Wave speed describes how fast a wave pattern travels through space. For many periodic waves, the fundamental relationship is
\[
v=f\lambda,
\]
where \(v\) is wave speed (m/s), \(f\) is frequency (Hz), and \(\lambda\) is wavelength (m). This comes from the idea that one wavelength
passes a fixed point each cycle, so distance per cycle is \(\lambda\) and cycles per second is \(f\).
Waves on a string. For a stretched string with tension \(T\) (newtons) and linear mass density \(\mu\) (kg/m),
the wave speed is
\[
v=\sqrt{\frac{T}{\mu}}.
\]
Increasing tension makes the string “stiffer” and increases wave speed, while increasing \(\mu\) makes the string heavier and slows the wave.
This formula is widely used in musical instruments: tightening a guitar string changes how waves travel and affects resonance frequencies.
Sound and waves in fluids. For small compressions in a fluid (including air), the speed of sound depends on how hard the medium is to compress
(bulk modulus \(B\), in pascals) and how much mass must be accelerated (density \(\rho\), kg/m³):
\[
v=\sqrt{\frac{B}{\rho}}.
\]
Large \(B\) (stiff medium) increases the speed; large \(\rho\) decreases it. For example, water has a much larger \(B\) than air, and sound travels
faster in water.
Phase vs group speed. In dispersive media, different wavelengths travel at different speeds, so the phase velocity \(v_p\) and group velocity
\(v_g\) are not the same. The simple formulas above assume ideal, non-dispersive behavior where one characteristic speed is meaningful.
University-level treatments introduce dispersion relations \( \omega(k) \), with \(v_p=\omega/k\) and \(v_g=d\omega/dk\).
This calculator focuses on the standard introductory wave-speed models and provides a plot to see how \(v\) changes when you vary one parameter
(frequency, wavelength, tension, density, etc.), keeping the other required values fixed.
