Sound Wavelength and Frequency Guide: Calculate Acoustic Wavelengths

The Sound Wavelength Formula

Sound waves follow the same basic wave equation as light, but with a crucial difference: sound travels much slower than light and its speed depends on the medium.

Where:

• λ (lambda) = wavelength in meters (m)
• v = speed of sound in the medium (m/s)
• f = frequency in hertz (Hz)

Unlike light, which travels at a constant speed in vacuum, sound speed varies significantly depending on:

• The medium (air, water, solid materials)
• Temperature (especially in gases)
• Humidity (slight effect in air)
• Pressure (minimal effect under normal conditions)

Speed of Sound in Different Media

Speed of Sound in Air

The speed of sound in air depends primarily on temperature. The standard formula is:

Common values:

For most practical calculations, using 343 m/s (at 20°C/68°F) is a reasonable approximation.

Speed of Sound in Liquids

Speed of Sound in Solids

Calculating Sound Wavelengths: Worked Examples

Example 1: Middle C (C4) in Air

Problem: Calculate the wavelength of middle C (262 Hz) in air at room temperature.

Solution:

• Frequency: f = 262 Hz
• Speed of sound at 20°C: v = 343 m/s
• λ = v/f = 343/262 = 1.31 m

Answer: 1.31 meters (about 4.3 feet)

Example 2: Concert Pitch A4 (440 Hz)

Problem: What is the wavelength of the A above middle C?

Solution:

• Frequency: f = 440 Hz
• Speed of sound: v = 343 m/s
• λ = 343/440 = 0.780 m = 78.0 cm

Answer: 78 cm (about 31 inches)

Example 3: Bass Frequency (60 Hz)

Problem: Calculate the wavelength of a deep bass note at 60 Hz.

Solution:

• λ = 343/60 = 5.72 m

Answer: 5.72 meters (about 18.8 feet)

This explains why bass frequencies require large speakers and can "wrap around" obstacles.

Example 4: High Treble (10,000 Hz)

Problem: What is the wavelength of a 10 kHz treble tone?

Solution:

• λ = 343/10,000 = 0.0343 m = 3.43 cm

Answer: 3.43 cm (about 1.35 inches)

High frequencies have short wavelengths and are easily absorbed or blocked by small obstacles.

Example 5: Ultrasound in Water (5 MHz)

Problem: Calculate the wavelength of medical ultrasound at 5 MHz in body tissue (v ≈ 1,540 m/s).

Solution:

• f = 5 MHz = 5,000,000 Hz
• v = 1,540 m/s
• λ = 1,540/5,000,000 = 0.000308 m = 0.308 mm

Answer: 0.31 mm

This short wavelength allows ultrasound to image fine anatomical details.

Example 6: Sonar in Seawater

Problem: A submarine sonar operates at 20 kHz in seawater. What is the wavelength?

Solution:

• f = 20,000 Hz
• v = 1,531 m/s (seawater)
• λ = 1,531/20,000 = 0.0766 m = 7.66 cm

Answer: 7.66 cm

Human Hearing and Wavelength Range

The human ear can detect frequencies from approximately 20 Hz to 20,000 Hz. Here are the corresponding wavelengths in air at 20°C:

The enormous range in wavelengths (from 17 meters to 1.7 cm) explains many acoustic phenomena:

• Bass wavelengths are larger than most rooms, so they wrap around obstacles and are hard to control
• High frequencies have wavelengths smaller than a hand, making them easy to block or absorb
• Diffraction becomes significant when obstacles are comparable in size to the wavelength

Musical Notes and Wavelengths

Musical pitch is directly related to frequency, and thus to wavelength. Here are the wavelengths for notes across the piano keyboard (at A4 = 440 Hz, standard concert pitch):

Low Register

Middle Register

High Register

Octave Relationships

Each octave doubles the frequency and halves the wavelength:

• A2 (110 Hz): λ = 3.12 m
• A3 (220 Hz): λ = 1.56 m (half of A2)
• A4 (440 Hz): λ = 0.78 m (half of A3)
• A5 (880 Hz): λ = 0.39 m (half of A4)

Practical Applications

Speaker Design and Placement

Speaker size and bass response are directly related to wavelength:

• Subwoofers (20-200 Hz) must move large volumes of air to reproduce wavelengths of 1.7-17 meters
• Woofers (200-2000 Hz) handle wavelengths from 17 cm to 1.7 m
• Tweeters (2000-20,000 Hz) reproduce wavelengths from 1.7 to 17 cm

The "quarter wavelength rule" suggests that speakers become inefficient when their diameter is less than λ/4 at the lowest frequency they reproduce.

