Solve the wave equation

by the method of separation of variables. The boundary conditions for this system that describes the vibration of a rectangular membrane of area a · b is that at the edges the membrane cannot vibrate, that is:

Find the natural frequencies of this membrane and show that there is a degeneracy in the natural frequencies.

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Method of separation of variables.Let U(x, y)=X(x) Y(y) \cdot T(t)Then the ware equation becomeT Y \frac{\partial^{2} x}{\partial x^{2}}+X T \frac{\partial^{2} Y}{\partial y^{2}}-\frac{X Y}{c^{2}} \frac{\partial^{2} T}{\partial t^{2}}=0 \text {. }divide by X Y T\frac{1}{x} \frac{\partial^{2} x}{\partial x^{2}}+\frac{1}{Y} \frac{\partial^{2} Y}{\partial y^{2}}-\frac{1}{T c^{2}} \frac{\partial^{2} T}{\partial t^{2}}=0 \text {. }which implies each term is a constant\begin{aligned}\Rightarrow \frac{1}{x} \frac{\partial^{2} x}{\partial x^{2}} & =k_{x}^{2} \\\frac{\partial^{2} x^{3}}{\partial x^{2}} & =k_{x}^{2} X \\\Rightarrow X & =A e^{-k_{x} x}+B e^{-k_{x} x} \\\text { similarly } Y & =C e^{k_{y} y}+D e^{-k_{y} y} \\\text { and } & T=E e^{\omega c t}+F e^{-w c t} \\\Rightarrow U & =Y T \\U(0, y, t)=0 & =(A+B) Y T=0 . \\A & =-B\end{aligned}U(a, y, t)=A\left(e^{k_{x} a}-e^{-k_{x} a}\right) Y T=0 .\Rightarrow k_{x} should be complex number let k_{x}=i k\begin{array}{l} \Rightarrow e^{i k a}-e^{-i k a}=0 \\\Rightarrow 2 i \frac{\left(e^{i k a}-e^{-i k a}\right)}{2 i}=0 \\2 i \quad \sin (k a)=0 \\\Rightarrow=\frac{n \pi}{a} \cdot \\\Rightarrow U(x, y, t)=A \sin \left(\frac{n \pi}{a} x\right) Y T \\\Rightarrow \operatorname{similarly} Y(y)=C \sin \left(\frac{n \pi}{b} \cdot y\right) \\\Rightarrow\left.=A C \sin \left(\frac{n \pi x}{a}\right) \sin \left(\frac{n i y}{b}\right) E e^{\omega c t}+F e^{-w c t}\right)\end{array}In order to find \omega, E, F we need move boundary conditions.period of \sin \left(\frac{n \pi x}{a}\right) and \sin \left(\frac{n \pi y}{b}\right) ane T_{x}=2 a and T_{y}=2 b so frequncies are \frac{1}{2 a} and \frac{1}{2 b}. ...