1. **Problem Statement:** Find the deflection $u(x,y,t)$ of a rectangular membrane with sides $a$ and $b$, wave speed $c=1$, 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 membrane vibration satisfies the 2D 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 edges $x=0,a$ and $y=0,b$. 3. **Separation of Variables and Eigenmodes:** The solution can be expressed as a sum of eigenmodes: $$u(x,y,t) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \cos(c \pi t \sqrt{(m/a)^2 + (n/b)^2}) \sin\left(\frac{m \pi x}{a}\right) \sin\left(\frac{n \pi y}{b}\right)$$ 4. **Initial Conditions:** - Initial displacement: $u(x,y,0) = f(x,y) = \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right)$ - Initial velocity: $\frac{\partial u}{\partial t}(x,y,0) = 0$ 5. **Determining Coefficients:** Since initial velocity is zero, the time-dependent part is cosine only. The initial displacement matches exactly one eigenmode with $m=6$, $n=2$. Thus, all $A_{mn} = 0$ except $A_{6,2}$. 6. **Coefficient $A_{6,2}$:** Using orthogonality of sine functions, $$A_{6,2} = \frac{4}{ab} \int_0^a \int_0^b f(x,y) \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right) dy dx = 1$$ 7. **Final Solution:** $$u(x,y,t) = \cos\left(\pi t \sqrt{\left(\frac{6}{a}\right)^2 + \left(\frac{2}{b}\right)^2}\right) \sin\left(\frac{6\pi x}{a}\right) \sin\left(\frac{2\pi y}{b}\right)$$ This represents the membrane vibrating in the $(6,2)$ mode with frequency $\omega = \pi \sqrt{(6/a)^2 + (2/b)^2}$. **Summary:** The deflection is a single mode oscillation with the given initial shape and zero initial velocity.