11. Partial Differential Equations

The one-dimensional wave equation is a second-order linear partial differential equation (PDE) that describes the propagation of waves, such as sound waves, light waves, and water waves, in a medium along a single spatial dimension. It is a fundamental equation in mathematical physics and engineering and serves as a prototype for many more complex wave-like phenomena. The standard form of the wave equation is: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ where:

• $u(x,t)$ is the displacement at position $x$ and time $t$,
• $c$ is the speed of wave propagation,
• $\frac{\partial^2 u}{\partial t^2}$ is the second-order time derivative,
• $\frac{\partial^2 u}{\partial x^2}$ is the second-order spatial derivative.

The one-dimensional wave equation is significant in various fields, such as physics and engineering. It illustrates how waves propagate through a medium, which can be visualized easily: consider a stringed musical instrument, where vibrating strings create sound waves. The formula itself relates the second derivatives of displacement with respect to time and position, indicating how changes at one position influence changes over time.

Imagine a child playing with a long rope. When the child shakes one end of the rope, a wave travels along it. The rope's movements at any point ($u(x,t)$) depend on how fast the wave travels ($c$) and how long the rope is. As the rope shakes, it essentially demonstrates the core principles outlined by the wave equation.

Let us derive the wave equation for a vibrating string under tension. Assumptions:

• The string is perfectly flexible and homogeneous.
• The motion is only in the vertical plane (transverse).
• No damping or external forces act on the string. Let:
• $T$ be the constant tension in the string,
• $\mu$ be the mass per unit length,
• $u(x,t)$ be the transverse displacement at point $x$ and time $t$. Consider an infinitesimal element of the string between $x$ and $x + \Delta x$. By Newton's second law: $\mu \Delta x \cdot \frac{\partial^2 u}{\partial t^2} = T \sin(\theta_2) - T \sin(\theta_1)$ Using the small-angle approximation: $\sin(\theta) \approx \tan(\theta) = \frac{\partial u}{\partial x}$, we get: $\mu \Delta x \cdot \frac{\partial^2 u}{\partial t^2} = T\left(\frac{\partial u}{\partial x}(x + \Delta x) - \frac{\partial u}{\partial x}(x)\right)$ Dividing by $\Delta x$ and taking the limit as $\Delta x \to 0$: $\frac{\partial^2 u}{\partial t^2} = \frac{T}{\mu} \frac{\partial^2 u}{\partial x^2}$ Thus: $\frac{\partial^2 u}{\partial t^2} = \frac{T}{\mu} \frac{\partial^2 u}{\partial x^2}$ Let $c^2 = \frac{T}{\mu}$, then we obtain the standard wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

This section explains the derivation of the one-dimensional wave equation by starting with a dynamic system: a vibrating string under tension. The assumptions help simplify the model, assuming no external force or damping, leading to a clean mathematical representation using Newton's laws and trigonometric approximations. The transition from intuitive forces to the final wave equation shows how physical concepts result in precise mathematical formulations.

Think of a tight rope used in a circus act. As the performer bounces on one end, waves travel along the rope influenced by the tension ('T') and the weight of the rope itself ('μ'). As the waves propagate along the string, the physical understanding of this motion translates into mathematical expression through the wave equation, which retains key properties of the physical system.

The general solution is given by D'Alembert’s formula: $u(x,t) = f(x - ct) + g(x + ct)$ Where:

• $f$ and $g$ are arbitrary functions,
• $f(x - ct)$ represents a wave traveling to the right,
• $g(x + ct)$ represents a wave traveling to the left. This form shows that the wave travels without changing its shape at speed $c$.

The general solution, expressed through D'Alembert's formula, showcases how waves can propagate in both directions at a consistent speed without altering their shape. The functions $f$ and $g$ allow for flexible descriptions of the waveforms, accommodating various scenarios in wave propagation, such as reflections and superposition of waves.

Visualize a calm lake disturbed by a stone thrown into it. The ripples (or waves) move outward, maintaining their shape while traveling. In the mathematical context, the right and left waves captured in D'Alembert’s formula model this behavior accurately, showing how disturbances propagate across a medium like water.

