How It Works
The wave equation v = f × λ links three quantities: wave velocity v (m/s), frequency f (Hz), and wavelength λ (m). Knowing any two determines the third. For sound traveling in air at 20 °C, the speed of sound is about 343 m/s; in water it's roughly 1,480 m/s, and in steel it can exceed 5,000 m/s. The same identity rearranges to λ = v / f when solving for wavelength and to f = v / λ when solving for frequency.
Example Problem
A speaker emits a 440 Hz tone (concert A) in air at 20 °C. The speed of sound in air at this temperature is 343 m/s. What is the wavelength of the sound wave?
- Identify the known values: v = 343 m/s and f = 440 Hz.
- Rearrange the wave equation v = f × λ to solve for λ: λ = v / f.
- Substitute the numbers: λ = 343 / 440.
- Compute the wavelength: λ ≈ 0.78 m (78 cm).
- Sanity check the units: m/s ÷ 1/s = m, so the result is in meters as expected.
At 20 °C, the speed of sound in air is approximately v ≈ 331 + 0.6 × T (T in °C), so warmer air carries sound faster and shifts wavelengths upward at fixed frequency.
Key Concepts
The wave equation v = f × λ holds for every kind of mechanical and electromagnetic wave — sound in air, ripples on water, seismic P-waves, even light in a vacuum (where v = c). Sound propagates as longitudinal pressure waves in fluids and as both longitudinal and transverse waves in solids. The speed depends on the medium's bulk modulus and density (v = √(K/ρ) for fluids) and rises with temperature in gases. Doubling frequency halves wavelength at fixed velocity; doubling temperature in a gas raises sound speed by roughly √2 in the absolute scale.
Applications
- Architectural acoustics: matching room dimensions to half-wavelengths of bass frequencies for clean low-end response.
- Sonar and ultrasound: knowing v in water (~1,480 m/s) and the operating frequency determines the wavelength and resolution.
- Medical imaging: ultrasound transducers operate at 1–20 MHz; in soft tissue (v ≈ 1,540 m/s) this gives sub-millimeter wavelengths.
- Music acoustics: relating instrument string length and air-column length to fundamental wavelengths.
- Non-destructive testing: ultrasonic flaw detection in steel uses v ≈ 5,900 m/s to size defects by transit time.
Common Mistakes
- Mixing units — entering frequency in kHz but expecting wavelength in m without converting. Always normalize to Hz and m, or use the unit dropdowns.
- Using the speed of sound in air for a problem in water or steel — sound speed varies by an order of magnitude across media.
- Forgetting temperature dependence — at 0 °C v ≈ 331 m/s, at 20 °C ≈ 343 m/s, at 40 °C ≈ 355 m/s. The same frequency gives different wavelengths.
- Confusing angular frequency ω (rad/s) with ordinary frequency f (Hz) — the wave equation v = f × λ uses ordinary frequency.
- Applying the equation to dispersive waves without a frequency-dependent v — for shallow-water surface waves and capillary waves, v itself depends on λ.
Frequently Asked Questions
How do you calculate the speed of a sound wave?
Multiply the wave's frequency by its wavelength: v = f × λ. If a sound wave has frequency 440 Hz and wavelength 0.78 m, its speed is 440 × 0.78 ≈ 343 m/s — the speed of sound in air at 20 °C.
What is the formula for wave speed?
v = f × λ, where v is wave velocity, f is frequency, and λ is wavelength. The same identity rearranges to λ = v / f and f = v / λ when you need to solve for the other variables.
What is the speed of sound in air?
At 20 °C the speed of sound in dry air is about 343 m/s (768 mph). The approximate temperature dependence is v ≈ 331 + 0.6 × T, where T is in °C, so it's 331 m/s at 0 °C and 355 m/s at 40 °C.
Why is the speed of sound different in water than in air?
Sound speed in a medium is v = √(K/ρ), where K is bulk modulus and ρ is density. Water is much stiffer than air (high K) despite being much denser, and stiffness wins — sound travels about 4.3× faster in water (~1,480 m/s) than in air (~343 m/s).
What is the wavelength of a 440 Hz sound in air?
Using v = 343 m/s for air at 20 °C, λ = v / f = 343 / 440 ≈ 0.78 m or about 78 centimeters. Higher pitches give shorter wavelengths in the same medium.
Does the speed of sound depend on frequency?
For ordinary sound in air, water, and most solids the speed is essentially the same across the audible range — sound is non-dispersive in these media. Speed does vary with the medium's temperature, density, and stiffness, but not with frequency.
Reference: Tipler, Paul A. Physics For Scientists and Engineers. Worth Publishers. Halliday, Resnick, Walker. Fundamentals of Physics.
Worked Examples
Three sound-wave speed problems across music, medicine, and ocean acoustics. Click 'Load this example' to populate the inputs above.
MUSIC ACOUSTICS
Wavelength of concert A (440 Hz) in air
A piano tuner plays concert A at 440 Hz in a room at 20 °C, where the speed of sound is 343 m/s. What is the wavelength of the tone?
- v = 343 m/s, f = 440 Hz, solve for λ
- λ = v / f = 343 / 440
Wavelength λ ≈ 0.78 m (about 78 cm).
Bass notes have much longer wavelengths — a 55 Hz low A has λ ≈ 6.2 m, which is why subwoofers couple poorly to small rooms.
MEDICAL ULTRASOUND
Wavelength of 5 MHz ultrasound in soft tissue
A diagnostic ultrasound probe emits 5 MHz pulses through soft tissue where the speed of sound is approximately 1,540 m/s. What is the wavelength?
- v = 1,540 m/s, f = 5,000,000 Hz, solve for λ
- λ = 1540 / 5,000,000
Wavelength λ ≈ 3.08 × 10⁻⁴ m ≈ 0.31 mm.
Sub-millimeter wavelengths are why ultrasound can resolve organ boundaries — image resolution is roughly λ/2.
OCEAN ACOUSTICS
Frequency of a 5 m wavelength sonar pulse in seawater
An ocean sonar system emits pulses with a 5 m wavelength in seawater (v ≈ 1,500 m/s). What is the operating frequency?
- v = 1,500 m/s, λ = 5 m, solve for f
- f = v / λ = 1500 / 5
Frequency f = 300 Hz.
Low-frequency long-wavelength sonar travels much farther in water but resolves objects more coarsely than high-frequency sonar.
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