The speed of sound depends on how quickly a medium can respond elastically to a compression and on how much inertia the medium has. A stiffer medium usually carries sound faster, while a denser medium usually slows it down. That is why sound travels much faster in water and steel than in air.
In air near everyday conditions, a common classroom approximation is
\[
v \approx 331 + 0.6T,
\]
where \(T\) is the temperature in °C and \(v\) is in m/s. This is a very convenient formula because it shows directly that sound speed increases with temperature. For example, at \(20^\circ\text{C}\),
\[
v \approx 331 + 0.6(20)=343\ \text{m/s}.
\]
This is the standard value often used in introductory acoustics problems.
A more general formula for gases is
\[
v = \sqrt{\frac{\gamma P}{\rho}},
\]
where \(\gamma\) is the adiabatic index, \(P\) is pressure, and \(\rho\) is density. This formula reflects the fact that sound in a gas is a compression wave. The gas must be compressible, but the response is fast enough that the process is treated approximately as adiabatic. For dry air, \(\gamma \approx 1.4\) is commonly used.
In liquids and solids, the same basic idea appears in the form
\[
v = \sqrt{\frac{E}{\rho}},
\]
where \(E\) stands for an effective elastic or bulk modulus. The modulus measures how resistant the material is to compression or deformation. A very large modulus means the medium is stiff, so the wave travels quickly. Steel, for example, has a much larger modulus than water, which is why sound can travel through steel at several thousand meters per second.
These formulas explain a number of real phenomena. If you see lightning and then hear thunder later, the time delay is due to the finite speed of sound in air. If sound speed in air is about \(343\ \text{m/s}\), then the distance to the lightning can be estimated from
\[
d = vt.
\]
A delay of about 3 seconds corresponds to roughly 1 km. This is one reason sound-speed calculations are so useful in everyday physics examples.
The calculator supports several common cases. For air, it uses the temperature approximation. For a custom gas, it uses the gas formula with \(\gamma\), \(P\), and \(\rho\). For water, steel, or a custom liquid/solid, it uses the elastic-medium formula. The interactive graph then shows how sound speed changes when the main parameter is varied, and the animation illustrates how a pulse moves along a fixed-distance track.
At more advanced level, sound speed can depend on humidity, composition, frequency, anisotropy, and other effects. For example, humid air slightly changes the result compared with dry air, and solids can support different kinds of waves with different speeds. Even so, the formulas used here are the standard starting point for sound-wave speed calculations in school and undergraduate physics.
