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Wavelength, Frequency
& Wave Speed

Master the relationship \( v = f \lambda \) and learn how wave speed, frequency, and wavelength connect every wave in the universe.

High School Physics

What Is a Wave?

A wave is a disturbance that transfers energy from one place to another without permanently displacing the medium. Waves are everywhere: sound, light, ocean swells, and even seismic tremors.

Direction
Transverse Wave

Three Key Quantities

Wavelength \( \lambda \)
The distance between two consecutive identical points on a wave (e.g., crest to crest). Measured in metres (m).
Frequency \( f \)
How many complete cycles pass a point each second. Measured in hertz (Hz), where \( 1\;\text{Hz} = 1\;\text{s}^{-1} \).
Wave Speed \( v \)
How fast the wave pattern travels through the medium. Measured in metres per second (m/s).

Visualising \( \lambda \) and \( f \)

\( \lambda \)
\( A \)
One full cycle in time = Period T
\( f = \frac{1}{T} \)
Crest
Trough

Why This Matters

Understanding the wave equation lets you predict behaviour across all wave types: sound in air, light in a vacuum, seismic waves in rock, and radio transmissions through space. One elegant relationship governs them all.

Sound
Concert A = 440 Hz. Knowing \( v = 343\;\text{m/s} \) in air gives \( \lambda \approx 0.78\;\text{m} \).
Light
Red light: \( \lambda \approx 700\;\text{nm} \). With \( v = 3 \times 10^8\;\text{m/s} \), \( f \approx 4.3 \times 10^{14}\;\text{Hz} \).

The Wave Equation

\( v = f \, \lambda \)
\( v \)
Wave speed (m/s)
\( f \)
Frequency (Hz)
\( \lambda \)
Wavelength (m)

Deriving \( v = f\lambda \)

1
Speed is distance over time: \( v = \dfrac{d}{t} \).
2
In one period \( T \) the wave travels exactly one wavelength \( \lambda \). So \( v = \dfrac{\lambda}{T} \).
3
Since \( f = \dfrac{1}{T} \), substitute: \( v = \lambda \cdot \dfrac{1}{T} = f\lambda \). ✓

Rearranging for \( f \) and \( \lambda \)

Find \( v \)
\( v = f \lambda \)
Find \( f \)
\( f = \dfrac{v}{\lambda} \)
Find \( \lambda \)
\( \lambda = \dfrac{v}{f} \)

Frequency vs Wavelength

High f, short λ
Low f, long λ

Speed Depends on the Medium

Wave speed is set by the properties of the medium (density, elasticity, temperature), not by the source. Changing the source changes \( f \) and \( \lambda \) together, keeping \( v \) constant in a given medium.

Medium Sound Speed Example
Air (\(20\,^\circ\text{C}\)) 343 m/s Normal speech
Water 1 480 m/s Sonar
Steel 5 960 m/s Railway vibrations
Vacuum (light) \( 3 \times 10^{8}\;\text{m/s} \) All EM waves

How to Apply the Equation

1
Identify knowns. Which two of \( v \), \( f \), and \( \lambda \) are given?
2
Choose the rearrangement. Solve \( v = f\lambda \) for the unknown variable.
3
Check units. Convert to SI if needed: metres, seconds, hertz.
4
Substitute and solve. Plug in numbers and compute.
5
Sanity check. Does the answer have reasonable magnitude and correct units?

Example 1: Finding Speed

A wave on a string has a frequency of \( f = 10\;\text{Hz} \) and a wavelength of \( \lambda = 0.5\;\text{m} \). What is its speed?

\( \lambda = 0.5\;\text{m} \)
\( f = 10\;\text{Hz} \)

Example 2: Finding Wavelength

A sound wave in air travels at \( v = 343\;\text{m/s} \). If a tuning fork vibrates at \( f = 440\;\text{Hz} \) (concert A), what is the wavelength?

1
Known: \( v = 343\;\text{m/s} \), \( f = 440\;\text{Hz} \).
2
Rearrange: \( \lambda = \dfrac{v}{f} \).
3
Substitute: \( \lambda = \dfrac{343}{440} \approx 0.780\;\text{m} \).

Example 3: Finding Frequency of Light

Green light has a wavelength of \( \lambda = 530\;\text{nm} \). Light travels at \( c = 3.00 \times 10^{8}\;\text{m/s} \). Find the frequency.

1
Convert units: \( 530\;\text{nm} = 530 \times 10^{-9}\;\text{m} = 5.30 \times 10^{-7}\;\text{m} \).
2
Rearrange: \( f = \dfrac{v}{\lambda} = \dfrac{c}{\lambda} \).
3
Substitute: \( f = \dfrac{3.00 \times 10^{8}}{5.30 \times 10^{-7}} \approx 5.66 \times 10^{14}\;\text{Hz} \).

Common Mistakes

✕ Forgetting Unit Conversion
Using nm, cm, or kHz directly in \( v = f\lambda \). Always convert to metres and Hz first.
✕ Confusing \( f \) and \( T \)
Plugging the period \( T \) where frequency \( f \) belongs. Remember: \( f = \frac{1}{T} \), they are reciprocals.
✕ Changing Speed with Source
Assuming a higher-pitched sound is faster. In a given medium, \( v \) stays constant: only \( f \) and \( \lambda \) change.

Summary Table

Quantity Symbol SI Unit Formula
Wave Speed \( v \) m/s \( v = f\lambda \)
Frequency \( f \) Hz (s\(^{-1}\)) \( f = v / \lambda \)
Wavelength \( \lambda \) m \( \lambda = v / f \)
Period \( T \) s \( T = 1 / f \)

Question 1 of 5

A wave has a frequency of \( 50\;\text{Hz} \) and a wavelength of \( 2\;\text{m} \). What is its speed?

Question 2 of 5

Sound travels at \( 343\;\text{m/s} \) in air. A note has a wavelength of \( 0.686\;\text{m} \). What is its frequency?

Question 3 of 5

If the frequency of a wave doubles while it stays in the same medium, what happens to the wavelength?

Question 4 of 5

A radio station broadcasts at \( f = 100\;\text{MHz} \). Radio waves travel at \( 3.00 \times 10^{8}\;\text{m/s} \). What is the wavelength?

Question 5 of 5

A wave has a period of \( T = 0.02\;\text{s} \) and travels at \( v = 340\;\text{m/s} \). What is its wavelength?