Suppose we are given:

• Initial displacement: $u(x,0) = \varphi(x)$
• Initial velocity: $\frac{\partial u}{\partial t}(x,0) = \psi(x)$ Then, D’Alembert’s solution becomes: $u(x,t) = \frac{1}{2} \left[ \varphi(x - ct) + \varphi(x + ct) \right] + \frac{1}{2c} \int_{x - ct}^{x + ct} \psi(s) \, ds$

In this section, initial conditions play a crucial role in determining how waves behave over time. The initial displacement and velocity of a wave provide essential information required for D’Alembert's formula to produce specific solutions tailored for different situations. The integral term indicates how initial velocities affect the wave’s evolution.

Imagine a guitar string that is plucked. The shape of the string at the instant it's plucked correlates to the initial displacement, while how fast and in what direction it moves when released represents the initial velocity. This initial state defines how the sound wave will proceed to develop.

There are common types of boundary conditions:

1. Fixed End Conditions (Dirichlet BC): $u(0,t) = 0$, $u(L,t) = 0$
2. Free End Conditions (Neumann BC): $\frac{\partial u}{\partial x}(0,t) = 0$, $\frac{\partial u}{\partial x}(L,t) = 0$
3. Mixed Conditions: One end fixed, other free.

Boundary conditions are critical when solving the wave equation in physical systems. They define how the wave interacts with the edges of the medium in which it propagates. The fixed-end conditions imply that energy cannot leave, while free-end conditions allow for energy to exit without constraint. Mixed conditions create complex interactions at the boundaries.

Consider a drum skin. When you hit the center, the waves travel outward. The edges act as boundaries: at a fixed drum edge, the tension fixes the wave whereas at an open edge, the wave can escape freely. Each boundary condition leads to different types of sound and resonance in the drum.

To solve using this method: Assume $u(x,t) = X(x)T(t)$ Substitute into the wave equation: $X(x)T''(t) = c^2 X''(x) T(t)$ Divide both sides: $\frac{T''(t)}{c^2 T(t)} = \frac{X''(x)}{X(x)} = -\lambda$ This gives two ODEs:

• $X'' + \lambda X = 0$
• $T'' + \lambda c^2 T = 0$ The form of solutions depends on boundary conditions.

The separation of variables technique provides a systematic way to tackle the wave equation by decomposing the solution into two distinct parts: one that depends solely on position ($X(x)$) and another on time ($T(t)$). The resulting ordinary differential equations (ODEs) can now be solved individually under specific boundary conditions, allowing for tailored responses based on how the edges of the medium behave.

Think of a classroom where a teacher interacts with students individually. By separating tasks – grading papers (position) and preparing lessons (time) – both parts can be managed more effectively than if they were mixed. Likewise, separating variables in the wave equation simplifies understanding and enables solutions that align with particular conditions.

Let $0 < x < L$, and BCs:

• $u(0,t) = 0$, $u(L,t) = 0$ Solving the spatial part: $X'' + \lambda X = 0, \, X(0)= X(L) = 0$ Solution: $X_n(x) = \sin\left(\frac{n \pi x}{L}\right), \; \lambda_n = \left(\frac{n \pi}{L}\right)^2$ Time part: $T_n(t) = A_n \cos\left(\frac{n \pi ct}{L}\right) + B_n \sin\left(\frac{n \pi ct}{L}\right)$ General solution: $u(x,t) = \sum_{n=1}^{\infty} \left[A_n \cos\left(\frac{n \pi ct}{L}\right) + B_n \sin\left(\frac{n \pi ct}{L}\right)\right] \sin\left(\frac{n \pi x}{L}\right)$ Coefficients $A_n$, $B_n$ determined from initial conditions using Fourier series.

In this example, we consider a string fixed at both ends. Each segment of the string can vibrate and contribute to the overall motion. By solving the spatial and temporal components separately, we determine how each mode of vibration (represented by $n$) combines through the superposition principle to produce the general solution. The role of the coefficients $A_n$ and $B_n$ is crucial as they adjust the contribution of each mode to meet the initial conditions set before vibration begins.

Picture a swing set with various swings. Each swing can move independently, but together they create a lively scene. The individual swings represent different harmonic modes, and their combined movement results in the overall behavior of the entire swing set. Similarly, the fixed string's multiple modes come together to form a rich tapestry of sound and vibration when plucked.

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