1. Overview
Progressive waves are fundamental mechanisms for the transfer of energy from one point in space to another. Unlike the bulk flow of matter (such as water flowing in a pipe), a progressive wave involves the oscillation of particles or fields about a fixed equilibrium position. The key characteristic of a progressive wave is that while the "disturbance" or "wavefront" moves through the medium, the medium itself does not travel with the wave.
In mechanical waves, such as those in ropes or springs, this transfer is achieved through inter-particle forces. In electromagnetic waves, the transfer occurs through the oscillation of electric and magnetic fields, allowing energy to travel even through a vacuum. Understanding progressive waves is essential for mastering subsequent topics in A-Level Physics, including superposition, stationary waves, and the Doppler effect.
Key Definitions
To achieve full marks in the Cambridge 9702 exam, definitions must be precise. Use the following bolded keywords in your answers:
- Progressive Wave: A wave that transfers energy from one position to another as a result of oscillations (or vibrations), without the transfer of matter.
- Displacement ($x$ or $y$): The distance of a point on the wave from its equilibrium (rest) position in a specified direction. (Unit: $\text{m}$)
- Amplitude ($A$): The maximum displacement of a particle from its equilibrium position. (Unit: $\text{m}$)
- Period ($T$): The time taken for one complete oscillation of a particle in the wave, or the time taken for the wave to travel one whole wavelength. (Unit: $\text{s}$)
- Frequency ($f$): The number of oscillations per unit time. (Unit: $\text{Hz}$ or $\text{s}^{-1}$)
- Wavelength ($\lambda$): The minimum distance between two points on a wave that are vibrating in phase with each other (e.g., the distance between adjacent crests or adjacent troughs). (Unit: $\text{m}$)
- Wave Speed ($v$): The speed at which the energy is transmitted (or the speed at which the wavefront travels) through the medium. (Unit: $\text{m s}^{-1}$)
- Phase Difference ($\phi$): The difference in the relative positions of the crests or troughs of two waves, or the difference in the stage of a cycle of two particles on the same wave, usually measured in degrees ($^\circ$) or radians ($\text{rad}$).
- Intensity ($I$): The power transmitted per unit area perpendicular to the direction of energy travel. (Unit: $\text{W m}^{-2}$)
Content
3.1 Wave Motion in Physical Systems
The syllabus requires you to describe wave motion using three specific models. In all these cases, notice that the particles return to their original positions after the wave has passed.
- Vibrations in Ropes: When one end of a rope is moved up and down, a transverse wave is created. The particles of the rope oscillate perpendicular to the direction of wave travel (energy transfer). You can see the "peak" move along the rope, but any specific piece of string only moves vertically.
- Vibrations in Springs (Slinkys):
- If the end is moved back and forth along the length of the spring, a longitudinal wave is formed. This consists of compressions (regions where coils are close together/high pressure) and rarefactions (regions where coils are spread out/low pressure).
- The coils oscillate parallel to the direction of energy transfer.
- Ripple Tanks: These are used to visualize water waves. A vibrating bar creates plane waves (straight wavefronts), while a dipping point creates circular waves.
- The wavefronts represent the peaks of the waves.
- The distance between two successive bright lines (where the water acts as a converging lens for the light above) is the wavelength.
3.2 The Wave Equation: Derivation
You are required to derive $v = f\lambda$ from first principles.
- Start with the definition of speed: $$\text{speed} = \frac{\text{distance}}{\text{time}}$$
- Apply this to one complete cycle of a wave: In the time it takes for a particle to complete one oscillation (the Period, $T$), the wave travels a distance equal to one Wavelength ($\lambda$).
- Substitute these specific terms into the speed equation: $$v = \frac{\lambda}{T}$$
- Use the relationship between frequency and period: Since frequency is the number of cycles per second, $f = \frac{1}{T}$.
- Final substitution: Replacing $\frac{1}{T}$ with $f$ gives: $$v = f \lambda$$ (This equation is on the Data Sheet, but the derivation is frequently examined.)
3.3 Phase Difference ($\phi$)
Phase difference tells us how much one point "lags" or "leads" another. It is a measure of the fraction of a cycle that separates two points.
