The wave equation, given by $\partial_t^2 u-c^2 \nabla^2 u=f$, is a crucial partial differential equation in physics defined by its second-order time derivative, which enables the description of propagating waves unlike the diffusion equation. Its application depends on the physical context: for transversal waves on a string or a membrane, the equation arises from applying Newton's second law under a small deviation approximation ( $|\nabla u| \ll 1$ ), with the wave speed squared determined by the ratio of tension to density ( $c^2=S / \rho_{\ell}$ or $c^2=\sigma / \rho_A$ ). In contrast, for electromagnetic fields, the wave equation for the magnetic field $B$ is a direct and exact consequence of Maxwell's equations, where the wave speed is precisely the speed of light, $c=1 / \sqrt{\varepsilon_0 \mu_0}$.

Key takeaways

1. General Form and Distinction

   • The general, sourced wave equation is:

     $$ \partial_t^2 u-c^2 \nabla^2 u=f $$

   • Defining Feature: The appearance of the second-order time derivative $\left(\partial_t^2 u\right)$ fundamentally distinguishes it from the diffusion equation, leading to propagating wave solutions.

   • Parameters: $c$ is the wave velocity, and $f$ is the source term.

2. Derivations and Wave Speed (c)

   The text provides derivations for three key physical scenarios, showing how the wave speed $c$ is determined by the system's material properties.

   Scenario	Nature of Derivation	Resulting PDE Term	Wave Speed Squared ( $c^2$ )	Key Dependencies
   Transversal String (1D)	Approximative (Small Deviation)	$\begin{aligned}& \partial_t^2 u- \\& \frac{S}{\rho_t} \partial_x^2 u\end{aligned}$	$c^2=\frac{S}{\rho_{\ell}}$	$c \propto \sqrt{\text { Tension/Linear Density }}$
   Transversal Membrane (2D)	Approximative (Small Deviation)	$\begin{aligned}& \partial_t^2 u- \\& \frac{\sigma}{\rho_A} \nabla^2 u\end{aligned}$	$c^2=\frac{\sigma}{\rho_A}$	$c \propto$$\sqrt{\text { Surface Tension/Surface Density }}$
   Electromagnetic Field (B) (3D)	Exact (Maxwell's Equations)	$\begin{aligned}& \partial_t^2 B- \\& c^2 \nabla^2 B\end{aligned}$	$\begin{aligned}& c^2= \\& \frac{1}{\varepsilon_0 \mu_0}\end{aligned}$	$c$ is the Speed of Light (constant)

3. Role of Approximations

   • The derivations for the string and membrane are based on the assumption of small transversal deviations ( $|\nabla u| \ll 1$ ), which simplifies $\sin (\theta)$ or the normalization factor $\alpha$ to approximately 1.
   • The wave equation for electromagnetic fields is derived exactly from Maxwell's equations without requiring any small-deviation approximation.

🎬Demos

Dynamic Visualization of Wave Equation Principles-Analyzing Force Balance and Traveling Wave Components and the Effects of Damping

🫧Cue Column

<aside> 🔎

1. From Extensive Properties to the Continuity Equation
2. Derivation of the Diffusion and Heat Equations from the Continuity Principle
3. The Wave Equation-Derivation and Physical Applications and Wave Speed Determination
4. Boundary and Initial Conditions for Partial Differential Equations-Types and Uniqueness and Stationary States
5. Fluid Momentum and the Continuity Equation-Derivation of the Cauchy and Navier-Stokes Equations
6. The Wave Equation-Derivation and Physical Applications and Wave Speed Determination
7. Boundary and Initial Conditions for Partial Differential Equations-Types and Uniqueness and Stationary States </aside>