QuestionAugust 2, 2025

1. (a) Derive the equation for Vibrating Membrane as: (partial ^2u)/(partial t^2)=c^2[(partial ^2u)/(partial x^2)+(partial ^2u)/(partial y^2)] where c^2=(T)/(m)

1. (a) Derive the equation for Vibrating Membrane as: (partial ^2u)/(partial t^2)=c^2[(partial ^2u)/(partial x^2)+(partial ^2u)/(partial y^2)] where c^2=(T)/(m)
1. (a) Derive the equation for Vibrating Membrane as: (partial ^2u)/(partial t^2)=c^2[(partial ^2u)/(partial x^2)+(partial ^2u)/(partial y^2)] where c^2=(T)/(m)

Solution
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Answer

\frac {\partial ^{2}u}{\partial t^{2}}=c^{2}\left(\frac {\partial ^{2}u}{\partial x^{2}}+\frac {\partial ^{2}u}{\partial y^{2}}\right) where c^{2}=\frac {T}{m} Explanation 1. Understand the Physical System A vibrating membrane is modeled by considering small displacements u(x, y, t) from equilibrium. 2. Apply Newton's Second Law For a small element of the membrane, the net force equals mass times acceleration. The tension T acts along the edges, and the mass per unit area is m. 3. Express Forces in Terms of Displacement The force due to tension in the x-direction is proportional to \frac{\partial^2 u}{\partial x^2}, and similarly for the y-direction. 4. Formulate the Wave Equation Combine these forces using Newton's second law: \[ m \frac{\partial^2 u}{\partial t^2} = T \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \] 5. Simplify Using Given Relation Divide through by m and use c^2 = \frac{T}{m}: \[ \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \]

Explanation

1. Understand the Physical System<br /> A vibrating membrane is modeled by considering small displacements $u(x, y, t)$ from equilibrium.<br /><br />2. Apply Newton's Second Law<br /> For a small element of the membrane, the net force equals mass times acceleration. The tension $T$ acts along the edges, and the mass per unit area is $m$.<br /><br />3. Express Forces in Terms of Displacement<br /> The force due to tension in the $x$-direction is proportional to $\frac{\partial^2 u}{\partial x^2}$, and similarly for the $y$-direction. <br /><br />4. Formulate the Wave Equation<br /> Combine these forces using Newton's second law: <br />\[ m \frac{\partial^2 u}{\partial t^2} = T \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \]<br /><br />5. Simplify Using Given Relation<br /> Divide through by $m$ and use $c^2 = \frac{T}{m}$:<br />\[ \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \]
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