# Wave Parameters

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

This sub-topic covers the core physical parameters used to describe all progressive waves, the relationships between them and the meaning of phase difference. These parameters are foundational for all further wave topics in your A-Level course.

**Prerequisites:** [Basic classification of waves (transverse/longitudinal)](https://www.owlsprep.com/study/cie-9702-u7-introduction-to-waves/); Radian measure for angles

## Learning objectives

- Define and recall all core wave parameters correctly for exam questions
- Convert between period and frequency, and apply the wave equation to solve problems
- Calculate phase difference between two points on a progressive wave
- Distinguish between the parameters measurable from displacement-position and displacement-time graphs

## Core Wave Parameter Definitions

The following core parameters are routinely tested in definition questions in both Paper 1 and Paper 2 of CIE 9702. You must be able to recall and write each definition exactly to earn full marks.

| Parameter | Definition | SI Unit |
| --- | --- | --- |
| Displacement | Distance + direction from equilibrium | m |
| Amplitude | Maximum displacement from equilibrium | m |
| Wavelength | Distance between consecutive in-phase points | m |
| Period | Time for one full oscillation | s |
| Frequency | Number of oscillations per second | Hz (s⁻¹) |
| Wave speed | Distance travelled per unit time | m s⁻¹ |

**Worked example:** A wave completes 20 full oscillations in 5.0 seconds. State the period and frequency of the wave.

1. Period T is time per oscillation, so divide total time by number of oscillations:
2. $$T = \frac{5.0}{20} = 0.25 \text{ s}$$
3. Frequency is the reciprocal of period:
4. $$f = \frac{1}{T} = \frac{1}{0.25} = 4.0 \text{ Hz}$$

> **Exam tip:** Always mention equilibrium for displacement/amplitude and 'consecutive' for wavelength to get full marks for definitions.

## The Universal Wave Equation

All waves (mechanical and electromagnetic) obey the fundamental wave equation, derived directly from the definitions of wave speed, period and frequency.

**Derivation:** Derive the wave equation $v = f\lambda$

*Starting from:* In one period $T$, a wave travels exactly one wavelength $\lambda$

1. Speed equals distance divided by time:
2. $$v = \frac{\lambda}{T}$$
3. Substitute $f = \frac{1}{T}$:
4. $$v = \lambda \times f = f\lambda$$

*Conclusion:* This equation applies to all waves, regardless of type.

**Worked example:** Calculate the speed of red light with wavelength 700 nm and frequency $4.3 \times 10^{14}$ Hz in a vacuum.

1. Convert wavelength to SI units (metres):
2. $$700 \text{ nm} = 700 \times 10^{-9} = 7.0 \times 10^{-7} \text{ m}$$
3. Substitute into $v = f\lambda$:
4. $$v = (4.3 \times 10^{14}) \times (7.0 \times 10^{-7}) = 3.01 \times 10^8 \text{ m s}^{-1}$$

> **Exam tip:** Always convert all quantities to SI units before substitution to avoid order of magnitude errors.

## Phase and Phase Difference

Phase describes the position of an oscillation in its cycle, while phase difference measures how far out of step two oscillations are.

**Phase Difference** — The difference in phase between two points on a wave, measuring how much one oscillation is ahead or behind another.

*Example:* Two points one wavelength apart have a phase difference of $2\pi$ radians (360°) and are in phase.

$$\text{Phase difference } \Delta\phi = \frac{\Delta x}{\lambda} \times 2\pi \text{ (radians)}$$

**Worked example:** Two points on a wave of wavelength 1.2 m are separated by 0.3 m. Calculate the phase difference in radians.

1. Use the phase difference formula:
2. $$\Delta\phi = \frac{\Delta x}{\lambda} \times 2\pi$$
3. Substitute values:
4. $$\Delta\phi = \frac{0.3}{1.2} \times 2\pi = \frac{\pi}{2} \text{ radians}$$

> **Exam tip:** Always check if the question asks for radians or degrees, and give your answer in the required unit.

## Graphical Representations of Waves

There are two standard graphs used to represent waves, each showing different core parameters. Examiners regularly test the distinction between these two graph types.

| Graph Type | Axes | Parameters Read Directly |
| --- | --- | --- |
| Displacement-Position | y: Displacement, x: Position along wave | Amplitude, Wavelength |
| Displacement-Time | y: Displacement, x: Time of oscillation | Amplitude, Period |

**Worked example:** A displacement-time graph shows one full oscillation from $t=0$ to $t=0.1$ s, with maximum displacement 2.0 cm. Find amplitude and frequency.

1. Amplitude is maximum displacement, read directly from the graph:
2. $$A = 2.0 \text{ cm} = 0.020 \text{ m}$$
3. Period is time for one full oscillation: $T = 0.1$ s
4. Calculate frequency as reciprocal of period:
5. $$f = \frac{1}{T} = \frac{1}{0.1} = 10 \text{ Hz}$$

> **Exam tip:** Remember: period comes from time graphs, wavelength comes from position graphs.

## Common pitfalls

- **Wrong:** Defining wavelength as 'distance between two points in phase' without 'consecutive'
  - Why it fails: Points multiple wavelengths apart are also in phase, so the definition is incomplete
  - Correct: Always include 'consecutive' in your wavelength definition to earn full marks
- **Wrong:** Trying to read period from a displacement-position graph, or wavelength from a displacement-time graph
  - Why it fails: Position graphs show the wave at one instant, so they contain no time period information
  - Correct: Stick to the rule: period from time graphs, wavelength from position graphs
- **Wrong:** Using non-SI units like nanometres or centimetres directly in the wave equation
  - Why it fails: This leads to incorrect orders of magnitude, even if the method is correct
  - Correct: Always convert all units to metres and seconds before substituting into calculations
- **Wrong:** Leaving phase difference in degrees when the question asks for radians
  - Why it fails: Examiners do not award marks for answers in the wrong unit, even if the value is correct
  - Correct: Double-check the required unit for phase difference at the end of your calculation

## Cheatsheet

| Quantity | Relationship | SI Unit |
| --- | --- | --- |
| Frequency $f$ | $f = 1/T$ | Hz |
| Period $T$ | $T = 1/f$ | s |
| Wave speed | $v = f\lambda$ | m s⁻¹ |
| Phase difference | $\Delta\phi = (\Delta x / \lambda) \times 2\pi$ | radians |
| Displacement-position graph | Reads $A$, $\lambda$ | n/a |
| Displacement-time graph | Reads $A$, $T$ | n/a |

## What's next

Wave parameters are the foundation for all subsequent wave topics in CIE A-Level Physics, including transverse/longitudinal waves, interference, diffraction, standing waves and the electromagnetic spectrum. A strong mastery of these definitions and relationships will make every more complex wave topic much easier to understand, as most exam questions require you to apply these core concepts to new scenarios. Common extended response questions combine wave parameter calculations with wave phenomena like superposition and double-slit interference. Next, you will build on this knowledge to explore the properties of different wave types.

- [Electromagnetic Spectrum](https://www.owlsprep.com/study/cie-9702-u7-electromagnetic-spectrum/)
- [Wave Equation](https://www.owlsprep.com/study/cie-9702-u7-wave-equation/)
- [Wave Intensity](https://www.owlsprep.com/study/cie-9702-u7-wave-intensity/)

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