# Plotting One-Dimensional Wave Equations

Recently, I have been trying to plot (or graph) the below one-dimensional wave equation:

$$T(x,y) = \sum_{n=1}^{\infty} \frac{4T_0}{\pi n \, \sinh(\pi n)} \sin \left(\frac{n \pi}{S} x \right) \sinh \left(\frac{n \pi}{S} y \right)$$

Note that $$T_0$$ is a constant and $$S$$ is an arbitrary (side) length.

With that said, I’ve been wanting to plot the above equation out and check whether or not does it fulfill the boundary conditions of $$T(0,y) = T(S,y) = T(x,0) = 0$$ and $$T(x,S) = T_0$$

I did try to manipulate the below Mathematica code which graphs a Fourier Series (along with piecewise functions) to graph my equation above. However, coding quickly became progressively difficult when dealing with $$x$$, $$y$$, $$S$$, and $$T_0$$.

fApprox[max_, t_] := (1/2) +    Sum[ Sin[2 n Pi]/(Pi n)  Cos[       n Pi t] + ((-1)^n - Cos[2 Pi n])/(Pi n) Sin[n Pi t], {n, 1, max}] f[t_] := Piecewise[{{0, 0 < t < 1}, {1, 1 < t < 2}}]; Manipulate[  Plot[{f[t], fApprox[nTerms, t]}, {t, 0, 2},   PlotRange -> {Automatic, {-0.3, 1.3}},   PlotStyle -> {{Thick, Blue}, Red},   Exclusions -> None   ],  {{nTerms, 5, "How many terms?"}, 1, 30, 1, Appearance -> "Labeled"},  TrackedSymbols :> {nTerms}  ] 

Source: Graphing a Fourier Series

Therefore, my question is how can I graph my one-dimensional wave equation and check whether or not does it fulfill the given boundary conditions on Mathematica? Is there a way for Mathematica to accommodate this many variables and arbitrary constants?

Thank you for reading through this as well as presenting your assistance! I sincerely appreciate any help offered by this community.

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