Mathematical equation: Traveling wave method for wave equation

Updated: APR 2016

Based on the idea of ordinary differential equations, the general solution of partial differential equations is obtained, and then the arbitrary functions and coefficients are determined by boundary conditions. However, this idea is only feasible for a few partial differential equations.

One-dimensional wave equation | d ' Alembert formula

\ (\dfrac{\partial^2u}{\partial t^2}=a^2\dfrac{\partial^2u}{\partial x^2}\)

For variable substitution,

\ (\xi =x+at\)

\ (\eta = x-at\)

To calculate the partial derivative of class two, the surrogate equation gets

\ (\dfrac{\partial^2u}{\partial \xi \partial\eta}=0\)

Integral gets results with two arbitrary functions

\ (U (\xi,\eta) =f_1 (\xi) +f_2 (\eta) \)

That

\ (U (x, y) =f_1 (x+at) +f_2 (x-at) \)

This is the general solution of one-dimensional wave equation.

For the general initial conditions,

\ (U|_{t=0}=\varphi (x), \quad–\infty<x<+\infty\)

\ (\left.\dfrac{\partial u}{\partial t}\right|_{t=0}=\psi (x), \quad–\infty<x<+\infty\)

You can solve it.

\ (U (x,t) =\dfrac{1}{2}[\varphi (x+at) +\varphi (x-at)]+\dfrac{1}{2a}\int_{x-at}^{x+at}\psi (\zeta) d\zeta\)

This is the d ' Alembert formula of the infinite Long chord free vibration (homogeneous).
Note that the opposite line waves may synthesize standing waves:

(Image source: wikipedia.org)

The function value U (x,t) of the "dependency interval" point (x,t) is determined by the initial conditions in the interval \ ([x-at,x+at]\) on the x axis, which is the interval of this point.

The interval \ ([x_1,x_2]\) on the "decision area" x-axis determines the value U (x,t) of all points (x,t) in the x-axis, line \ (x=x_1+at\), straight line \ (x=x_2-at\) in the x-t plane, This area is the decision area for this interval.

The interval \ ([x_1,x_2]\) on the "affected area" x-axis affects the value U (x,t) of all points (x,t) in the area of the x-axis, line \ (x=x_1-at\), line \ (x=x_2+at\) within the x-t plane, This area is the affected area of this interval.

Three-dimensional wave equation | Poisson formula

Three-dimensional wave equation

\ (\dfrac{\partial^2 u}{\partial t^2}=a^2\nabla^2 u\)

corresponding to generalized initial conditions

\ (U|_{t=0}=\varphi (\textbf{r}) \)

\ (\left.\dfrac{\partial u}{\partial t}\right|_{t=0}=\psi (\textbf{r}), \quad–\infty<x<+\infty\)

If the system spherical symmetry, that is, U and \ (\theta, \varphi\) independent. At this time Jiewei

\ (U (r,t) =\dfrac{f_1 (r+at) +f_2 (r-at)}{r}\)

The general situation,

Mathematical equation: Traveling wave method for wave equation