# C.2 Travelling waves

> IB Physics HL · IB HL Physics 2025+
> Source: https://www.owlsprep.com/study/ib-physics-hl-u3-c-2-travelling-waves/

This sub-topic introduces travelling waves, which transfer energy without transferring net matter. You will learn to classify wave types, use the fundamental wave equation, write displacement functions, and calculate intensity for travelling waves.

**Prerequisites:** [Simple harmonic motion](https://www.owlsprep.com/study/ib-physics-hl-u2-t01-simple-harmonic-motion/)

## Learning objectives

- Describe the core properties of travelling waves
- Distinguish between transverse and longitudinal travelling waves
- Use the fundamental wave equation to solve for unknown wave parameters
- Write the displacement function for sinusoidal travelling waves
- Apply intensity relationships for waves from point sources

## Fundamentals of Travelling Waves

**Travelling Wave** — A disturbance that propagates through a medium (or vacuum for electromagnetic waves) transferring energy from one point to another, with no net transfer of matter.

*Example:* A water wave moving across a lake transfers energy to the shore without moving water from the middle of the lake to the shore.

All travelling waves are categorized by the direction of particle oscillation relative to the direction of wave propagation, into two main types:

**Transverse vs Longitudinal Waves** — Transverse waves have particle oscillation perpendicular to the direction of wave travel; longitudinal waves have oscillation parallel to the direction of travel.

*Example:* Transverse: electromagnetic waves, waves on a plucked guitar string; Longitudinal: sound waves, pressure waves in a slinky.

**Worked example:** Classify each of the following as transverse or longitudinal: (a) Light from the Sun, (b) Sound from a speaker, (c) Primary earthquake P-waves

1. Recall the classification rule: transverse = oscillation perpendicular to propagation, longitudinal = oscillation parallel to propagation.
2. (a) Light is an electromagnetic wave with electric and magnetic fields oscillating perpendicular to direction of travel: this is
3. transverse.
4. (b) Sound propagates as alternating compressions and rarefactions of air, with particles oscillating parallel to propagation: this is
5. longitudinal.
6. (c) Primary P-waves from earthquakes propagate as compression waves with oscillation parallel to travel direction: this is
7. longitudinal.

> **Exam tip:** IB questions often ask you to label compressions and rarefactions on a longitudinal wave diagram: always mark regions of high density and low density clearly.

## Key Parameters and the Fundamental Wave Equation

Travelling waves are described by four core parameters that relate through the fundamental wave equation, the most used relationship in wave physics:

- Wavelength $\lambda$: distance between two consecutive identical points on the wave (units: m)
- Period $T$: time for one full wave to pass a fixed point (units: s)
- Frequency $f = 1/T$: number of full waves passing a point per second (units: Hz)
- Wave speed $v$: speed at which the wave propagates (units: m s⁻¹)

$$v = f \$$

**Worked example:** A red laser has a wavelength of 650 nm in vacuum, where light travels at $3.00 \times 10^8$ m s⁻¹. Calculate the frequency of the laser light.

1. Rearrange the fundamental wave equation to solve for frequency:
2. $$f = \frac{v}{\lambda}$$
3. Convert wavelength to SI units: $\lambda = 650 \text{ nm} = 650 \times 10^{-9} \text{ m}$
4. Substitute values:
5. $$f = \frac{3.00 \times 10^8}{650 \times 10^{-9}} \approx 4.62 \times 10^{14} \text{ Hz}$$

**Check your understanding**

Check your understanding:

1. A wave has frequency 10 Hz and wavelength 2 m. What is its speed?

   - 5 m s⁻¹
   - 10 m s⁻¹
   - 20 m s⁻¹
   - 0.2 m s⁻¹

   *Why:* Using $v = f\lambda = 10 \times 2 = 20$ m s⁻¹

*Calculator:* allowed

## Displacement Function for Sinusoidal Travelling Waves

For a sinusoidal travelling wave, we can write the displacement of a particle at position $x$ and time $t$ as a function, using the additional parameters wave number $k = 2\pi/\lambda$ and angular frequency $\omega = 2\pi f$.

**Displacement Function for Travelling Waves** — For a wave of amplitude $A$, phase constant $\phi$: travelling in the +x direction: $y(x,t) = A \sin(kx - \omega t + \phi)$; travelling in the -x direction: $y(x,t) = A \sin(kx + \omega t + \phi)$.

*Notation:* $y(x,t)$

> **Sign Memory Hook**
>
> Minus for positive x, plus for negative x: 'MiP-PoN' to remember

**Worked example:** A sinusoidal wave travelling in the positive x-direction has amplitude 0.2 m, wavelength 4.0 m, frequency 2.5 Hz, and zero phase constant. Write the displacement function.

