Problem: Let $u(x,t)$ be the solution of
\begin{gather*}
\frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = 0, \quad x \in (-\infty ,\infty ), \ t > 0 \\
u(x,0) = \sin x, \quad x \in (-\infty ,\infty ) \\
\frac{\partial u}{\partial t} (x,0) = \cos x, \quad x \in (-\infty ,\infty ),
\end{gather*}
for some positive real number $c$.
Let the domain of dependence of the solution $u$ at the point $P(3,2)$ be the line segment on the $x$-axis with end points $Q$ and $R$.
If the area of the triangle $PQR$ is $8$ square units, then the value of $c^2$ is _____________.
Solution:
The given PDE is hyperbolic (wave equation). The two characteristic lines are
\[
x - ct \quad \text{ and }\quad x + ct.
\]
Therefore, the triangle that will be made by the line segment is given below. The lines will intersect the $x$-axis at $x-ct$ and $x+ct$. Look at the figure below.
Therefore, the area of the triangle will be
\begin{align*}
\text{ area of } \triangle PQR & = 8 = \frac{1}{2}\times \text{ base } \times \text{ height } \\
& \implies 8 = \frac{1}{2} \times (2ct) \times 2 \\
& \implies c = \frac{4}{t}.
\end{align*}
As $t = 2$, the value of $c$ will be
\[
c = 2 \implies c^2 = 4.
\]
