Toronto Math Forum
APM3462018S => APM346Lectures => Topic started by: Adam Gao on February 05, 2018, 08:49:23 PM

I believe on section 2.6, example 1 there is a misreferencing of equation (4) (D'Alembert's) in place of equation (5) (General Solution to wave equation). In Example 1, it says:
"Plugging (4) we see that $$\phi(ct)+\psi(−ct)=p(t)$$ as $$t>0, t>0 $$", (4) referring to D'Alembert's formula. However, I believe it results from plugging $$p(t)$$ into (5), the general solution $$u(x,t)=\phi(x+ct)+\psi(x−ct)$$
Again, later, it says:
"Then plugging $$x:=x+ct$$ into (6) and $$x:=x−ct$$ into (9) and adding we get from (4) that $$\begin{multline}
u(x,t)=
\underbracket{\frac{1}{2}g(x+ct)+
\frac{1}{2c}\int_0^{x+ct}h(x')\,dx'}_{=\phi(x+ct)}+ \\
\underbracket{p(tx/c)\frac{1}{2}g(ctx)
\frac{1}{2c}\int_0^{ctx} h(x')\,dx'}_{=\psi(xct)}. \qquad
\end{multline}$$
Again, I believe this results from plugging the results into (5) and not (4).
I apologize if this correction seems a bit trivial but I personally had a bit of difficulty understanding where the results came from. Or maybe I am wrong and am missing something.

I noticed this too. I seems to me a problem as well.

Yeah I think it means to say that you use the initial boundary condition to get $ \phi(ct)+\psi(−ct)=p(t) $. Then you solve for $ \psi(x), x <0 $ and plug d'Alembert's solution for $ \phi(x) $ into the equation

Adam, thanks. Fixed