1. Calculate wave number $k$:
2. $$k = \frac{2\pi}{\lambda} = \frac{2\pi}{4.0} = 0.5\pi \text{ m}^{-1}$$
3. Calculate angular frequency $\omega$:
4. $$\omega = 2\pi f = 2\pi(2.5) = 5\pi \text{ rad s}^{-1}$$
5. Use the form for +x direction, with $A=0.2$ and $\phi=0$:
6. $$y(x,t) = 0.2 \sin(0.5\pi x - 5\pi t)$$

*Calculator:* allowed

## Energy Transfer and Intensity

The primary role of a travelling wave is to transfer energy from a source to surrounding space. Intensity describes how much energy a wave carries per unit area per unit time.

**Intensity Relationships** — Intensity $I = P/A$ where $P$ is power and $A$ is cross-sectional area. For a point source spreading uniformly in 3D, intensity follows the inverse square law $I \propto 1/r^2$, where $r$ is distance from the source. Intensity is also proportional to the square of amplitude: $I \propto A^2$.

**Worked example:** At 1 m from a point source, the intensity is 200 W m⁻² and amplitude is 0.4 m. Calculate the intensity at 2 m from the source, and find the new amplitude.

1. Use inverse square law for intensity: $I_1/I_2 = r_2^2/r_1^2$
2. $$\frac{200}{I_2} = \frac{2^2}{1^2} = 4 \implies I_2 = 50 \text{ W m}^{-2}$$
3. Use $I \propto A^2$, so $A_2 = A_1 \sqrt{I_2/I_1}$
4. $$A_2 = 0.4 \sqrt{50/200} = 0.4 \times 0.5 = 0.2 \text{ m}$$

> **Exam tip:** You will often be asked to relate intensity to amplitude and distance: always remember both proportionalities, they are common exam question topics.

## Common pitfalls

- **Wrong:** Mixing up the sign of the phase term for direction of travel, saying $(kx - \omega t)$ travels in the negative x-direction.
  - Why it fails: The sign convention is counter-intuitive for many students, leading to frequent errors.
  - Correct: Remember the mnemonic: minus for positive x, plus for negative x (MiP-PoN).
- **Wrong:** Claiming travelling waves transfer matter along with energy.
  - Why it fails: Demonstrations of water waves or sound waves can give the false impression that bulk matter moves with the wave.
  - Correct: Travelling waves only transfer energy; particles oscillate around their equilibrium position with no net displacement.
- **Wrong:** Changing frequency when a wave moves between media, assuming $v = f\lambda$ means all three variables change.
  - Why it fails: Frequency is determined by the source of the wave, not the medium.
  - Correct: Frequency stays constant when a wave moves between media; only speed and wavelength change.
- **Wrong:** Using a linear relationship between intensity and amplitude, saying if intensity doubles, amplitude doubles.
  - Why it fails: It is easy to misremember the proportionality between intensity and amplitude.
  - Correct: Intensity is proportional to amplitude squared, so $A_2 = A_1 \sqrt{I_2/I_1}$.

## Cheatsheet

| Quantity | Symbol | Key Relationship | Note |
| --- | --- | --- | --- |
| Wave speed | $v$ | $v = f\lambda = \omega/k$ | Constant for a given medium |
| Wavelength | $\lambda$ | $\lambda = v/f = 2\pi/k$ | Distance between identical wave points |
| Frequency | $f$ | $f = 1/T = \omega/2\pi$ | Constant across different media |
| Wave number | $k$ | $k = 2\pi/\lambda$ | Spatial frequency of the wave |
| +x direction wave | $y(x,t)$ | $A\sin(kx - \omega t + \phi)$ | Standard displacement form |
| -x direction wave | $y(x,t)$ | $A\sin(kx + \omega t + \phi)$ | Sign of $kx$ reversed |
| Intensity | $I$ | $I \propto 1/r^2 \propto A^2$ | Inverse square law for point sources |

## What's next

Travelling waves are the foundation for all further wave topics in IB Physics HL, and the parameters and relationships you learned here will be used in every subsequent wave topic. The next core topic is standing waves, which form when two identical travelling waves travel in opposite directions and superpose, creating a stationary wave pattern. After mastering standing waves, you will move on to more complex wave phenomena including interference, diffraction, refraction, and the Doppler effect, all of which are heavily weighted in IB exams, appearing in both multiple choice and extended response questions. Mastery of this sub-topic is essential to score full marks on all wave-related exam questions.

- [C.5 Doppler effect](https://www.owlsprep.com/study/ib-physics-hl-u3-c-5-doppler-effect/)
- [C.3 Wave phenomena](https://www.owlsprep.com/study/ib-physics-hl-u3-c-3-wave-phenomena/)
- [C.4 Standing waves and resonance](https://www.owlsprep.com/study/ib-physics-hl-u3-c-4-standing-waves-and/)

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