Progressive Waves: Study Notes for 9702 Physics

Hello future physicist! Waves might seem like a tricky topic, but they are essential for understanding everything from light and sound to radio communication. This chapter, "Progressive Waves," is all about how energy moves efficiently through space or a medium. Don't worry if concepts like 'phase difference' sound complicated—we'll break them down using simple analogies! Let's dive in.

1. What is a Progressive Wave? (7.1)

A progressive wave is a wave that travels (or propagates) through a medium (like water or air) or through a vacuum (like light), transferring energy from one point to another.

The key concept: Energy is transferred, but matter is not.

Analogy: Imagine a crowd doing "The Wave" in a stadium. The wave (energy/disturbance) moves around the stadium, but the individual people (particles of the medium) only move up and down in their seats. They return to their original spot once the wave passes.

Characteristics of Progressive Wave Motion:
  • The wave moves the disturbance (the pattern), carrying energy with it.
  • The particles of the medium oscillate about their fixed equilibrium positions.
  • The speed of the wave ($v$) is the speed at which the wave profile moves.


Key Takeaway: Progressive waves are dynamic—they are the mechanism by which energy travels without permanently displacing the material they travel through.

2. Defining the Key Wave Parameters (7.1)

To describe any wave mathematically, we need six main quantities. These are the tools you must master for wave calculations.

i. Displacement (\(x\) or \(y\))

The displacement of a vibrating particle is its distance and direction from its equilibrium position (the position it rests at when the wave is absent). Displacement constantly changes over time.

ii. Amplitude (\(A\))

The amplitude is the maximum displacement of a particle from its equilibrium position.
Think of it as the "height" of the wave. A bigger amplitude usually means a more energetic wave.

iii. Wavelength (\(\lambda\), lambda)

The wavelength is the distance between two consecutive points on the wave that are vibrating in phase (moving identically).
Commonly measured from crest to crest or trough to trough. Unit: metres (m).

iv. Period (\(T\))

The period is the time taken for one complete oscillation of a particle in the wave, or the time taken for one full wavelength to pass a fixed point.
Unit: seconds (s).

v. Frequency (\(f\))

The frequency is the number of complete oscillations made by a particle (or cycles completed by the wave) per unit time.
$f$ is the reciprocal of $T$: $$f = \frac{1}{T}$$ Unit: hertz (Hz), which is equivalent to $s^{-1}$.

vi. Phase Difference (\(\phi\))

The phase difference is a measure of how 'out of step' two points on a wave, or two separate waves, are. It is measured in degrees (\(^\circ\)) or radians (rad).

  • One complete cycle (one wavelength $\lambda$) corresponds to 360\(^\circ\) or \(2\pi\) radians.
  • If two particles are at the same point in their cycle (e.g., both at a crest), they are in phase. Phase difference = $0, 360^\circ, 2\pi$ rad, etc.
  • If one particle is at a crest and the other is at a trough, they are antiphase (completely out of step). Phase difference = $180^\circ$ or $\pi$ rad.

Memory Aid: To find the phase difference between two points separated by distance $x$ on a wave of wavelength $\lambda$: $$\text{Phase Difference } (\phi) = \frac{\text{Distance } (x)}{\text{Wavelength } (\lambda)} \times 360^\circ$$


Key Takeaway: $T$ and $f$ describe the time aspects of oscillation; $\lambda$ and $A$ describe the spatial aspects; $\phi$ compares the timing of different points.

3. Types of Progressive Waves (7.2)

We classify waves based on the direction the particles vibrate relative to the direction the energy travels.

i. Transverse Waves

In a transverse wave, the oscillation of the particles in the medium is perpendicular (at a 90\(^\circ\) angle) to the direction of energy transfer.

  • They consist of crests (max upward displacement) and troughs (max downward displacement).
  • Examples: All Electromagnetic waves (light, radio, gamma rays), waves on a rope, most water waves.
ii. Longitudinal Waves

In a longitudinal wave, the oscillation of the particles in the medium is parallel (in the same direction) to the direction of energy transfer.

  • They consist of regions of compression (where particles are squashed together, high density/pressure) and rarefaction (where particles are spread out, low density/pressure).
  • Example: Sound waves, compressional waves in a spring.

Comparing Graphical Representations (7.2(2)):

When you see a graph of displacement vs. distance for a longitudinal wave (like sound), the crests represent the point of maximum displacement, but these points correspond to the *centre* of a rarefaction or compression, depending on whether the displacement is defined relative to the wave direction. For AS/A Level, focus on knowing that longitudinal waves are defined by compressions and rarefactions, while transverse waves are defined by crests and troughs.


Quick Review Box: Transverse vs. Longitudinal
Transverse: Up/Down motion, Energy moves Sideways. (PERpendicular)
Longitudinal: Back/Forth motion, Energy moves Back/Forth. (PARallel)

4. The Fundamental Wave Equation (7.1)

The speed of a wave, $v$, links the spatial characteristics ($\lambda$) and the time characteristics ($f$ or $T$).