Room Acoustics and Standing Waves

Room modes (standing waves) occur when room dimensions match half-wavelengths of sound frequencies:

Example: A room that is 5 meters long will have an axial mode at:

f = 343 / (2 × 5) = 34.3 Hz

And harmonic modes at 68.6 Hz, 102.9 Hz, 137.2 Hz, etc.

To avoid acoustic problems:

• Room dimensions should not have simple integer ratios (avoid 1:1, 1:2, 2:3)
• Recommended ratios include 1:1.4:1.9 or 1:1.6:2.3
• Bass traps should be sized to absorb the longest problematic wavelengths

Bass Traps and Acoustic Treatment

Effective bass absorption requires treatment that's thick relative to the wavelength:

• To absorb 100 Hz (λ = 3.43 m), traps should be at least λ/4 = 86 cm deep
• Typical 10 cm acoustic panels only effectively absorb above ~860 Hz
• Corner placement doubles effective depth (pressure doubles at boundaries)

Diffuser Design

Acoustic diffusers scatter sound waves. Their effectiveness depends on wavelength:

• Diffuser well depth should be approximately λ/4 for the target frequency
• Overall diffuser dimensions should span several wavelengths
• A diffuser designed for 1000 Hz (λ = 34 cm) needs wells about 8.5 cm deep

Microphone and Speaker Spacing

Phase issues occur when sounds arrive at different times. Critical distances relate to wavelength:

• 3:1 rule: Multiple microphones should be three times as far apart as each is from its source
• Comb filtering: Occurs when direct and reflected sounds are separated by λ/2 (180° out of phase)
• Stereo speaker spacing: Typically 2-4 meters apart, comparable to mid-frequency wavelengths

Concert Hall Design

Concert halls are designed with wavelength considerations in mind:

• Reverberation time depends on room volume and surface absorption at various frequencies
• Early reflections should arrive within 20-40 ms (7-14 meter path difference)
• Diffusion is achieved with surfaces having features comparable to wavelengths (10 cm to 2 m)
• Bass buildup must be controlled with absorptive materials thick enough for long wavelengths

Ultrasound and Infrasound

Ultrasound (Above 20 kHz)

Frequencies above human hearing have very short wavelengths:

In medical ultrasound, resolution is approximately equal to the wavelength, so higher frequencies give better resolution but less penetration.

Infrasound (Below 20 Hz)

Very low frequencies have enormous wavelengths:

Infrasound can travel vast distances and is used for detecting nuclear tests, volcanic eruptions, and meteorites.

Temperature Effects on Sound Wavelength

Since sound speed changes with temperature, wavelength also changes for a given frequency:

Example: A4 (440 Hz) at Different Temperatures

This is why orchestras tune to the oboe after warming up—the instruments' air columns have warmed, changing the effective wavelengths and thus the pitch.

Wavelength and Diffraction

Sound waves bend around obstacles when the obstacle size is comparable to or smaller than the wavelength. This diffraction effect explains many acoustic phenomena:

• Bass bending around corners: A 100 Hz wave (λ = 3.4 m) easily bends around doorways and furniture
• Treble being blocked: A 10,000 Hz wave (λ = 3.4 cm) is blocked by objects larger than a few centimeters
• Head shadowing: The human head (about 20 cm) significantly attenuates frequencies above ~1,700 Hz, helping with sound localization

The transition frequency where an obstacle becomes significant is roughly when its size equals the wavelength:

Example: A 50 cm wide speaker cabinet significantly affects sound above:

f = 343 / 0.5 = 686 Hz

Quick Reference: Sound Wavelength Formulas

Use our sound wavelength calculator to quickly convert between frequency and wavelength in various media and temperatures.

