Frequency to Wavelength Conversion: Complete Guide with Examples

The Frequency to Wavelength Formula

The conversion between frequency and wavelength is based on the fundamental wave equation. To convert frequency to wavelength, use this formula:

Where:

• λ (lambda) = wavelength
• v = wave speed (velocity)
• f = frequency

To convert wavelength back to frequency, simply rearrange the formula:

The key to successful conversion is knowing the correct wave speed for your application. Different types of waves travel at different speeds through different media.

Wave Speed Values for Common Applications

Before converting frequency to wavelength, you must know the wave speed. Here are the most commonly used values:

Electromagnetic Waves (Light, Radio, X-rays, etc.)

All electromagnetic waves travel at the speed of light in vacuum:

For most practical calculations, you can use c ≈ 3 × 10⁸ m/s as an approximation.

Sound Waves

Sound speed varies with the medium and conditions:

For sound in air, the speed varies with temperature according to:

Frequency Units and Conversions

Frequency is measured in hertz (Hz), where 1 Hz = 1 cycle per second. Depending on the application, you may encounter various prefixes:

Converting Between Frequency Units

To convert to Hz (required for wavelength calculation), multiply by the appropriate factor:

• kHz to Hz: Multiply by 1,000
• MHz to Hz: Multiply by 1,000,000
• GHz to Hz: Multiply by 1,000,000,000
• THz to Hz: Multiply by 1,000,000,000,000

Example: Convert 2.4 GHz to Hz

2.4 GHz = 2.4 × 10⁹ Hz = 2,400,000,000 Hz

Wavelength Units and Conversions

Wavelength results may need to be converted to more practical units depending on the wave type:

Quick Conversion Formulas for Light

For electromagnetic waves, these simplified formulas give wavelength directly in useful units:

Step-by-Step Conversion Process

Follow these steps for accurate frequency to wavelength conversion:

Step 1: Identify the Wave Type

Determine whether you're working with electromagnetic waves (light, radio, X-rays) or mechanical waves (sound, water waves). This determines the wave speed to use.

Step 2: Determine the Wave Speed

• For electromagnetic waves: Use c = 299,792,458 m/s
• For sound in air: Calculate using temperature, or use 343 m/s for 20°C
• For sound in other media: Look up the appropriate speed

Step 3: Convert Frequency to Hz

Ensure your frequency is in hertz (Hz). Convert from kHz, MHz, GHz, or THz as needed.

Step 4: Apply the Formula

Calculate λ = v / f

Step 5: Convert to Desired Units

Convert the result from meters to nm, μm, cm, or other units as appropriate.

Worked Examples: Electromagnetic Waves

Example 1: Visible Light (Green)

Problem: Convert 560 THz to wavelength.

Solution:

1. Wave type: Electromagnetic (light)
2. Wave speed: c = 299,792,458 m/s
3. Convert frequency: 560 THz = 560 × 10¹² Hz = 5.6 × 10¹⁴ Hz
4. Calculate: λ = 299,792,458 / (5.6 × 10¹⁴) = 5.354 × 10⁻⁷ m
5. Convert to nm: 5.354 × 10⁻⁷ m × 10⁹ nm/m = 535.4 nm

Answer: 535 nm (green light)

Example 2: WiFi Signal (2.4 GHz)

Problem: What is the wavelength of a 2.4 GHz WiFi signal?

Solution:

1. Wave type: Electromagnetic (radio/microwave)
2. Wave speed: c = 299,792,458 m/s
3. Convert frequency: 2.4 GHz = 2.4 × 10⁹ Hz
4. Calculate: λ = 299,792,458 / (2.4 × 10⁹) = 0.1249 m
5. Convert to cm: 0.1249 m × 100 = 12.49 cm

Answer: 12.5 cm

Example 3: FM Radio (100 MHz)

Problem: Calculate the wavelength for an FM station at 100 MHz.

Solution:

1. Wave type: Electromagnetic (radio)
2. Wave speed: c = 299,792,458 m/s
3. Convert frequency: 100 MHz = 100 × 10⁶ Hz = 10⁸ Hz
4. Calculate: λ = 299,792,458 / 10⁸ = 2.998 m

Answer: 3.0 meters

Example 4: X-ray Radiation

Problem: Medical X-rays have frequencies around 3 × 10¹⁷ Hz. What is the wavelength?

