# Wave Equation

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

This module explains the core wave equation $v = f\lambda$, derives it from first principles, shows how to apply it to CIE exam problems, and clarifies common misconceptions about wave speed.

**Prerequisites:** [Basic wave properties (wavelength, frequency, period)](https://www.owlsprep.com/study/cie-9702-u7-wave-properties/)

## Learning objectives

- Recall and apply $v = f\lambda$ to solve standard wave problems
- Derive the wave equation from first principles of wave properties
- Distinguish between wave propagation speed and oscillating particle speed

## Derivation of the Wave Equation

The wave equation follows directly from the fundamental definitions of speed, frequency, wavelength and period. Speed is always distance divided by time, so we can adapt this definition for waves.

**Wave Period** — Time taken for one complete wave cycle to pass a fixed point

*Notation:* T

*Example:* A wave of 5 Hz frequency has a period of 0.2 s

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

*Starting from:* Basic definitions of speed, period and frequency

1. In one full period $T$, a wave travels exactly one wavelength $\lambda$:
2. Wave speed equals distance travelled divided by time taken:
3. $$v = \frac{\text{distance}}{\text{time}} = \frac{\lambda}{T}$$
4. We know frequency $f$ is the inverse of period: $f = \frac{1}{T}$
5. Substitute $\frac{1}{T} = f$ into the speed equation:

*Conclusion:* This gives the standard wave equation: $v = f\lambda$

> **Exam tip:** When asked for a derivation, always start from basic definitions to earn full marks in CIE exams.

## Applying the Wave Equation

For any wave, if you know two of the three quantities ($v$, $f$, $\lambda$), you can calculate the third by rearranging the wave equation. Always use SI units (metres for wavelength, hertz for frequency) to get the correct speed in m/s.

**Worked example:** A microwave has frequency 12 GHz. Given the speed of microwaves is $3.0 \times 10^8$ m/s, calculate the wavelength.

1. Convert frequency from GHz to Hz (SI unit):
2. $$12 \text{ GHz} = 12 \times 10^9 \text{ Hz} = 1.2 \times 10^{10} \text{ Hz}$$
3. Rearrange $v = f\lambda$ to solve for $\lambda$:
4. $$\lambda = \frac{v}{f}$$
5. Substitute values and calculate:
6. $$\lambda = \frac{3.0 \times 10^8}{1.2 \times 10^{10}} = 0.025 \text{ m} = 2.5 \text{ cm}$$

**Check your understanding**

Test your unit conversion and rearrangement skills

1. A sound wave of wavelength 1.7 m travels at 340 m/s. What is its frequency?

   - 20 Hz
   - 200 Hz
   - 578 Hz
   - 0.005 Hz

   *Why:* Correct! $f = v/\lambda = 340 / 1.7 = 200$ Hz.

> **Exam tip:** Unit conversion for prefixes (M, G, c, k) is a common trap: always check your units before substituting.

## Wave Speed vs Particle Speed

One of the most commonly tested misconceptions in CIE exams is confusing the speed of wave propagation through the medium with the speed of the individual particles of the medium.

> **warning**
>
> Wave speed depends **only** on the properties of the medium. Changing frequency does not change wave speed in a fixed medium — it changes wavelength proportionally instead.

**Worked example:** A wave generator on a fixed tension string produces 5 Hz waves with wavelength 0.4 m. If frequency is increased to 10 Hz, what is the new wave speed?

1. The tension (medium property) is unchanged, so wave speed stays constant. Calculate original speed:
2. $$v = f\lambda = 5 \times 0.4 = 2 \text{ m/s}$$
3. The new wave speed is still 2 m/s; the new wavelength becomes 0.2 m.

> **Exam tip:** If a question says the wave is travelling in the same medium, wave speed is constant regardless of frequency change.

## Common pitfalls

- **Wrong:** Forgetting to convert units from MHz/cm to SI units (Hz/m)
  - Why it fails: This leads to answers off by orders of magnitude, which are marked wrong
  - Correct: Always confirm wavelength is in metres and frequency in hertz before substituting into the equation
- **Wrong:** Claiming increasing frequency increases wave speed in a fixed medium
  - Why it fails: Wave speed depends only on medium properties, not frequency of the wave
  - Correct: Remember $v$ is constant for a fixed medium, so increasing $f$ decreases $\lambda$ proportionally
- **Wrong:** Deriving the wave equation by just writing $v=f\lambda$ with no steps
  - Why it fails: CIE awards marks for the derivation process starting from first principles
  - Correct: Start from $v = \lambda/T$, substitute $f = 1/T$ to get the final result
- **Wrong:** Confusing wave propagation speed with particle oscillation speed
  - Why it fails: Particles oscillate around equilibrium so their speed changes over a cycle, while wave speed is constant
  - Correct: Wave speed describes how fast energy travels through the medium, not how fast particles move

## Cheatsheet

| Quantity | Symbol | SI Unit | Relation |
| --- | --- | --- | --- |
| Wave speed | $v$ | m/s | $v = f\lambda$ |
| Frequency | $f$ | Hz | $f = v/\lambda = 1/T$ |
| Wavelength | $\lambda$ | m | $\lambda = v/f$ |
| Period | $T$ | s | $T = 1/f = \lambda/v$ |

## What's next

The wave equation is the foundational relation for all advanced wave topics in CIE A-Level Physics. It is used constantly to solve problems involving standing waves, interference, diffraction gratings, and electromagnetic radiation. A solid understanding of this equation and its underlying assumptions is required for almost all wave-related questions, including many high-mark extended response problems. Mastery of this sub-topic makes all subsequent wave topics much easier to grasp.

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

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