the vibrating string

Where does the wave equation actually come from? The cleanest answer is a guitar or violin string. Stretch a thin string tight between two fixed pegs, pluck it, and it quivers. If you write down Newton's law for a tiny piece of that string, the wave equation falls out almost by itself. This is the original problem (Euler, d'Alembert, Bernoulli, 1700s) that launched the whole subject.

Here is the derivation in words. Take a small segment of string between x and x + dx. The string has mass per length rho and is pulled by a tension T. When the string is curved, the tension at the two ends pulls in slightly different directions, and the leftover sideways force is proportional to the curvature u_xx times the tension. Newton's second law says (mass) times (acceleration) equals (force): rho times u_tt equals T times u_xx. Dividing by rho gives u_tt = (T/rho) u_xx, which is the wave equation with c^2 = T/rho. So the wave speed is set by how tight (T) and how heavy (rho) the string is — tighter and lighter means faster, which is exactly how you tune a guitar.

Once you add the two fixed ends (u = 0 at both ends for all time — a boundary condition), the string can only vibrate in special patterns called normal modes, the pure musical tones. The lowest is the fundamental; the higher ones are the overtones. This little problem is the perfect first laboratory for separation of variables, Fourier series, standing waves, and resonance, all of which were essentially invented to understand it. The same derivation, done for a stretched drum membrane, gives the two-dimensional wave equation u_tt = c^2 (Laplacian u).