# Wave characteristics

> IB Physics SL · IB DP Physics SL Core
> Source: https://www.owlsprep.com/study/ib-physics-sl-u3-wave-characteristics/

This sub-topic covers the fundamental properties and definitions for progressive mechanical waves, including classification of wave types, key parameters, graphical interpretation, and the core wave equation. You will learn to calculate and interpret core wave characteristics.

**Prerequisites:** [Basic properties of oscillations](https://www.owlsprep.com/study/ib-physics-sl-u2-simple-harmonic-motion/)

## Learning objectives

- Classify transverse and longitudinal waves correctly
- Extract wave parameters from displacement-time and displacement-distance graphs
- Apply the wave equation $v = f\lambda$ to solve problems
- Relate wave intensity to amplitude and distance from a source
- Calculate phase difference between points on a wave

## Types of Waves

Waves transfer energy from one point to another without transferring matter. They are classified based on the direction of particle oscillation relative to the direction of energy propagation.

**Transverse Wave** — A wave where particles of the medium oscillate perpendicular to the direction of energy transfer

*Example:* Electromagnetic waves, waves on a stretched string, seismic S-waves

**Longitudinal Wave** — A wave where particles of the medium oscillate parallel to the direction of energy transfer

*Example:* Sound waves, seismic P-waves, pressure waves in a slinky

**Worked example:** Classify each of the following as transverse or longitudinal: (a) X-rays, (b) ultrasound from a medical scanner, (c) ripples on a water surface.

1. Recall the two definitions: transverse = oscillation perpendicular to energy flow, longitudinal = oscillation parallel to energy flow.
2. Analyze (a): X-rays are a form of electromagnetic radiation, so they are transverse.
3. Analyze (b): Ultrasound is a high-frequency sound wave, so it is longitudinal.
4. Analyze (c): Water surface ripples have particles moving in circles, but the net oscillation is perpendicular to the direction of travel, so they are transverse.

> **Exam tip:** Always refer to the direction of oscillation relative to energy transfer, not any other frame of reference when classifying waves.

*Calculator:* forbidden

## Graphs and Key Wave Parameters

Two common graphs are used to represent wave motion, and it is critical to distinguish what information each provides.

| Graph Type | X-Axis | Extractable Parameters | Non-Extractable Parameters |
| --- | --- | --- | --- |
| Displacement-Time | Time $t$ | Amplitude $A$, Period $T$ | Wavelength $\lambda$ |
| Displacement-Distance | Position $x$ | Amplitude $A$, Wavelength $\lambda$ | Period $T$ |

Core key definitions for wave parameters are:

**Frequency** — Number of full cycles per second, $f = \frac{1}{T}$, measured in Hertz (Hz)

*Notation:* $f$

**Wavelength** — Shortest distance between two points on a wave that are in phase, measured in meters (m)

*Notation:* $\lambda$

**Worked example:** A displacement-time graph for a wave shows one full cycle takes 0.02 s. The wave speed is 340 m/s. Calculate the wavelength of the wave.

1. Extract period from the displacement-time graph: $T = 0.02$ s.
2. Calculate frequency from period:
3. $$f = \frac{1}{T} = \frac{1}{0.02} = 50 \text{ Hz}$$
4. Rearrange the wave equation to solve for wavelength:
5. $$\lambda = \frac{v}{f} = \frac{340}{50} = 6.8 \text{ m}$$

*Calculator:* allowed

## Wave Equation and Phase Difference

All travelling waves follow the fundamental wave equation that relates wave speed, frequency and wavelength. Wave speed is the speed at which energy propagates through the medium.

$$v = f \lambda$$

**Phase Difference** — The difference in oscillation phase between two points on a wave, measured in radians. One full wavelength separation gives a phase difference of $2\pi$ radians.

*Notation:* $\Delta \phi$

For two points separated by distance $\Delta x$, phase difference is calculated as:

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

**Worked example:** Two points on a wave are 0.25 m apart. The wavelength of the wave is 1.0 m. What is the phase difference between the two points?

1. Substitute the values into the phase difference formula:
2. $$\Delta \phi = 2\pi \times \frac{0.25}{1.0} = 0.5\pi = \frac{\pi}{2} \text{ radians}$$
3. This means the two points are one quarter of a cycle out of phase with each other.

