20.1. The Two-Dimensional Wave Equation

The transverse vibration u(x,y,t) of a rectangular membrane stretched tightly and fixed at the boundary is governed by the two-dimensional wave equation:

∂²u / ∂t² = c² (∂²u / ∂x² + ∂²u / ∂y²)

where:

• u(x,y,t): displacement of the membrane at point (x,y) and time t, • c: wave speed, depending on the tension and mass density of the membrane.

This section introduces the two-dimensional wave equation that describes how the displacement of a rectangular membrane—like a taut sheet or a vibrating plate—changes over time and space. The equation involves second derivatives with respect to time (t) and spatial dimensions (x and y), indicating that it models acceleration. Here, u(x,y,t) represents the displacement of the membrane at a specific point (x,y) and time (t), and c is the speed of the wave, which is influenced by how tense the membrane is and its mass density.

Imagine a trampoline—when you jump on it, the surface vibrates. The way it vibrates depends on how tightly the trampoline is stretched and how heavy you are. Similarly, the two-dimensional wave equation helps us predict how the membrane vibrates under disturbance.

The rectangular membrane has dimensions 0 < x < a, 0 < y < b, and its edges are held fixed, leading to boundary conditions:

u(0,y,t) = u(a,y,t) = u(x,0,t) = u(x,b,t) = 0

In this chunk, we learn about the boundary conditions that apply to our rectangular membrane. These conditions stipulate that the displacement (u) at the edges of the membrane remains zero at all times. This means that the edges of the membrane do not move—they're fixed in place—while the middle can oscillate. This is crucial for accurately modeling physical behavior since the boundary conditions help define how the system will respond to disturbances.

Think of a tent anchored to the ground at its corners. Those anchor points (the edges of the tent) stay put, but the middle can flap up and down when the wind blows. The fixed edges of the membrane behave similarly, providing a stable reference point as the center vibrates.

Additionally, the initial conditions are:

u(x,y,0) = f(x,y), ∂u / ∂t |_{t=0} = g(x,y) where f(x,y) is the initial shape and g(x,y) is the initial velocity distribution.

Initial conditions are important in any wave equation and refer to the state of the system at the beginning of observation (the moment t=0). Here, f(x,y) describes how the membrane is shaped initially (like its form before any disturbance), and g(x,y) indicates how fast different parts of the membrane are initially moving. Together, these conditions provide the needed information to understand how the wave will evolve over time.

Imagine plucking a guitar string. The initial shape of the string after being plucked is not flat; it has an initial displacement. The speed it starts moving at (how fast it moves up and down) gives us the initial velocity. These initial conditions will determine how the sound waves produced travel and how the guitar resonates.

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