- In Phase: Points are at the same stage of their cycle (e.g., both at maximum positive displacement). Phase difference is $0, 2\pi, 4\pi \dots$ rad or $0^\circ, 360^\circ, 720^\circ \dots$
- Antiphase: Points are at exactly opposite stages of their cycle (e.g., one at maximum positive displacement, the other at maximum negative). Phase difference is $\pi, 3\pi, 5\pi \dots$ rad or $180^\circ, 540^\circ \dots$
Calculating Phase Difference: If two points on a wave are separated by a distance $x$: $$\phi = \frac{x}{\lambda} \times 360^\circ \quad (\text{for degrees})$$ $$\phi = \frac{x}{\lambda} \times 2\pi \quad (\text{for radians})$$
If two waves are compared on a time axis (displacement-time graph) and are separated by time $t$: $$\phi = \frac{t}{T} \times 360^\circ \quad \text{or} \quad \phi = \frac{t}{T} \times 2\pi$$
3.4 Cathode-Ray Oscilloscope (CRO)
A CRO is essentially a voltmeter that plots a graph of Voltage (y-axis) against Time (x-axis).
- Y-gain (Vertical Control): This is the "sensitivity" of the vertical axis. It is measured in Volts per division ($\text{V/div}$ or $\text{V/cm}$).
- To find the Amplitude, count the number of vertical divisions from the center line to a peak and multiply by the Y-gain.
- Time-base (Horizontal Control): This is the "speed" of the horizontal sweep. It is measured in Seconds per division ($\text{s/div}$, $\text{ms/div}$, or $\mu\text{s/div}$).
- To find the Period ($T$), count the number of horizontal divisions for one full cycle and multiply by the time-base setting.
- Once $T$ is found, calculate frequency using $f = 1/T$.
Worked Example 1 — CRO Analysis
A signal from a microphone is displayed on a CRO. The Y-gain is set to $5.0 \text{ mV/div}$ and the time-base is set to $0.20 \text{ ms/div}$. The trace shows a wave with a peak-to-peak height of $6.0$ divisions, and $4.0$ complete cycles occupy $10.0$ horizontal divisions. Calculate the amplitude and frequency of the sound wave.
Step 1: Calculate Amplitude The peak-to-peak height is $6.0$ divisions. The amplitude $A$ is half of the peak-to-peak height: $$A = \frac{6.0 \text{ div}}{2} = 3.0 \text{ div}$$ Convert divisions to Volts using the Y-gain: $$A = 3.0 \text{ div} \times 5.0 \times 10^{-3} \text{ V/div} = 0.015 \text{ V} = 15 \text{ mV}$$
Step 2: Calculate Period ($T$) $4.0$ cycles occupy $10.0$ divisions. Divisions for $1$ cycle $= \frac{10.0}{4.0} = 2.5 \text{ divisions}$. $$T = 2.5 \text{ div} \times 0.20 \times 10^{-3} \text{ s/div} = 5.0 \times 10^{-4} \text{ s}$$
Step 3: Calculate Frequency ($f$) $$f = \frac{1}{T} = \frac{1}{5.0 \times 10^{-4} \text{ s}} = 2000 \text{ Hz} = 2.0 \text{ kHz}$$ Answer: $A = 15 \text{ mV}, f = 2.0 \text{ kHz}$
3.5 Energy and Intensity
Progressive waves transfer energy. The rate at which this energy is transferred is the Power ($P$).
1. Intensity Definition Intensity is the power per unit area: $$I = \frac{P}{A}$$ (This must be memorised. Units: $\text{W m}^{-2}$)
2. Inverse Square Law For a point source emitting waves in all directions, the energy spreads out over the surface of a sphere. The surface area of a sphere is $4\pi r^2$, where $r$ is the distance from the source. $$I = \frac{P}{4\pi r^2}$$ Since $P$ and $4\pi$ are constants for a given source: $$I \propto \frac{1}{r^2}$$ If you double your distance from a light source, the intensity decreases by a factor of $2^2 = 4$.
3. Intensity and Amplitude The energy of an oscillating particle is proportional to the square of its amplitude ($E \propto A^2$). Because intensity is proportional to the energy transferred per second: $$I \propto A^2$$ This is a critical relationship for exams. If the amplitude of a wave is tripled, the intensity increases by a factor of $3^2 = 9$.
