# Progressive waves

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u7-progressive-waves/

This sub-topic introduces core properties of progressive waves, which transfer energy without transferring net matter. We cover transverse vs longitudinal classification, key wave parameters, the fundamental wave equation, and phase difference required for CIE A-Level exams.

**Prerequisites:** [Simple Harmonic Motion](https://www.owlsprep.com/study/cie-9702-u6-simple-harmonic-motion/)

## Learning objectives

- Define progressive waves and explain their key property of energy transfer
- Distinguish between transverse and longitudinal progressive waves
- Recall and apply the wave equation $v = f\lambda$ to solve problems
- Calculate phase difference between two points on a progressive wave

## What is a Progressive Wave?

All waves are either progressive (travelling) or stationary (standing). Progressive waves move continuously through a medium (or vacuum for electromagnetic waves) transferring energy from one region to another.

**Progressive Wave** — A travelling wave that propagates energy through space/medium, with no net transfer of matter. Each particle in the medium oscillates about a fixed equilibrium position.

*Example:* Sound waves travelling through air, ocean waves on the water surface

> **info**
>
> Particles in the medium do not travel with the wave, they only oscillate around their original position. This is the key property examiners test repeatedly.

**Worked example:** Explain why a buoy floating on the ocean does not move towards the shore with incoming waves, even as the wave travels towards land.

1. 1. Ocean surface waves are progressive mechanical waves, which only transfer energy, not matter.
2. 2. Each water molecule oscillates vertically around a fixed equilibrium position close to its original location.
3. 3. The buoy moves with the local water molecules, so it oscillates up and down in place and does not travel horizontally with the wave.

## Transverse vs Longitudinal Waves

Progressive mechanical waves are classified based on the direction of particle oscillation relative to the direction the wave travels.

**Transverse Wave** — Oscillation of particles is perpendicular to the direction of wave propagation.

*Example:* Electromagnetic waves, waves on a stretched string, water surface waves

**Longitudinal Wave** — Oscillation of particles is parallel to the direction of wave propagation.

*Example:* Sound waves, compression waves in a slinky spring

> **mnemonic**
>
> Transverse = oscillation Perpendicular, Longitudinal = oscillation paraLLel → PP for both to remember.

**Worked example:** A sound wave travels through air from a speaker to a listener. State and explain whether it is transverse or longitudinal.

1. 1. Sound propagates as a series of compressions (high pressure) and rarefactions (low pressure).
2. 2. Air molecules oscillate back and forth parallel to the direction the sound wave is travelling, creating these pressure variations.
3. 3. Since oscillation is parallel to propagation direction, sound is a longitudinal wave.

**Check your understanding**

Test your classification understanding

1. Which of the following is a longitudinal wave?

   - Visible light
   - Sound from a violin
   - Waves on a guitar string
   - Gamma rays

   *Answer:* Sound from a violin

   *Why:* Correct! All sound waves are longitudinal. Light, gamma rays and string waves are transverse.

## Key Parameters and the Wave Equation

All progressive waves can be described by standard parameters that define their size, timing, and speed:

- **Amplitude ($A$):** Maximum displacement of a particle from equilibrium (unit: m).
- **Wavelength ($\lambda$):** Distance between two consecutive points that are in phase (unit: m).
- **Period ($T$):** Time for one full wave cycle to pass a point (unit: s). $T = 1/f$
- **Frequency ($f$):** Number of full cycles passing a point per second (unit: Hz). $f = 1/T$
- **Wave speed ($v$):** Distance the wave travels per unit time (unit: m/s).

The fundamental relationship between these parameters applies to all progressive waves, regardless of type:

$$v = f \lambda$$

**Worked example:** A middle A note on a piano has a frequency of 440 Hz. If the speed of sound in air is 330 m/s, calculate the wavelength of this sound wave.

