1. The problem is to understand the expression for the function $$u(x,y,t) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \cos\left(c\pi t \left(\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2\right)\right) \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right)$$. 2. This expression represents a double infinite series summing over indices $m$ and $n$ starting from 1 to infinity. 3. Each term in the sum is a product of three parts: - $A_{mn}$: coefficients that depend on $m$ and $n$. - A cosine term: $$\cos\left(c\pi t \left(\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2\right)\right)$$ which depends on time $t$, constants $c$, $a$, $b$, and indices $m$, $n$. - Two sine terms: $$\sin\left(\frac{m\pi x}{a}\right)$$ and $$\sin\left(\frac{n\pi y}{b}\right)$$ which depend on spatial variables $x$ and $y$ and constants $a$, $b$. 4. This type of series often appears in solutions to partial differential equations like the wave equation or heat equation in rectangular domains, where $a$ and $b$ represent the domain lengths in $x$ and $y$ directions. 5. The cosine term represents oscillations in time with frequencies determined by $$c\pi \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$$. 6. The sine terms represent spatial modes that satisfy boundary conditions (e.g., zero at boundaries) in $x$ and $y$ directions. 7. The double summation means the solution is a superposition of all these modes, each weighted by $A_{mn}$. 8. In summary, this step expresses the solution as a sum of separable modes in space and time, each mode oscillating with a specific frequency and spatial pattern.