Wave Equation: Solutions and d'Alembert's Formula

• Deals with the wave equation and methods to solve it

Physical Context of Wave Equation

• Common situations include:
  • Small vibrations of a stretched string
  • Acoustics
  • Electromagnetism

Mathematical Context of Wave Equation

• Wave Equation involves a scalar function of space and time: $u(x,t)$
• Wave Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $
  • $c$ represents the wave speed
  • It is a linear, second-order PDE (Partial Differential Equation)

Solution Methods for Wave Equations

• Can be solved with:
  • Direct method, using d'Alembert's formula
  • Separation of variables, leading to Fourier series

d'Alembert's Formula Derivation

• Define new variables: $\xi = x + ct$ and $\eta = x - ct$
  • Partial derivatives transform as follows: $\frac{\partial}{\partial x} = \frac{\partial}{\partial \xi} + \frac{\partial}{\partial \eta}$ and $\frac{\partial}{\partial t} = c\frac{\partial}{\partial \xi} - c\frac{\partial}{\partial \eta}$
  • Second-order partial derivatives transform as follows: $\frac{\partial^2}{\partial x^2} = (\frac{\partial}{\partial \xi} + \frac{\partial}{\partial \eta})^2 = \frac{\partial^2}{\partial \xi^2} + 2\frac{\partial^2}{\partial \xi \partial \eta} + \frac{\partial^2}{\partial \eta^2}$
  • Second-order partial derivatives transform as follows: $\frac{\partial^2}{\partial t^2} = c^2(\frac{\partial}{\partial \xi} - \frac{\partial}{\partial \eta})^2 = c^2(\frac{\partial^2}{\partial \xi^2} - 2\frac{\partial^2}{\partial \xi \partial \eta} + \frac{\partial^2}{\partial \eta^2})$
• Wave equation transforms to: $\frac{\partial^2 u}{\partial \xi \partial \eta} = 0$
  • Integrating with respect to $\xi$ gives: $\frac{\partial u}{\partial \eta} = b(\eta)$
  • Integrating with respect to $\eta$ gives: $u(\xi, \eta) = \int b(\eta) d\eta + A(\xi) = B(\eta) + A(\xi)$
  • General solution: $u(x,t) = A(x + ct) + B(x - ct)$
• The general solution Consists of two arbitrary functions
  • $A(x + ct)$ representing a wave moving to the left
  • $B(x - ct)$ representing a wave moving to the right

Finding Arbitrary Functions

• Initial conditions are necessary:
  • Initial displacement: $u(x,0) = f(x)$
  • Initial velocity: $\frac{\partial u}{\partial t}(x,0) = g(x)$
• From the initial displacement: $u(x,0) = A(x) + B(x) = f(x)$
• The partial derivative with respect to time is: $\frac{\partial u}{\partial t}(x,t) = cA'(x + ct) - cB'(x - ct)$
• Initial velocity condition yields: $\frac{\partial u}{\partial t}(x,0) = cA'(x) - cB'(x) = g(x)$
• Rearranging, $A'(x) - B'(x) = \frac{1}{c}g(x)$
• Integrating with respect to x: $A(x) - B(x) = \frac{1}{c}\int_0^x g(s) ds + k$
• Solve the system of equations:
  • $A(x) + B(x) = f(x)$
  • $A(x) - B(x) = \frac{1}{c}\int_0^x g(s) ds + k$
• Solving for A(x) and B(x):
  • $A(x) = \frac{1}{2}f(x) + \frac{1}{2c}\int_0^x g(s) ds + \frac{k}{2}$
  • $B(x) = \frac{1}{2}f(x) - \frac{1}{2c}\int_0^x g(s) ds - \frac{k}{2}$
• Substituting back into the general solution $u(x,t) = A(x + ct) + B(x - ct)$
  • Substituting functions A and B, yields: $= \frac{1}{2}[f(x + ct) + f(x - ct)] + \frac{1}{2c}\int_0^{x+ct} g(s) ds - \frac{1}{2c}\int_0^{x-ct} g(s) ds$

d'Alembert's Formula

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