Editing Wave equation (section)

===Derivation===
{{see also|Acoustic wave equation}}
The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a [[String vibration|string vibrating]] in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of [[Tension (physics)|tension]].<ref name=Tipler>Tipler, Paul and Mosca, Gene. ''[https://books.google.com/books?id=upa42dyhf38C&pg=PA470 Physics for Scientists and Engineers, Volume 1: Mechanics, Oscillations and Waves; Thermodynamics]'', pp.&nbsp;470–471 (Macmillan, 2004).</ref>

Another physical setting for derivation of the wave equation in one space dimension uses [[Hooke's law]]. In the [[theory of elasticity]], Hooke's law is an approximation for certain materials, stating that the amount by which a material body is deformed (the [[deformation (mechanics)|strain]]) is linearly related to the force causing the deformation (the [[stress (mechanics)|stress]]).

====Hooke's law====
The wave equation in the one-dimensional case can be derived from [[Hooke's law]] in the following way: imagine an array of little weights of mass {{mvar|m}} interconnected with massless springs of length {{mvar|h}}. The springs have a [[stiffness|spring constant]] of {{mvar|k}}:
: [[Image:array of masses.svg|300px]]

Here the dependent variable {{math|''u''(''x'')}} measures the distance from the equilibrium of the mass situated at {{mvar|x}}, so that {{math|''u''(''x'')}} essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The resulting force exerted on the mass {{mvar|m}} at the location {{math|''x'' + ''h''}} is:
<math display="block">\begin{align}
 F_\text{Hooke}  &= F_{x+2h} - F_x = k [u(x + 2h, t) - u(x + h, t)] - k[u(x + h,t) - u(x, t)].
\end{align}</math>

By equating the latter equation with

<math display="block">\begin{align}
 F_\text{Newton} &= m \, a(t) = m \, \frac{\partial^2}{\partial t^2} u(x + h, t),
\end{align}</math>

the equation of motion for the weight at the location {{math|''x'' + ''h''}} is obtained:
<math display="block">\frac{\partial^2}{\partial t^2} u(x + h, t) = \frac{k}{m} [u(x + 2h, t) - u(x + h, t) - u(x + h, t) + u(x, t)].</math>
If the array of weights consists of {{mvar|N}} weights spaced evenly over the length {{math|1=''L'' = ''Nh''}} of total mass {{math|1=''M'' = ''Nm''}}, and the total [[stiffness|spring constant]] of the array {{math|1=''K'' = ''k''/''N''}}, we can write the above equation as

<math display="block">\frac{\partial^2}{\partial t^2} u(x + h, t) = \frac{KL^2}{M} \frac{[u(x + 2h, t) - 2u(x + h, t) + u(x, t)]}{h^2}.</math>

Taking the limit {{math|''N'' → ∞, ''h'' → 0}} and assuming smoothness, one gets
<math display="block">\frac{\partial^2 u(x, t)}{\partial t^2} = \frac{KL^2}{M} \frac{\partial^2 u(x, t)}{\partial x^2},</math>
which is from the definition of a [[second derivative]]. {{math|''KL''<sup>2</sup>/''M''}} is the square of the propagation speed in this particular case.

[[File:1d wave equation animation.gif|thumbnail|1-d standing wave as a superposition of two waves traveling in opposite directions]]

====Stress pulse in a bar====
In the case of a stress pulse propagating longitudinally through a bar, the bar acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness {{mvar|K}} given by
<math display="block">K = \frac{EA}{L},</math>
where {{mvar|A}} is the cross-sectional area, and {{mvar|E}} is the [[Young's modulus]] of the material. The wave equation becomes
<math display="block">\frac{\partial^2 u(x, t)}{\partial t^2} = \frac{EAL}{M} \frac{\partial^2 u(x, t)}{\partial x^2}.</math>

{{math|''AL''}} is equal to the volume of the bar, and therefore
<math display="block">\frac{AL}{M} = \frac{1}{\rho},</math>
where {{mvar|ρ}} is the density of the material. The wave equation reduces to
<math display="block">\frac{\partial^2 u(x, t)}{\partial t^2} = \frac{E}{\rho} \frac{\partial^2 u(x, t)}{\partial x^2}.</math>

The speed of a stress wave in a bar is therefore <math>\sqrt{E/\rho}</math>.