Solution:

1. Wave type: Electromagnetic (X-ray)
2. Wave speed: c = 299,792,458 m/s
3. Frequency already in Hz: f = 3 × 10¹⁷ Hz
4. Calculate: λ = 299,792,458 / (3 × 10¹⁷) = 9.99 × 10⁻¹⁰ m
5. Convert to Angstroms: 9.99 × 10⁻¹⁰ m × 10¹⁰ Å/m = 9.99 Å

Answer: 10 Å (1 nm)

Example 5: 5G mmWave (28 GHz)

Problem: Calculate the wavelength of 5G mmWave at 28 GHz.

Solution:

1. Wave type: Electromagnetic (millimeter wave)
2. Wave speed: c = 299,792,458 m/s
3. Convert frequency: 28 GHz = 28 × 10⁹ Hz
4. Calculate: λ = 299,792,458 / (28 × 10⁹) = 0.01071 m
5. Convert to mm: 0.01071 m × 1000 = 10.71 mm

Answer: 10.7 mm (hence "millimeter wave")

Worked Examples: Sound Waves

Example 6: Musical Note A4 (440 Hz)

Problem: Calculate the wavelength of concert pitch A (440 Hz) in air at 20°C.

Solution:

1. Wave type: Sound
2. Wave speed in air at 20°C: v = 343 m/s
3. Frequency: f = 440 Hz
4. Calculate: λ = 343 / 440 = 0.780 m
5. Convert to cm: 0.780 m × 100 = 78.0 cm

Answer: 78 cm (0.78 m)

Example 7: Ultrasound (5 MHz in tissue)

Problem: Medical ultrasound operates at 5 MHz. What is the wavelength in human tissue (v ≈ 1,540 m/s)?

Solution:

1. Wave type: Sound (ultrasound)
2. Wave speed in tissue: v = 1,540 m/s
3. Convert frequency: 5 MHz = 5 × 10⁶ Hz
4. Calculate: λ = 1,540 / (5 × 10⁶) = 3.08 × 10⁻⁴ m
5. Convert to mm: 3.08 × 10⁻⁴ m × 1000 = 0.308 mm

Answer: 0.31 mm

Example 8: Bass Frequency (60 Hz)

Problem: What is the wavelength of a 60 Hz bass tone in air at 20°C?

Solution:

1. Wave type: Sound
2. Wave speed in air at 20°C: v = 343 m/s
3. Frequency: f = 60 Hz
4. Calculate: λ = 343 / 60 = 5.72 m

Answer: 5.72 meters

This explains why bass waves bend around obstacles easily and require large speakers or subwoofers.

Example 9: Sonar in Seawater (50 kHz)

Problem: A sonar system operates at 50 kHz in seawater. Calculate the wavelength.

Solution:

1. Wave type: Sound (sonar)
2. Wave speed in seawater: v = 1,531 m/s
3. Convert frequency: 50 kHz = 50,000 Hz
4. Calculate: λ = 1,531 / 50,000 = 0.0306 m
5. Convert to cm: 0.0306 m × 100 = 3.06 cm

Answer: 3.06 cm

Frequency to Wavelength Conversion Tables

Electromagnetic Spectrum Quick Reference

Audio Frequency to Wavelength (Air at 20°C)

Common Mistakes to Avoid

Mistake 1: Using the Wrong Wave Speed

The most common error is using the speed of light for sound calculations or vice versa. Light travels at 299,792,458 m/s, while sound in air only travels at about 343 m/s—nearly a million times slower.

Example of error: Calculating a 1 kHz sound wavelength using speed of light would give 300 km instead of the correct 34.3 cm.

Mistake 2: Forgetting to Convert Units

If frequency is given in MHz and you forget to convert to Hz, your answer will be off by a factor of 10⁶.

Correct approach: Always convert frequency to Hz before applying the formula, then convert the wavelength result to appropriate units.

Mistake 3: Mixing Metric and Imperial Units

If wave speed is in m/s and you want wavelength in feet, you need an extra conversion step. Either convert wave speed to ft/s first, or convert the final answer from meters to feet (1 m = 3.281 ft).

Mistake 4: Ignoring Medium Effects

Light slows down in glass, water, and other materials. Sound speed varies with temperature and medium. Using vacuum/standard values without considering the actual medium introduces errors.

Mistake 5: Scientific Notation Errors

Be careful with powers of 10. GHz = 10⁹ Hz, not 10⁶. Double-check exponents, especially when calculating visible light wavelengths in nanometers.

