# D'alembert solution for the semi-infinite string

1. Nov 18, 2014

### bobred

1. The problem statement, all variables and given/known data
Find the solution of the wave equation using d'Alembert solution.

2. Relevant equations
$u(0,t)=0$
and $u(x,0)=0$
$$u_t(x,0)=\frac{x^2}{1+x^3}, \, x\geq0$$
$$u_t(x,0)=0, \, x<0$$

3. The attempt at a solution
For a semi infinite string we have the solution
$$u(x,t)=\frac{1}{2}\left( a(x-ct)+a(x+ct)-a(-x-ct)-a(-x+ct) \right)+\frac{1}{2c}\left( \int^{x+ct}_{x-ct} dy\, b(y) - \int^{-x+ct}_{-x-ct} dy\, b(y) \right)$$
with $u(x,0)=a(x)=0$ so
$$u(x,t)=\frac{1}{2c}\left( \int^{x+ct}_{x-ct} dy\, b(y) - \int^{-x+ct}_{-x-ct} dy\, b(y) \right)$$
where

$$b(y)=\frac{y^2}{1+y^3}$$

Is this right?

Last edited: Nov 18, 2014
2. Nov 23, 2014

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Jan 2, 2015

### bobred

Thsi is what I came up with
for $$x \geq ct$$
$$u(x,t)=\dfrac{1}{6c}\ln\left[\dfrac{1+(x+ct)^{3}}{1+(x-ct)^{3}}\right]$$
and $$x < ct$$
$$u(x,t)=\dfrac{1}{6c}\left[\ln\left(\dfrac{1+(x+ct)^{3}}{1+(ct-x)^{3}}\right)\right]$$