Wave Equation Formula
The wave equation v = fλ relates wave speed, frequency, and wavelength.
Learn how to calculate wave properties with worked examples.
The Formula
The wave equation is one of the most fundamental relationships in physics. It connects the speed of a wave to its frequency and wavelength.
Every wave — whether it is a sound wave, light wave, water wave, or radio wave — obeys this relationship. If you know any two of the three quantities, you can always find the third.
The formula tells us that faster waves either have higher frequencies or longer wavelengths (or both). Conversely, for a wave traveling at a fixed speed (like sound in air), increasing the frequency means the wavelength must decrease.
Variables
| Symbol | Meaning |
|---|---|
| v | Wave speed (measured in meters per second, m/s) |
| f | Frequency of the wave (measured in hertz, Hz, which means cycles per second) |
| λ | Wavelength — the distance between two consecutive crests or troughs (measured in meters, m) |
Key Relationships
The formula can be rearranged to solve for any variable:
- Wave speed: v = fλ
- Frequency: f = v / λ
- Wavelength: λ = v / f
Example 1
A guitar string vibrates at a frequency of 440 Hz (the note A4). The wave travels along the string at 320 m/s. What is the wavelength?
Identify the values: f = 440 Hz, v = 320 m/s
Rearrange the formula: λ = v / f
λ = 320 / 440
λ ≈ 0.727 m (about 72.7 cm)
Example 2
A radio station broadcasts at a wavelength of 3.0 m. Radio waves travel at the speed of light (3.0 × 10⁸ m/s). What is the broadcast frequency?
Identify the values: λ = 3.0 m, v = 3.0 × 10⁸ m/s
Rearrange: f = v / λ
f = 3.0 × 10⁸ / 3.0
f = 1.0 × 10⁸ Hz = 100 MHz (FM radio range)
Example 3
Sound in air has a frequency of 256 Hz and a wavelength of 1.34 m. What is the speed of sound?
Identify the values: f = 256 Hz, λ = 1.34 m
Apply the formula: v = fλ = 256 × 1.34
v ≈ 343 m/s (the speed of sound at room temperature)
When to Use It
Use the wave equation whenever you need to relate wave speed, frequency, and wavelength.
- Calculating the frequency of radio, microwave, or light waves
- Finding the wavelength of sound at different pitches
- Solving problems involving musical instruments and acoustics
- Working with electromagnetic spectrum problems
- Designing antennas where wavelength determines size