19.2. Derivation of the Two-Dimensional Wave Equation

Let:

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

In this segment, we define the essential variables and parameters used in our study of the wave equation for a vibrating membrane. Here, ρ represents the mass per unit area, which indicates how much mass is distributed over a given surface area of the membrane. T is the tension per unit length and is crucial in determining how much force is applied to the membrane when it vibrates. Finally, u(x, y,t) denotes the vertical displacement of the membrane at specific coordinates (x,y) and at a particular time (t), showing how far the membrane has moved from its rest position.

Think of a trampoline. The mass per unit area would be analogous to how thick the trampoline's surface is, the tension would relate to how tightly the surface is stretched, and the vertical displacement would represent how far down the trampoline dips when someone jumps on it.

Take a small rectangular element Δx×Δy. The vertical force due to tension is: (∂²u / ∂x² + ∂²u / ∂y²)F=T + ΔxΔy According to Newton's second law: ρΔxΔy = T + ΔxΔy (∂²u / ∂t²) Dividing both sides: ∂²u / ∂t² = (T / ρ)(∂²u / ∂x² + ∂²u / ∂y²)

Here, we analyze a small rectangular section of the membrane, denoted by Δx and Δy. By using the principles of tension in the membrane, we can express the forces acting on this small element. The equation incorporates the second partial derivatives of the displacement u, reflecting how the displacement varies in both the x and y directions. By applying Newton's second law, we relate the force acting on the small area to the mass times the acceleration, resulting in a equation that describes the motion of the membrane over time.

Imagine a small piece of a stretched rubber band. When you pluck it, the tension changes and the rubber band oscillates. The small piece of rubber responds to how it's stretched in different areas and the force acting on it, similar to how the small element of the membrane responds.

Or: ∂²u / ∂t² = (T / ρ)(∂²u / ∂x² + ∂²u / ∂y²) where c² = T / ρ is the square of the wave speed.

In this final part, we derive the wave equation that links the vertical displacement of the membrane to its tensions and properties. By simplifying our previous equations, we establish that the wave speed squared (c²) equals the ratio of tension (T) to the surface density (ρ). The overall equation we arrive at is a classic form of the two-dimensional wave equation, which describes how waves propagate in this medium over time and across the two-dimensional surface.

Think of a ripple moving across a pond when you throw a stone. The speed of that ripple relates to how tightly the water is disturbed (tension) versus how heavy the water is (mass per unit area). This is exactly how the wave speed in our derived equation works: it represents the balance between the surface tension of the membrane and its density.

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