343.2
m/s
1,235.52
km/h
100
m/s
1.0006
—
rad/m
343.2
m/s
1,235.52
km/h
100
m/s
1.0006
—
rad/m
The Wave Speed Calculator determines how fast a wave propagates through a medium using two methods: the fundamental wave equation $$v = f\lambda$$ and the string wave formula $$v = \sqrt{T/\mu}$$. Wave speed is a property of the medium, not the wave itself, and understanding it is crucial for acoustics, optics, seismology, and musical instrument design.
Sound travels at approximately 343 m/s in air at 20°C but speeds up to 1,480 m/s in water and 5,960 m/s in steel. Light in vacuum moves at $$c = 299{,}792{,}458$$ m/s but slows in glass and other materials. Seismic P-waves travel through Earth's crust at 5,000–8,000 m/s. These differences in wave speed have profound practical consequences — from sonar ranging to earthquake early warning systems.
For waves on strings and cables, the speed depends on the mechanical properties of the string: the tension $$T$$ pulling it taut and the linear mass density $$\mu$$ (mass per unit length). This relationship $$v = \sqrt{T/\mu}$$ explains why tightening a guitar string raises the pitch — increasing tension increases wave speed and therefore frequency for a given string length.
The calculator computes wave speed using two independent methods:
Method 1 — Wave equation:
$$v = f \lambda$$
This universal relationship applies to all waves: sound, light, water waves, seismic waves, etc.
Method 2 — String wave speed:
$$v = \sqrt{\frac{T}{\mu}}$$
where $$T$$ is the string tension in newtons and $$\mu$$ is the linear mass density in kg/m. This applies to transverse waves on stretched strings, ropes, and cables.
The Mach number is the ratio of the wave speed to the speed of sound in air (343 m/s): $$M = v / 343$$. A Mach number greater than 1 indicates supersonic propagation.
The wave speed tells you how fast energy is transported through the medium. If both methods give different values, it means the frequency-wavelength combination does not match the string's physical properties — one set of parameters may correspond to a different wave mode. For strings, higher tension or lower density increases speed, resulting in higher frequencies for a given wavelength.
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Results
A 440 Hz sound wave with λ = 0.78 m travels at approximately 343 m/s — the speed of sound in air at room temperature. The Mach number is essentially 1.0.
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Results
A guitar high-E string with 70 N tension and μ = 0.39 g/m has a wave speed of about 424 m/s. The string wave speed (√(T/μ)) closely matches the product fλ.
The speed of sound in an ideal gas is $$v = \sqrt{\gamma R T / M}$$, where $$\gamma$$ is the heat capacity ratio, $$R$$ is the gas constant, $$T$$ is absolute temperature, and $$M$$ is molar mass. In air at 20°C, this gives approximately 343 m/s. Warmer air transmits sound faster.
Wave speed is determined by how quickly the medium can transmit a disturbance, which depends on its elastic properties (stiffness) and inertial properties (density). Stiffer media propagate waves faster; denser media slow them down. The general form is $$v = \sqrt{\text{elastic property} / \text{inertial property}}$$.
The phase velocity of a wave can exceed $$c$$ in certain media (e.g., inside a waveguide), but the group velocity — which carries information and energy — never exceeds $$c$$ in vacuum, consistent with special relativity.
For sound in air, speed increases about 0.6 m/s per degree Celsius: $$v \approx 331.3 + 0.606 \cdot T_{°C}$$ m/s. In solids, higher temperature generally decreases elastic modulus, slightly reducing wave speed.
Phase velocity $$v_p = \omega/k$$ is the speed of individual wave crests. Group velocity $$v_g = d\omega/dk$$ is the speed of the wave envelope (wave packet) and determines how fast energy propagates. In dispersive media, $$v_p \neq v_g$$.
Increasing tension $$T$$ provides a greater restoring force, pulling displaced string segments back to equilibrium faster. Since $$v = \sqrt{T/\mu}$$, doubling the tension increases the wave speed by a factor of $$\sqrt{2} \approx 1.41$$.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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