Chapter 1: Wave Equation and Oscillatory Phenomena

Wave Equation and Oscillations

📖 Reading Time: 20-25 minutes 📊 Difficulty: Beginner 💻 Code Examples: 0 📝 Exercises: 0

🌐 EN | 🇯🇵 JP | Last sync: 2025-11-16

FM Dojo > Partial Differential Equations and Boundary Value Problems > Chapter 1

1.1 Derivation of the Wave Equation

We derive the one-dimensional wave equation from string vibrations.

📐 Theory

One-Dimensional Wave Equation:

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

where \(u(x,t)\) is the displacement and \(c\) is the wave propagation velocity

Derivation: For a string with tension \(T\) and linear density \(\rho\), we have \(c = \sqrt{T/\rho}\)

💻 Code Example 1: Numerical simulation of wave equation (finite difference method)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

"""
Example: Derivation:For a string with tension \(T\) and linear densit

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from IPython.display import HTML

# Parameters
L = 10.0 # string length
c = 1.0 # wave propagation velocity
T = 20.0 # time range
Nx = 200 # number of spatial grid points
Nt = 1000 # number of time steps

# Grid
x = np.linspace(0, L, Nx)
t = np.linspace(0, T, Nt)
dx = x[1] - x[0]
dt = t[1] - t[0]

# CFL condition verification
r = c * dt / dx # Courant number
print(f"Courant number r = {r:.3f} (stability condition: r ≤ 1)")

# Initial condition: triangular wave
def initial_displacement(x):
 u = np.zeros_like(x)
 peak = L / 2
 width = L / 5
 mask = np.abs(x - peak) < width
 u[mask] = 1 - np.abs(x[mask] - peak) / width
 return u

# Initialization
u = np.zeros((Nt, Nx))
u[0] = initial_displacement(x)
u[1] = u[0].copy() # initial velocity = 0

# finite difference method（explicit scheme）
for n in range(1, Nt-1):
 for i in range(1, Nx-1):
 u[n+1, i] = (2*(1-r**2)*u[n, i] - u[n-1, i] +
 r**2*(u[n, i+1] + u[n, i-1]))

 # Boundary conditions: u(0,t) = u(L,t) = 0
 u[n+1, 0] = 0
 u[n+1, -1] = 0

# Visualization
fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Snapshots
times = [0, Nt//4, Nt//2, 3*Nt//4]
for idx, n in enumerate(times):
 ax = axes[idx//2, idx%2]
 ax.plot(x, u[n], 'b-', linewidth=2)
 ax.axhline(0, color='gray', linewidth=0.5)
 ax.grid(True, alpha=0.3)
 ax.set_xlabel('Position x', fontsize=12)
 ax.set_ylabel('Displacement u(x,t)', fontsize=12)
 ax.set_title(f't = {t[n]:.2f}', fontsize=12)
 ax.set_ylim(-1.2, 1.2)

plt.suptitle('Numerical solution of wave equation (finite difference method)', fontsize=14)
plt.tight_layout()
plt.show()

# Spacetime plot
plt.figure(figsize=(12, 6))
plt.contourf(x, t, u, levels=50, cmap='RdBu_r')
plt.colorbar(label='Displacement u(x,t)')
plt.xlabel('Position x', fontsize=12)
plt.ylabel('Time t', fontsize=12)
plt.title('Wave propagation (Spacetime diagram)', fontsize=14)
plt.tight_layout()
plt.show()

print("\nProperties of wave equation:")
print("- Waves propagate left and right, and reflect at boundaries")
print("- Energy is conserved")
print(f"- Propagation velocity: c = {c} m/s")

1.2 d’Alembert’s Solution (Traveling Wave Solution)

The general solution of the wave equation in an infinite domain is given by d’Alembert’s formula.

📐 Theory

d’Alembert’s solution:

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

Superposition of left-traveling wave \(f(x-ct)\) and right-traveling wave \(g(x+ct)\)

💻 Code Example 2: Visualization of d’Alembert’s solution

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

# Parameters
c = 1.0
x = np.linspace(-10, 10, 500)
t_values = np.linspace(0, 5, 6)

# Initial waveforms
def f(xi):
 """left-traveling wave"""
 return np.exp(-xi**2)

def g(xi):
 """right-traveling wave"""
 return 0.5 * np.exp(-(xi-2)**2 / 0.5)

# Visualization
fig, axes = plt.subplots(3, 2, figsize=(14, 12))
axes = axes.flatten()

for idx, t in enumerate(t_values):
 ax = axes[idx]

 # d'Alembert's solution
 u_left = f(x - c*t)
 u_right = g(x + c*t)
 u_total = u_left + u_right

 ax.plot(x, u_left, 'b--', linewidth=1.5, alpha=0.7, label='left-traveling wave f(x-ct)')
 ax.plot(x, u_right, 'r--', linewidth=1.5, alpha=0.7, label='right-traveling wave g(x+ct)')
 ax.plot(x, u_total, 'k-', linewidth=2, label='total waveform u(x,t)')

 ax.axhline(0, color='gray', linewidth=0.5)
 ax.grid(True, alpha=0.3)
 ax.set_xlabel('Position x', fontsize=11)
 ax.set_ylabel('Displacement u', fontsize=11)
 ax.set_title(f't = {t:.2f}', fontsize=12)
 ax.set_ylim(-0.5, 1.5)
 ax.legend(fontsize=9)

plt.suptitle('d'Alembert's solution: superposition of traveling waves', fontsize=14)
plt.tight_layout()
plt.show()

print("=== Properties of d'Alembert's solution ===")
print("- Left-traveling wave: f(x-ct) moves to the right with velocity c")
print("- Right-traveling wave: g(x+ct) moves to the left with velocity c")
print("- Total waveform: u = f + g by the superposition principle")

Summary

← Series Top Chapter 2 →

Disclaimer

Back to Series Index Next Chapter →