Worked Example 2 — Intensity and Distance
A point source of sound has an intensity of $2.0 \times 10^{-6} \text{ W m}^{-2}$ at a distance of $3.0 \text{ m}$ from the source. (a) Calculate the intensity at a distance of $9.0 \text{ m}$. (b) If the amplitude of the wave at $3.0 \text{ m}$ is $A_0$, determine the amplitude at $9.0 \text{ m}$ in terms of $A_0$.
Part (a) Solution: Use the inverse square law: $I \propto \frac{1}{r^2}$ The distance has increased from $3.0 \text{ m}$ to $9.0 \text{ m}$ (a factor of $3$). Therefore, the intensity decreases by a factor of $3^2 = 9$. $$I_{\text{new}} = \frac{2.0 \times 10^{-6}}{9} = 2.22 \times 10^{-7} \text{ W m}^{-2}$$
Part (b) Solution: We know $I \propto A^2$ and $I \propto \frac{1}{r^2}$. Combining these: $A^2 \propto \frac{1}{r^2}$, which simplifies to $A \propto \frac{1}{r}$. Since the distance $r$ increased by a factor of $3$, the amplitude $A$ must decrease by a factor of $3$. Answer: (a) $2.2 \times 10^{-7} \text{ W m}^{-2}$, (b) $\frac{1}{3}A_0$
Key Equations
| Equation | Symbols | SI Units | Status |
|---|---|---|---|
| $v = f \lambda$ | $v$: speed, $f$: frequency, $\lambda$: wavelength | $\text{m s}^{-1}, \text{Hz}, \text{m}$ | Data Sheet |
| $f = \frac{1}{T}$ | $f$: frequency, $T$: period | $\text{Hz}, \text{s}$ | Data Sheet |
| $I = \frac{P}{A}$ | $I$: intensity, $P$: power, $A$: area | $\text{W m}^{-2}, \text{W}, \text{m}^2$ | Memorise |
| $I \propto A^2$ | $I$: intensity, $A$: amplitude | $\text{W m}^{-2}, \text{m}$ | Memorise |
| $I \propto \frac{1}{r^2}$ | $I$: intensity, $r$: distance from source | $\text{W m}^{-2}, \text{m}$ | Memorise |
| $\phi = \frac{x}{\lambda} \times 2\pi$ | $\phi$: phase diff (rad), $x$: distance | $\text{rad}, \text{m}$ | Memorise |
Common Mistakes to Avoid
- ❌ Wrong: Using the peak-to-peak vertical distance as the amplitude on a CRO.
- ✓ Right: Amplitude is the distance from the equilibrium line to the peak. Always divide the peak-to-peak value by 2.
- ❌ Wrong: Forgetting to convert time-base units like $\text{ms}$ ($10^{-3} \text{ s}$) or $\mu\text{s}$ ($10^{-6} \text{ s}$) before calculating frequency.
- ✓ Right: Always convert to base SI units ($\text{seconds}$) first.
- ❌ Wrong: Confusing displacement-distance graphs with displacement-time graphs.
- ✓ Right: On a distance graph, the peak-to-peak horizontal distance is wavelength ($\lambda$). On a time graph, it is period ($T$).
- ❌ Wrong: Thinking that if intensity doubles, amplitude doubles.
- ✓ Right: Intensity is proportional to the square of the amplitude. If intensity doubles, amplitude increases by $\sqrt{2}$.
- ❌ Wrong: Stating that matter is transferred in a progressive wave.
- ✓ Right: Only energy and momentum are transferred; the particles of the medium only oscillate.
Exam Tips
- Phase Difference Units: Read the question carefully to see if the answer should be in degrees or radians. If the question uses $\pi$, it is almost certainly radians. Ensure your calculator is in the correct mode.
- The "Explain" Question: If asked to explain why a wave is "progressive," your answer must include: "It transfers energy from one point to another without the transfer of the medium."
- Significant Figures: In the 9702 exam, always provide your final answer to the same number of significant figures as the least precise data given in the question (usually 2 or 3 s.f.).
- Intensity of EM Waves: For electromagnetic waves, the speed $v$ is always $c = 3.00 \times 10^8 \text{ m s}^{-1}$ in a vacuum or air. You may need to use this constant to find $f$ or $\lambda$ even if it isn't explicitly stated in the question.
- Graph Interpretation: When calculating frequency from a CRO trace, use as many cycles as possible to reduce the percentage uncertainty in your measurement of the horizontal distance, then divide by the number of cycles to find $T$.