Practical Applications

Antenna Design

Radio antennas are often sized as fractions of the wavelength:

• Quarter-wave antenna: Length = λ/4
• Half-wave dipole: Length = λ/2
• Full-wave loop: Circumference = λ

For a 2.4 GHz WiFi signal (λ = 12.5 cm), a quarter-wave antenna is about 3.1 cm long.

Audio and Acoustics

Room acoustics depend heavily on wavelength:

• Bass traps must be sized to absorb long wavelengths (60 Hz = 5.7 m)
• Diffusers scatter frequencies with wavelengths similar to their feature size
• Room modes occur when dimensions match half-wavelengths of bass frequencies

Fiber Optics

Optical fiber communications use specific wavelengths where glass has minimal absorption:

• 850 nm: Multimode fiber (353 THz)
• 1310 nm: Single-mode fiber (229 THz)
• 1550 nm: Long-haul fiber, lowest loss (193 THz)

Medical Imaging

Different wavelengths/frequencies are suited to different imaging tasks:

• Ultrasound (1-18 MHz): Higher frequency = better resolution but less penetration
• MRI RF pulses (10-300 MHz): Chosen to match hydrogen precession frequency
• CT X-rays (~10¹⁸ Hz): High frequency penetrates tissue for cross-sectional imaging

Converting Wavelength to Frequency

The reverse conversion uses the rearranged formula:

Example: Yellow Light (580 nm)

Problem: What is the frequency of yellow light with wavelength 580 nm?

Solution:

1. Convert wavelength to meters: 580 nm = 580 × 10⁻⁹ m = 5.8 × 10⁻⁷ m
2. Wave speed: c = 299,792,458 m/s
3. Calculate: f = 299,792,458 / (5.8 × 10⁻⁷) = 5.17 × 10¹⁴ Hz
4. Convert to THz: 5.17 × 10¹⁴ Hz = 517 THz

Answer: 517 THz

Online Calculator

For quick conversions without manual calculation, use our wavelength calculator. It handles all unit conversions automatically and works for electromagnetic waves, sound waves, and custom wave speeds.

Simply enter your frequency value, select the unit (Hz, kHz, MHz, GHz, THz), choose whether you're working with light (speed of light) or sound (with temperature adjustment), and get instant results in multiple wavelength units.

Pre-Calculated Frequency to Wavelength Conversion Table

For quick reference, the following table provides pre-calculated wavelength values for commonly encountered frequencies across the electromagnetic spectrum and sound. These values are calculated using the exact speed of light (299,792,458 m/s) for electromagnetic waves and 343 m/s for sound in air at 20°C.

Electromagnetic Wave Conversions

Sound Wave Conversions (Air at 20°C)

Metric Prefix Reference for Frequency and Wavelength

Mastering metric prefixes is essential for efficient frequency-to-wavelength conversion. Each step between adjacent prefixes represents a factor of 1,000 (10³). The following table provides a comprehensive reference for all metric prefixes commonly encountered in wave physics.

An important relationship to remember: as the frequency prefix increases (Hz to kHz to MHz to GHz to THz), the wavelength prefix decreases (km to m to cm to mm to μm). This is a direct consequence of the inverse relationship between frequency and wavelength at a constant wave speed.

Step-by-Step Conversion Examples at a Glance

The following table compiles a variety of conversion examples showing each step of the process. These cover different wave types, frequency ranges, and unit systems to serve as templates for your own calculations.

These examples demonstrate three important points. First, always verify that your wave speed is appropriate for the medium (electromagnetic waves use c, sound waves use the medium-specific speed). Second, consistent unit conversion is the most common source of errors. Third, expressing the final result in practical units (cm instead of m, or mm instead of m) makes the answer more meaningful for the application at hand.

Summary

Converting frequency to wavelength is straightforward once you know the process:

1. Identify the wave type and determine the appropriate wave speed
2. Convert frequency to hertz if necessary
3. Apply the formula: λ = v / f
4. Convert the result to desired units

Key wave speeds to remember:

• Light in vacuum: c = 299,792,458 m/s ≈ 3 × 10⁸ m/s
• Sound in air (20°C): 343 m/s
• Sound in water: ~1,480 m/s

The inverse relationship between frequency and wavelength means:

• Higher frequency = shorter wavelength
• Lower frequency = longer wavelength

Frequently Asked Questions