الوصفMembrane exampleA.gif |
English: Graphical representation of the solution to the wave equation for 2D membrane bounded by a rectangular region given by:
![{\displaystyle \nabla ^{2}u={\frac {1}{c^{2}}}u_{tt}\ (0\leq x\leq \pi ),\ (0\leq y\leq \pi ),\ (t\geq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1098d29ed462fd0d3209b16437da93fc556dbf71)
where c = 1 subject to the boundary conditions:
![{\displaystyle {\begin{aligned}u(0,y,t)=0,&\ u(\pi ,y,t)=0\\u(x,0,t)=0,&\ u(x,\pi ,t)=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76f099781467e1d769b783a5e6362948a1d5c0a5)
with the initial displacement of the membrane given by:
![{\displaystyle {\begin{aligned}u(x,y,0)&=x(\pi -x)y(\pi -y)\\u_{t}(x,y,0)&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54125be99780485f83567b7c655940b32ddc0314)
The solution is:
![{\displaystyle u(x,y,t)=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }C_{mn}\sin[(2n-1)x]\sin[(2m-1)y]\cos(\omega _{mn}t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a80241a844f4b7017ab0be96f50deac048e9fa68)
where:
![{\displaystyle \omega _{mn}={\sqrt {m^{2}+n^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4f012052f0bda54c300c6854a486e38c469913)
![{\displaystyle {\begin{aligned}C_{mn}&={\frac {4}{\pi ^{2}}}\int _{0}^{\pi }\sin(mx)\int _{0}^{\pi }x(x-\pi )y(y-\pi )\sin(ny)\ dy\ dx\\&={\frac {4\times 4\left(-1+(-1)^{m}\right)\left(-1+(-1)^{n}\right)}{\pi ^{2}m^{3}n^{3}}};\ n,m=1,2,\ldots \\&={\frac {64}{\pi ^{2}(2m-1)^{3}(2n-1)^{3}}};\ n,m=1,2,\ldots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a5ef5d81e92df975b6c0b5031005e745abaa69c)
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