1. **Problem Statement:** (a) Write down the governing equation for the deflection of a rectangular membrane. (b) Find the deflection of a rectangular membrane of sides $a$ and $b$ with wave speed squared $c^2=1$, initial displacement $f(x,y) = \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right)$, and initial velocity zero. 2. **Governing Equation:** The deflection $u(x,y,t)$ of a rectangular membrane satisfies the two-dimensional wave equation: $$\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)$$ with boundary conditions $u=0$ on the edges $x=0,a$ and $y=0,b$ (fixed edges). 3. **Initial Conditions:** $$u(x,y,0) = f(x,y) = \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right)$$ $$\frac{\partial u}{\partial t}(x,y,0) = 0$$ 4. **Solution Method:** The solution can be expressed as a product of spatial eigenfunctions and time-dependent terms: $$u(x,y,t) = X(x) Y(y) T(t)$$ where $$X(x) = \sin\left(\frac{m\pi x}{a}\right), \quad Y(y) = \sin\left(\frac{n\pi y}{b}\right)$$ for integers $m,n$. 5. **Eigenfrequencies:** The natural frequencies are given by $$\omega_{m,n} = c \pi \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$$ 6. **Applying Initial Conditions:** Since the initial displacement matches the eigenfunction with $m=6$, $n=2$, and initial velocity is zero, the solution is $$u(x,y,t) = \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right) \cos\left(\omega_{6,2} t\right)$$ 7. **Substitute $c^2=1$:** $$\omega_{6,2} = \pi \sqrt{\left(\frac{6}{a}\right)^2 + \left(\frac{2}{b}\right)^2}$$ **Final answer:** $$u(x,y,t) = \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right) \cos\left(\pi t \sqrt{\frac{36}{a^2} + \frac{4}{b^2}}\right)$$ This describes the time evolution of the membrane's deflection with the given initial conditions.