Editing Wave equation (section)

====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]]