Derivation (7.1(4)):

Speed is defined as distance divided by time. For a wave, the smallest repeating distance is the wavelength (\(\lambda\)), and the time taken for that distance to pass is the period (\(T\)).

$$v = \frac{\text{Distance}}{\text{Time}} = \frac{\lambda}{T}$$

Since frequency $f = 1/T$, we can substitute $1/T$ with $f$:

$$\mathbf{v = f\lambda}$$

This is one of the most important equations in the waves chapter!

Using the Wave Equation (7.1(5)):

If a wave has a frequency of 500 Hz and a wavelength of 0.6 m, its speed is: $$v = f\lambda = (500) \times (0.6) = 300 \text{ m s}^{-1}$$

Did you know? The speed of light in a vacuum ($c$) is constant ($\approx 3.00 \times 10^8 \text{ m s}^{-1}$). For light, the wave equation becomes $c = f\lambda$. This means if the frequency ($f$) changes, the wavelength ($\lambda$) MUST change too, keeping $c$ constant.


Key Takeaway: The speed of a wave ($v$) is determined by the medium it travels through, and it is related to frequency and wavelength by \(v = f\lambda\).

5. Energy Transfer and Wave Intensity (7.1)

Progressive waves are defined by their ability to transfer energy (7.1(6)). When you feel the warmth of the sun (light wave) or hear a sound (sound wave), you are detecting transferred wave energy.

Intensity ($I$) (7.1(7))

The intensity of a wave describes how concentrated the energy transfer is.

Definition: Intensity is the power transferred per unit area perpendicular to the direction of wave propagation.

$$I = \frac{\text{Power}}{\text{Area}}$$

$$\mathbf{I = \frac{P}{A}}$$

Units: Watts per square metre (\(\text{W m}^{-2}\)).

Analogy: If you shine a powerful flashlight (high power) onto a small spot (small area), the intensity is high (bright). If you shine the same light onto a large wall (large area), the intensity is low (dim).

Intensity and Amplitude Relationship (7.1(7))

The energy transferred by a wave is strongly related to how big the oscillation is (its amplitude).

The intensity of a progressive wave is proportional to the square of its amplitude: $$\mathbf{I \propto A^2}$$

This means if you double the amplitude ($A \times 2$), the intensity ($I$) increases by a factor of four ($2^2 = 4$).

Applying the relationship:
If a sound wave's amplitude is halved, its intensity becomes only a quarter of the original intensity.
If you compare two waves: $$\frac{I_1}{I_2} = \frac{A_1^2}{A_2^2}$$


Common Mistake Alert: Students sometimes confuse $I \propto A$ with $I \propto A^2$. Remember, intensity and power are related to energy, and energy often depends on squared terms in Physics (like $E_k = \frac{1}{2}mv^2$). Always square the amplitude!

6. Using the Cathode-Ray Oscilloscope (CRO) (7.1)

The Cathode-Ray Oscilloscope (CRO) is a vital piece of equipment for measuring alternating signals, including sound waves converted into electrical signals by a microphone. It displays a graph of voltage (or displacement) versus time.

Determining Amplitude and Frequency using a CRO (7.1(3)):

The CRO screen is a grid. The two crucial controls are:

i. Y-Gain (Vertical Axis) for Amplitude
  • The Y-gain setting determines how many volts correspond to one vertical division (or 'box') on the screen. (Units often: Volts/div or $\text{V}/\text{cm}$).
  • To determine Amplitude (\(A\)): Count the maximum number of vertical divisions from the equilibrium line to a crest.
  • $$\text{Amplitude } (A) = \text{Peak Vertical Displacement (Divisions)} \times \text{Y-Gain Setting}$$
ii. Time-Base (Horizontal Axis) for Period and Frequency
  • The Time-base setting determines how much time corresponds to one horizontal division (or 'box') on the screen. (Units often: s/div or $\text{ms}/\text{cm}$).
  • Step 1: Determine the Period (\(T\)): Count the number of horizontal divisions taken for one complete cycle (e.g., from crest to the next crest).
  • $$T = \text{Horizontal Displacement (Divisions)} \times \text{Time-Base Setting}$$
  • Step 2: Determine the Frequency (\(f\)): Once $T$ is found, use the inverse relationship.
  • $$f = \frac{1}{T}$$

Example: If the Time-Base is set to 5 ms/div and one full wave cycle covers 4 horizontal divisions:
$T = 4 \text{ div} \times 5 \times 10^{-3} \text{ s}/\text{div} = 0.020 \text{ s}$.
$f = 1 / 0.020 \text{ s} = 50 \text{ Hz}$.


Key Takeaway: The CRO is the experimental link. The Y-gain gives you amplitude (size), and the Time-base gives you the period (timing).