave with "twice the frequency," ensure the wavelength is exactly half the original (assuming speed is constant). Use a ruler to mark the peaks to ensure the amplitude remains constant unless the question states otherwise.