Civil engineers frequently encounter problems involving vibrating surfaces—such as bridges, building floors, or membranes like drums or architectural fabrics. Understanding how these structures respond to external forces and oscillate over time is crucial for ensuring stability, safety, and performance. The behavior of such systems is modeled using partial differential equations, particularly the two-dimensional wave equation. This chapter presents the mathematical modelling of a vibrating membrane and develops the two-dimensional wave equation governing its motion.

A membrane is a thin, flexible surface stretched tightly across a frame—like a drumhead. When disturbed, it vibrates in complex patterns depending on the initial force, boundary constraints, and its tension and mass. Let us consider:

• A rectangular membrane lying in the xy-plane, occupying the domain 0<x<a, 0<y<b.
• It is tightly stretched and fixed along the boundary.
• Let u(x, y,t) represent the vertical displacement of the membrane at position (x,y) and time t.

Assumptions:
- The membrane is homogeneous and isotropic.
- Tension T is uniform across the surface.
- The motion is small (linearization valid).
- No external force acts during motion (free vibration).

Let:
- ρ: mass per unit area (surface density),
- T: tension per unit length (N/m),
- u(x, y,t): vertical displacement.

Small Element Analysis
Take a small rectangular element Δx×Δ y. The vertical force due to tension is:

\[ F = T \frac{\partial^{2}u}{\partial x^{2}} + T \frac{\partial^{2}u}{\partial y^{2}} \Delta x \Delta y \]

According to Newton's second law:

\[ \rho \Delta x \Delta y \frac{\partial^{2}u}{\partial t^{2}} = T \frac{\partial^{2}u}{\partial x^{2}} + T \frac{\partial^{2}u}{\partial y^{2}} \Delta x \Delta y \]

Dividing both sides:

\[ \frac{\partial^{2}u}{\partial t^{2}} = \frac{T}{\rho} \left( \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} \right) \]

Or:

\[ \frac{\partial^{2}u}{\partial t^{2}} = c^{2} \nabla^{2}u \]

where \( c^{2} =\frac{T}{\rho} \) is the square of the wave speed.

The two-dimensional wave equation can be expressed as:

\[ \frac{\partial^{2}u}{\partial t^{2}} = c^{2} \nabla^{2}u = c^{2} \left( \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} \right) \]

This is the two-dimensional wave equation, a second-order linear PDE describing wave motion in a rectangular membrane.

Boundary Conditions (Dirichlet)
Since the membrane is fixed at the boundary:
- u(0,y,t)=u(a, y,t)=0, ∀0<y0
- u(x,0,t)=u(x,b,t)=0, ∀0<x0

Initial Conditions
At t=0:
- u(x, y,0)=f(x,y) (initial shape)
- \frac{\partial u}{\partial t}(x, y,0)=g(x,y) (initial velocity)