**Check your understanding**

Test your understanding:

1. What is the phase difference between two points 2 wavelengths apart?

   - $2\pi$ radians
   - $4\pi$ radians
   - $\pi$ radians
   - $2$ radians

   *Why:* One wavelength = $2\pi$ radians, so two wavelengths = $2 \times 2\pi = 4\pi$ radians.

*Calculator:* allowed

## Wave Intensity

Intensity measures the rate of energy transfer per unit area by a wave. It is a commonly tested relationship between intensity and amplitude.

**Intensity** — Power transferred per unit area perpendicular to the wave direction, measured in $\text{W m}^{-2}$

*Notation:* $I$

Intensity is proportional to the square of the wave amplitude: $I \propto A^2$. For a point source emitting waves equally in all directions, intensity also follows the inverse square law with distance from the source: $I \propto \frac{1}{r^2}$, where $r$ is distance from the source.

**Worked example:** At a distance of 2 m from a point source, the amplitude of a sound wave is 0.1 m. What is the amplitude at a distance of 4 m from the source?

1. Combine the two proportionalities: $I \propto \frac{A^2}{r^2}$. Intensity is proportional to $A^2$ and inversely proportional to $r^2$, so:
2. $$\frac{A_1^2}{r_1^2} = \frac{A_2^2}{r_2^2} \implies \frac{A_1}{r_1} = \frac{A_2}{r_2}$$
3. Rearrange to solve for the new amplitude $A_2$:
4. $$A_2 = A_1 \times \frac{r_1}{r_2} = 0.1 \times \frac{2}{4} = 0.05 \text{ m}$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Extracting wavelength directly from a displacement-time graph
  - Why it fails: Displacement-time graphs plot against time, not distance, so wavelength cannot be read directly
  - Correct: Use the graph to get period, calculate frequency, then use $v=f\lambda$ to find wavelength
- **Wrong:** Rearranging the wave equation as $\lambda = \frac{f}{v}$
  - Why it fails: Common algebraic error that loses easy marks in exams
  - Correct: Remember $v = f \times \lambda$, so $\lambda = \frac{v}{f}$
- **Wrong:** Claiming intensity is proportional to amplitude
  - Why it fails: The relationship is non-linear, and this is a common multiple-choice trap
  - Correct: Memorize $I \propto A^2$, intensity scales with the square of amplitude
- **Wrong:** Confusing transverse and longitudinal sound waves
  - Why it fails: Many students mix up sound with electromagnetic waves
  - Correct: All sound waves in gases/liquids are longitudinal, only electromagnetic waves are transverse
- **Wrong:** Stating one wavelength corresponds to $\pi$ radians phase difference
  - Why it fails: Confusing full cycle phase difference with half cycle
  - Correct: One full wavelength (one full cycle) = $2\pi$ radians, half wavelength = $\pi$ radians

## Cheatsheet

| Parameter | Symbol | Key Relationship | Units |
| --- | --- | --- | --- |
| Amplitude | $A$ | Max displacement from equilibrium | m |
| Wavelength | $\lambda$ | Distance between in-phase points | m |
| Period | $T$ | $T = 1/f$ | s |
| Frequency | $f$ | $f = 1/T$ | Hz |
| Wave speed | $v$ | $v = f\lambda$ | m/s |
| Phase difference | $\Delta\phi$ | $\Delta\phi = 2\pi (\Delta x/\lambda)$ | radians |
| Intensity | $I$ | $I \propto A^2$, $I \propto 1/r^2$ | W/m² |
| Transverse wave | - | Oscillation ⊥ energy direction | - |
| Longitudinal wave | - | Oscillation ∥ energy direction | - |

## What's next

Wave characteristics form the foundation for all further topics in wave behaviour for IB Physics SL. All wave phenomena including refraction, reflection, interference, diffraction and standing waves build on these core definitions and relationships. This sub-topic is regularly tested in both Paper 1 (multiple choice) and Paper 2 (structured questions) so mastering these concepts will give you a strong base for more complex wave problems. You will next apply these characteristics to specific wave behaviours and interactions.

- [Standing Waves](https://www.owlsprep.com/study/ib-physics-sl-u3-standing-waves/)
- [Wave interference](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-interference/)
- [Fields](https://www.owlsprep.com/study/ib-physics-sl-u4-overview/)

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