Animated: a plucked string splitting, reflecting, and recombining

D’Alembert’s solution \(u = F(x - ct) + G(x + ct)\) says a plucked string’s bump splits into two half-height waves running in opposite directions. At the fixed ends each reflects upside down; after one full period \(2L/c\) the string is back where it started. The leapfrog solve (solid) runs at Courant number 1, where it reproduces d’Alembert’s formula (dashed) exactly at the grid points.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.pde import WaveEquation1D, dalembert_solution
from mathematicskit.pde.visualizers import animate_solution

# In Jupyter or JupyterLite, show animations as an HTML/JavaScript player
# (no ffmpeg needed).
plt.rcParams["animation.html"] = "jshtml"


def pluck(x):
    return np.maximum(0.0, 1.0 - np.abs(x - 0.3) / 0.1)


string = WaveEquation1D(pluck, n=201, c=1.0)
sol = string.solve(2.0, dt=string.dx, method="leapfrog")

One full period of the string

anim = animate_solution(sol, exact=lambda x, t: dalembert_solution(pluck, x, t, length=1.0), max_frames=81)
anim

Once Loop Reflect

plt.show()