Step 1: Change of Variables

This section introduces the change of variables technique as a fundamental step in deriving D'Alembert's solution for the one-dimensional wave equation.

The section explains how to apply a change of variables to simplify the one-dimensional wave equation. By introducing new variables, the original equation can be transformed into a simpler form that facilitates finding the general solution. This process is crucial for understanding wave propagation in various physical systems.

Detailed Summary

This section focuses on the derivation of D'Alembert's solution for the one-dimensional wave equation by employing a change of variables. The wave equation is given by:

where $u(x, t)$ is the wave's displacement, and $c$ is the speed of wave propagation.

To solve the wave equation, we introduce new variables:

• $\xi = x + ct$
• $\eta = x - ct$

With these transformations, we apply the chain rule for derivatives:

1. First Derivatives:

   • $\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \xi} \frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial \eta} \frac{\partial \eta}{\partial x} = u_\xi + u_\eta$

2. Second Derivatives:

   • $\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial \xi^2} \left(\frac{\partial \xi}{\partial x}\right)^2 + 2 \frac{\partial^2 u}{\partial \xi \partial \eta} \frac{\partial \xi}{\partial x} \frac{\partial \eta}{\partial x} + \frac{\partial^2 u}{\partial \eta^2} \left(\frac{\partial \eta}{\partial x}\right)^2 = u_{\xi\xi} + 2 u_{\xi\eta} + u_{\eta\eta}$

These derivatives help simplify the wave equation to the form:

$u_{\xi\xi} - u_{\eta\eta} = 0,$

which implies a general solution of the form:

$u(\xi, \eta) = F(\xi) + G(\eta),$

The next steps involve returning to original variables to obtain D’Alembert’s final solution for the wave equation. This transformation is vital for understanding wave mechanics and has practical applications in various fields such as acoustics and string vibrations.