Hyperbolic Partial Differential Equations

Hyperbolic PDEs describe wave propagation and vibrations. The classic example is the 1D wave equation.

1D Wave Equation¶

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

(1)

$u(x, t)$ : displacement
$c$ : wave speed

General Solution¶

The general solution is:

u(x, t) = f(x - ct) + g(x + ct)

(2)

where $f$ and $g$ are determined by initial and boundary conditions.

Example: Vibrating String¶

A string of length $L$ fixed at both ends:

$u(0, t) = u(L, t) = 0$
Initial displacement $u(x, 0) = f(x)$
Initial velocity $u_t(x, 0) = g(x)$

Solution (Fourier series):

u(x, t) = \sum_{n=1}^\infty \left[A_n \cos\left(\frac{n\pi c t}{L}\right) + B_n \sin\left(\frac{n\pi c t}{L}\right)\right] \sin\left(\frac{n\pi x}{L}\right)

(3)

Python Example¶

import numpy as np
import matplotlib.pyplot as plt
L = 1
c = 1
x = np.linspace(0, L, 100)
t = 0.5
u = np.sin(np.pi * x) * np.cos(np.pi * c * t)
plt.plot(x, nu)
plt.xlabel('x')
plt.ylabel('u(x, t)')
plt.title('Vibrating String at t=0.5')
plt.show()

Expand this section with d’Alembert’s solution, characteristics, higher dimensions, and numerical methods (finite difference, finite element, etc.).