From the arguments leading to the wave equation (2.1.1), show that in static equilibrium the displacement \(\zeta(x, y)\) of an elastic membrane obeys the two-dimensional Laplace equation $$ \nabla_{2}^{2} \zeta=\frac{\partial^{2} \zeta}{\partial x^{2}}+\frac{\partial^{2} \zeta}{\partial y^{2}}=0 $$ [Thus, given a uniformly stretched membrane, one may place blocks or clamps so as to fix some desired displacement along a closed boundary contour of arbitrary geometry. The apparatus is then an analog computer for solving Laplace's equation (in two dimensions) for geometries too complicated to handle analytically. This scheme is particularly useful in electrostatic problems, where the displacement \(\zeta\) represents the electrostatic potential \(V .\) Moreover, the trajectories of electrons in vacuum tubes can be plotted by rolling small ball bearings on such a model made from a rubber membrane.]

The wave equation is a fundamental concept in physics that describes the propagation of waves through different media. At its core, it is a second-order partial differential equation. In the context of an elastic membrane, such as a drumhead or a rubber sheet, the wave equation can be written as
\[\rho \frac{\text{d}^2 \text{zeta}}{\text{d} t^2} = \text{sigma} \left(\frac{\text{d}^2 \text{zeta}}{\text{d} x^2} + \frac{\text{d}^2 \text{zeta}}{\text{d} y^2}\right)\]
where \text{rho} represents the density and \text{sigma} is the tension within the membrane. The displacement of the membrane at a point \text{(x, y)} is represented by \text{zeta(x, y)}.
In a dynamic scenario, waves are created on the membrane as a result of forces acting upon it, causing displacements that depend on both time and space. It's interesting to note how the wave equation interconnects with static equilibrium conditions to derive the Laplace's equation, which is integral in describing static scenarios.

Static equilibrium is a situation where a system is at rest or its center of mass moves at constant velocity, and all forces acting upon it are balanced. In the case of a wave equation governing an elastic membrane, reaching static equilibrium means that the time-dependent part of the displacement is zero because the membrane is not moving. Mathematically, we express this as
\[\frac{\text{d}^2 \text{zeta}}{\text{d} t^2} = 0\]
This condition greatly simplifies the wave equation and allows us to derive the Laplace's equation for the given system. When we impose the static equilibrium condition on the wave equation for an elastic membrane, it eliminates the time derivative, setting the stage for a spatial analysis of membrane displacement without the dynamic effects of wave propagation. This ties directly into applications such as predicting the shape of a membrane under certain boundary conditions or solving complex electrostatic problems.

The displacement of an elastic membrane, denoted by \text{zeta(x, y)}, refers to the deviation from a membrane's rest position due to applied forces. When the membrane is in static equilibrium—meaning no further movement occurs—the resulting shape of the membrane is governed by the Laplace's equation:
\[abla_{2}^{2} \text{zeta} = \frac{\text{d}^2 \text{zeta}}{\text{d} x^2} + \frac{\text{d}^2 \text{zeta}}{\text{d} y^2} = 0\]
This equation illustrates that the sum of the second partial derivatives of \text{zeta} with respect to \text{x} and \text{y} equals zero, indicating a surface where the upward and downward forces balance out. The fascinating aspect of this equation is its practical applications in fields like electrostatics and fluid dynamics. You can visualize this concept by imagining a stretched rubber sheet that has been pinned at the edges; the sheet will naturally adopt a shape that satisfies the Laplace's equation for given boundary conditions.