Wave Features and the Wave Equation
Waves are not just about what type they are — they have measurable features. How tall is the wave? How long is each cycle? How many cycles pass per second? And how fast does the whole pattern move? These are the questions that let us predict, measure and control waves in science and engineering.
Before You Begin
You are watching ocean waves at the beach. Some waves are small and gentle; others are tall and powerful. Some waves arrive close together; others are spaced far apart.
Write down your answers before reading on:
- What word would you use to describe how tall a wave is?
- What word would you use to describe the distance from one wave crest to the next?
- If ten wave crests pass a point in five seconds, how could you describe how frequent the waves are?
Know
- The definitions of amplitude, wavelength, frequency, period and wave speed
- The units for each wave feature: metres, hertz, seconds, metres per second
- The wave equation: v = f × λ
Understand
- How changing frequency or wavelength affects wave speed in a given medium
- That wave speed depends on the medium, not on amplitude
- The relationship between frequency and period
Can Do
- Label amplitude and wavelength on a wave diagram
- Calculate wave speed given frequency and wavelength
- Calculate frequency given wave speed and wavelength
Amplitude and Wavelength
The size and spacing of a wave
Newtons Laws
Two waves can be the same type and speed, but look completely different if their amplitude and wavelength differ.
Amplitude is the maximum distance a particle moves from its rest (equilibrium) position. In a water wave, amplitude is the height from the calm water level to the crest. In a sound wave, amplitude relates to how loud the sound is — greater amplitude means a more powerful vibration and a louder sound. In a light wave, amplitude relates to brightness.
Wavelength (symbol: lambda, the Greek letter) is the length of one complete wave cycle — for example, the distance from one crest to the next crest, or from one trough to the next trough. It is always measured in metres (m).
Frequency and Period
How often and how long
Frequency (symbol: f) is the number of complete waves that pass a fixed point every second. It is measured in hertz (Hz). 1 Hz means one wave per second. 1000 Hz means one thousand waves per second.
Period (symbol: T) is the time it takes for one complete wave to pass a fixed point. It is measured in seconds (s).
Frequency and period are reciprocals of each other:
For example, if a wave has a frequency of 5 Hz, its period is 1/5 = 0.2 seconds. That means one complete wave passes every 0.2 seconds.
| Feature | Symbol | Unit | What it measures |
|---|---|---|---|
| Amplitude | A | metres (m) | Maximum displacement from rest |
| Wavelength | lambda | metres (m) | Length of one complete wave cycle |
| Frequency | f | hertz (Hz) | Waves per second |
| Period | T | seconds (s) | Time for one wave to pass |
| Wave speed | v | metres per second (m/s) | How fast the wave travels |
The Wave Equation
Connecting speed, frequency and wavelength
The wave equation relates the three most important measurable properties of a wave:
Where:
- v = wave speed (m/s)
- f = frequency (Hz)
- lambda = wavelength (m)
This equation tells us that for waves travelling in the same medium (at the same speed), if frequency increases, wavelength must decrease — and vice versa. This is why high-pitched sounds (high frequency) have shorter wavelengths than low-pitched sounds.
Worked examples
Example 1: A water wave has a frequency of 2 Hz and a wavelength of 1.5 m. What is its speed?
v = f × λ = 2 × 1.5 = 3 m/s
Example 2: A sound wave travels at 340 m/s in air and has a frequency of 680 Hz. What is its wavelength?
lambda = v / f = 340 / 680 = 0.5 m
Example 3: A radio wave travels at 300 000 000 m/s and has a wavelength of 100 m. What is its frequency?
f = v / lambda = 300 000 000 / 100 = 3 000 000 Hz = 3 MHz
Common Misconceptions
"Higher frequency means faster wave speed." No — in a given medium, wave speed is constant. Higher frequency means shorter wavelength, not higher speed. The wave equation (v = f × λ) shows that if v stays the same, f and lambda are inversely proportional.
"Amplitude and wavelength are related — a taller wave must have a longer wavelength." No — amplitude and wavelength are independent properties. You can have a wave with high amplitude and short wavelength, or low amplitude and long wavelength.
Tsunami Warning and Surf Forecasting
Australia's coastline is monitored by the Bureau of Meteorology and Geoscience Australia for tsunami threats. When an earthquake occurs under the ocean, the resulting tsunami is a wave with extremely long wavelength (sometimes over 100 km in deep water) and very high energy. In the deep ocean, a tsunami travels at speeds over 800 km/h — yet its amplitude may be less than 1 m, making it almost invisible to ships.