1. 1. Recall the wave equation $v = f\lambda$, rearrange to solve for $\lambda$:
2. $$\lambda = \frac{v}{f}$$
3. 2. Substitute the given values:
4. $$\lambda = \frac{330}{440} = 0.75 \text{ m}$$
5. 3. The wavelength of the sound wave is 0.75 m.

## Phase Difference

Phase describes the current position in the oscillation cycle of a particle on a wave. Phase difference is the difference in phase between two points on the same wave, measured in radians or degrees.

For two points separated by distance $\Delta x$ on a wave of wavelength $\lambda$, the phase difference $\Delta \phi$ is calculated as:

$$\Delta \phi = \frac{2\pi \Delta x}{\lambda}$$

> **tip**
>
> If two points are 1 wavelength apart, they are in phase ($\Delta \phi = 2\pi$). If they are half a wavelength apart, they are completely out of phase ($\Delta \phi = \pi$ radians = 180°).

**Worked example:** Calculate the phase difference between two points 15 cm apart on a progressive wave of wavelength 60 cm.

1. 1. Confirm units are consistent: $\Delta x = 15$ cm, $\lambda = 60$ cm, so no conversion needed.
2. 2. Substitute into the phase difference formula:
3. $$\Delta \phi = \frac{2\pi \times 15}{60} = \frac{30\pi}{60} = 0.5\pi \text{ radians (or 90\textdegree)}$$

## Common pitfalls

- **Wrong:** Claiming progressive waves transfer matter from one point to another.
  - Why it fails: This is the most common misconception examiners test for this topic.
  - Correct: Always state that progressive waves transfer energy, with no net transfer of matter.
- **Wrong:** Mixing up oscillation direction for transverse vs longitudinal waves.
  - Why it fails: Classification depends on direction relative to wave travel, not absolute direction.
  - Correct: Use the PP mnemonic: Perpendicular = Transverse, Parallel = Longitudinal.
- **Wrong:** Leaving $\Delta x$ and $\lambda$ in different units when calculating phase difference.
  - Why it fails: The ratio of distances will be wrong if units don't match, leading to incorrect phase difference.
  - Correct: Always convert both values to the same unit (usually metres) before calculation.
- **Wrong:** Rearranging $v = f\lambda$ incorrectly as $\lambda = f/v$.
  - Why it fails: Simple algebraic error common under exam pressure, costing easy marks.
  - Correct: Check via units: wavelength is length, so $v$ (m/s) / $f$ (1/s) = m, which is correct.

## Cheatsheet

| Parameter | Symbol | Key Definition | Relationship |
| --- | --- | --- | --- |
| Amplitude | $A$ | Max displacement from equilibrium | - |
| Wavelength | $\lambda$ | Distance between two in-phase points | - |
| Period | $T$ | Time for one full wave cycle | $T = 1/f$ |
| Frequency | $f$ | Cycles per second (Hz) | $f = 1/T$ |
| Wave speed | $v$ | Distance travelled per second | $v = f\lambda$ |
| Phase difference | $\Delta\phi$ | Difference in oscillation stage | $\Delta\phi = 2\pi\Delta x/\lambda$ |

## What's next

Progressive waves are the foundation for all other wave topics in the CIE 9702 syllabus. The core concepts of wavelength, frequency, phase difference, and the wave equation are used repeatedly to analyze more complex wave phenomena including standing waves, interference, diffraction, and polarization. These topics are heavily assessed in both multiple choice (Paper 1) and structured written (Paper 2) exams, so mastering the fundamentals of progressive waves is critical for achieving high marks. Next we will explore how two travelling progressive waves combine to form standing waves.

- [Transverse and Longitudinal Waves](https://www.owlsprep.com/study/cie-9702-u7-transverse-and-longitudinal-waves/)
- [Wave Parameters](https://www.owlsprep.com/study/cie-9702-u7-wave-parameters/)
- [Wave Equation](https://www.owlsprep.com/study/cie-9702-u7-wave-equation/)

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