Give a brief physical interpretation of the following partial differential equations. Classify each equation as elliptic, hyperbolic, or parabolic by comparing it to the general form of a linear partial differential equation with two independent variables.

(a) \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\)

(b) \(\frac{\partial u}{\partial t}=\kappa \frac{\partial^{2} u}{\partial x^{2}}\).

(c) \(\frac{\partial^{2} u}{\partial t^{2}}=\alpha^{2} \frac{\partial^{2} u}{\partial x^{2}}\).