Wave Equation

The 1D wave equation governs the propagation of waves (sound, light, strings):

$\frac{\partial^2 u}{\partial t^2} = c^2\, \frac{\partial^2 u}{\partial x^2}$

where $c$ is the wave speed.

d'Alembert's Solution

For an infinite domain with initial displacement $f(x)$ and initial velocity $g(x)$, the general solution is:

$u(x, t) = \frac{f(x - ct) + f(x + ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s)\, ds$

This has a beautiful physical interpretation: the wave splits into a right-traveling copy and a left-traveling copy of the initial displacement, each at speed $c$.

For zero initial velocity ($g = 0$):

$u(x, t) = \frac{f(x - ct) + f(x + ct)}{2}$

Standing Waves

On a finite domain $[0, L]$ with Dirichlet boundary conditions, the normal modes are standing waves:

$u_n(x, t) = \sin\!\left(\frac{n\pi x}{L}\right) \cos\!\left(\frac{n\pi c t}{L}\right)$

The angular frequencies are $\omega_n = n\pi c / L$, giving the harmonic series of the string.

Dispersion Relation

For a non-dispersive medium, the dispersion relation is:

$\omega = c\, k$

The phase velocity (speed of a wave crest) equals the group velocity (speed of energy transport):

$v_{\text{phase}} = \frac{\omega}{k} = c$

Your Task

Implement dalembert(x, t, c, f_func, g_func=None) using d'Alembert's formula. If g_func is provided, integrate numerically using 100 midpoint steps; if g_func=None, use zero initial velocity. Implement standing_wave(x, t, c, L, n=1) for the $n$-th normal mode. Implement wave_phase_velocity(omega, k) returning $\omega/k$.