As the wave approaches shallow water, its speed decreases but its amplitude increases dramatically — this is why tsunamis become devastating near coastlines. Australian scientists use the wave equation and deep-ocean pressure sensors to predict arrival times and wave heights, giving coastal communities critical warning time.
Similarly, surf forecasts for Australia's famous beaches use wave period (the time between waves) and wavelength data from offshore buoys to predict wave quality — longer period swells (10-15 seconds) produce better surfing conditions because they carry more energy.
✍ Copy Into Your Books
▾Wave Features
- Amplitude: max displacement from rest (m)
- Wavelength: length of one cycle (m)
- Frequency: waves per second (Hz)
- Period: time for one wave (s)
Wave Equation
- v = f × λ
- v = wave speed (m/s)
- f = frequency (Hz)
- lambda = wavelength (m)
Key Relationships
- T = 1/f and f = 1/T
- Same medium = same speed
- Higher f means shorter lambda
- Amplitude does not affect speed
Calculate Wave Speed
Wave Feature Detective
Test Your Understanding
1. What is the unit of frequency?
2. A wave has a frequency of 4 Hz. What is its period?
3. A water wave has a wavelength of 3 m and a speed of 6 m/s. What is its frequency?
4. In a given medium, the speed of sound is constant. If the frequency of a sound wave is doubled, what happens to its wavelength?
5. A student measures the amplitude of a wave as the distance from a crest to the next trough. What is wrong with this method?
Short Answer Questions
1. Define amplitude and wavelength. Explain how each is measured and state its unit. 4 MARKS
2. A sound wave in air has a frequency of 440 Hz and travels at 340 m/s. Calculate its wavelength. Then calculate its period. Show all working. 4 MARKS
3. Explain why tsunami waves in the deep ocean can travel at over 800 km/h with very small amplitude, yet become devastatingly large when they reach shallow water near the coast. Use the wave equation and the concept of wave speed changing in different media in your answer. 4 MARKS
Revisit Your Thinking
Go back to your Think First answer. Has your understanding changed?
- Can you now define all five wave features with their units?
- Can you rearrange the wave equation to solve for any variable?
Answers
▾MCQ 1
C — Frequency is measured in hertz (Hz), which means cycles per second.
MCQ 2
B — Period T = 1/f = 1/4 = 0.25 seconds.
MCQ 3
A — Using v = f × λ, rearranged: f = v / lambda = 6 / 3 = 2 Hz.
MCQ 4
D — Since v = f × λ and v is constant in a given medium, if f doubles, lambda must halve to keep the product the same.
MCQ 5
B — Amplitude is the maximum displacement from the rest position (centre line), not the total distance from crest to trough. Crest to trough would measure twice the amplitude.
Short Answer 1
Model answer: Amplitude is the maximum displacement of a particle from its rest position. It is measured as the distance from the centre line of the wave to either a crest or a trough. Its unit is metres (m). Wavelength is the distance between two consecutive corresponding points on a wave, such as from one crest to the next crest, or one trough to the next trough. It is measured in metres (m).
Short Answer 2
Model answer: Wavelength: lambda = v / f = 340 / 440 = 0.77 m (to 2 decimal places). Period: T = 1/f = 1/440 = 0.0023 s (or 2.3 milliseconds). The wavelength tells us the physical length of each sound wave in air, while the period tells us how long each wave takes to pass a point.
Short Answer 3
Model answer: In the deep ocean, tsunami waves travel very fast (over 800 km/h) with extremely long wavelengths (often over 100 km). Their amplitude is small (less than 1 m) because the energy is spread over a vast depth. According to the wave equation (v = f × λ), as the wave approaches shallow water, the wave speed decreases because the water depth is shallower. Since the frequency remains constant, the wavelength must also decrease. However, the total energy of the wave is conserved, so as the wavelength shortens and speed drops, the amplitude must increase dramatically. This is why a barely noticeable wave in deep water can become a devastating wall of water near the coast.
Wave Jumper
Jump through the wave platforms while testing your knowledge of wave features and the wave equation. Can you solve them all?
Mark lesson as complete
Tick when you have finished all activities and checked your answers.