Complete Musical Notes and Wavelengths Table (C2 through C8)

The following table provides the frequency, wavelength in air (at 20°C, v = 343 m/s), and wavelength in feet for every note in the chromatic scale from C2 through C8. These values are based on the standard equal temperament tuning system with A4 = 440 Hz. The frequency of each note is calculated using f = 440 × 2^((n-49)/12), where n is the key number on an 88-key piano.

This table is invaluable for instrument builders, sound engineers, and acoustic designers. For example, a pipe organ builder needs to know that the C2 pipe must be approximately 5.24 meters (about 17 feet) long for an open pipe, or half that for a stopped (closed) pipe. Similarly, guitar luthiers use these wavelength values to calculate fret positions and optimize body cavity resonances.

Notice that the wavelength at C2 (5.24 m) is larger than many living rooms, while the wavelength at C8 (8.2 cm) is about the size of a tennis ball. This thousandfold range in wavelength across the piano keyboard has profound implications for how different registers of music interact with rooms and listeners.

Expanded Sound Speed in Various Materials

The speed of sound varies enormously across different materials, which directly affects wavelength calculations. The following expanded table covers a wide range of gases, liquids, solids, and biological tissues relevant to acoustics, engineering, and medical applications.

The wavelength columns for 1 kHz and 440 Hz (concert A) provide immediate practical reference values. For instance, a 1 kHz tone has a wavelength of 34.3 cm in air but 5.96 m in steel, meaning the same frequency produces a wavelength 17.4 times longer in steel. This is why ultrasonic inspection of steel uses very high frequencies (typically 1-10 MHz) to achieve useful resolution.

The biological tissue values are critical for medical ultrasound design. The average soft tissue speed of 1,540 m/s is the standard calibration value for ultrasound machines. The significant speed difference between soft tissue and bone (1,540 vs. 4,080 m/s) causes strong reflections at bone interfaces, which is why ultrasound imaging through bone is difficult.

Room Acoustics Reference: Treatment Sizes vs. Target Frequencies

Acoustic room treatment effectiveness is directly tied to wavelength. Absorbers and diffusers must be properly sized relative to the wavelengths they target. This comprehensive reference table helps acoustic designers and home studio builders select appropriate treatment dimensions for their target frequencies.

How to Use This Table

The table connects four critical pieces of information for acoustic treatment design:

• Wavelength: The physical size of the sound wave at that frequency. Obstacles and treatments smaller than the wavelength have little effect on the sound.
• λ/4 (Quarter Wavelength): The minimum depth of a porous absorber needed to effectively absorb that frequency. This is because maximum particle velocity (where absorption occurs) happens at λ/4 from a rigid wall boundary.
• Room Dimension for Mode: The room length, width, or height that produces a standing wave (axial mode) at that frequency. Calculated as L = v/(2f). If your room has a dimension matching this value, expect a resonance peak at that frequency.
• Treatment Type: The recommended acoustic treatment approach for that frequency range.

For example, if your room is 3.43 meters wide (about 11.3 feet), you will have a standing wave at 50 Hz. To treat this resonance with a porous absorber, you would need treatment at least 1.72 meters deep, which is impractical. Instead, membrane (panel) bass traps or Helmholtz resonators tuned to 50 Hz should be used in the corners where pressure is maximum. For mid-range frequencies above 250 Hz, standard 10-15 cm thick acoustic panels (mounted with an air gap) provide effective absorption.

Summary

Understanding sound wavelength is essential for acoustics, audio engineering, and music production:

• Basic formula: λ = v/f, where v is the speed of sound in the medium
• Speed of sound in air: 343 m/s at 20°C (increases with temperature)
• Human hearing range: 20 Hz to 20 kHz corresponds to wavelengths from 17 m to 1.7 cm
• Octave relationship: Doubling frequency halves the wavelength
• Room acoustics: Standing waves occur when dimensions equal half-wavelengths
• Acoustic treatment: Bass traps must be sized relative to target wavelengths
• Diffraction: Sound bends around obstacles smaller than its wavelength

Frequently Asked Questions