Re: d'Almberts Forumla Help

Quote:

Originally Posted by

**princessmath** I have a question regarding the image below.

Is this how you solve it.

1) Identify if its parabolic/hyperbolic/elliptic etc

2) Find the charatertics coordinates

3) Using the Initial condition, find the charatertics cooridnates for them

4) sketch the charatertics coodinates in terms of x and t?

Attachment 25007
Thank you

d'Alembert's formula gives us a simple solution when the intial velocity is zero.

$\displaystyle u(x,t)=\frac{1}{2}\left(f(x-ct)+f(x+ct)\right)$

where

$\displaystyle f(x)$

is the intial condition. In you equation c=1 so this gives the solution

$\displaystyle u(x,t)=\frac{1}{2}\left(u(x-t,0)+u(x+t,0) \right)$

$\displaystyle u(x,t)=\frac{1}{2}\left(\begin{cases}2(x-t)+2, \quad -1 \le x-t \le 0 \\ 2-(x-t), \quad 0 \le x-t \le 2 \\ 0, \quad x-t < -1 \text{ or } x-t > 2 \end{cases} \right) + \frac{1}{2}\left(\begin{cases}2(x+t)+2, \quad -1 \le x+t \le 0 \\ 2-(x+t), \quad 0 \le x+t \le 2 \\ 0, \quad x+t < -1 \text{ or } x+t > 2 \end{cases} \right)$

This probably is not the best notation LOL.