Editing Catenary (section)

===Elastic catenary===
In an [[Elasticity (physics)|elastic]] catenary, the chain is replaced by a [[Spring (device)|spring]] which can stretch in response to tension. The spring is assumed to stretch in accordance with [[Hooke's law]]. Specifically, if {{math|p}} is the natural length of a section of spring, then the length of the spring with tension {{mvar|T}} applied has length

<math display=block>s=\left(1+\frac{T}{E}\right)p\,,</math>

where {{mvar|E}} is a constant equal to {{mvar|kp}}, where {{mvar|k}} is the [[stiffness]] of the spring.<ref>[[#Routh|Routh]] Art. 489</ref> In the catenary the value of {{mvar|T}} is variable, but ratio remains valid at a local level, so<ref>[[#Routh|Routh]] Art. 494</ref>
<math display=block>\frac{ds}{dp}=1+\frac{T}{E}\,.</math>
The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.<ref>Following [[#Routh|Routh]] Art. 500</ref>

The equations for tension of the spring are

<math display=block>T \cos \varphi = T_0\,,</math>
and
<math display=block>T \sin \varphi = w_0 p\,,</math>

from which

<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{w_0 p}{T_0}\,,\quad T=\sqrt{T_0^2+w_0^2 p^2}\,,</math>

where {{mvar|p}} is the natural length of the segment from {{math|'''c'''}} to {{math|'''r'''}} and {{math|''w''<sub>0</sub>}} is the weight per unit length of the spring with no tension. Write
<math display=block>a = \frac{T_0}{w_0}</math>
so
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{p}{a} \quad\text{and}\quad T=\frac{T_0}{a}\sqrt{a^2+p^2}\,.</math>

Then 
<math display=block>\begin{align}
\frac{dx}{ds} &= \cos \varphi = \frac{T_0}{T} \\[6pt]
\frac{dy}{ds} &= \sin \varphi = \frac{w_0 p}{T}\,,
\end{align}</math>
from which
<math display=block>\begin{alignat}{3}
\frac{dx}{dp} &= \frac{T_0}{T}\frac{ds}{dp} &&= T_0\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{a}{\sqrt{a^2+p^2}}+\frac{T_0}{E} \\[6pt]
\frac{dy}{dp} &= \frac{w_0 p}{T}\frac{ds}{dp} &&= \frac{T_0p}{a}\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{p}{\sqrt{a^2+p^2}}+\frac{T_0p}{Ea}\,.
\end{alignat}</math>

Integrating gives the parametric equations

<math display=block>\begin{align}
x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p + \alpha\,, \\[6pt]
y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2+\beta\,.
\end{align}</math>

Again, the {{mvar|x}} and {{mvar|y}}-axes can be shifted so {{mvar|α}} and {{mvar|β}} can be taken to be 0. So

<math display=block>\begin{align}
x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p\,, \\[6pt]
y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2
\end{align}</math>

are parametric equations for the curve. At the rigid [[Limit (mathematics)|limit]] where {{mvar|E}} is large, the shape of the curve reduces to that of a non-elastic chain.