Toronto Math Forum

APM346-2018S => APM346--Lectures => Topic started by: Adam Gao on February 05, 2018, 08:49:23 PM

Title: Section 2.6 Example 1 correction
Post by: Adam Gao on February 05, 2018, 08:49:23 PM
I believe on section 2.6, example 1 there is a mis-referencing 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 $$\begin{multline}
u(x,t)=
\underbracket{\frac{1}{2}g(x+ct)+
\frac{1}{2c}\int_0^{x+ct}h(x')\,dx'}_{=\phi(x+ct)}+ \\
\underbracket{p(t-x/c)-\frac{1}{2}g(ct-x)
-\frac{1}{2c}\int_0^{ct-x} h(x')\,dx'}_{=\psi(x-ct)}. \qquad
\end{multline}$$

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.
Title: Re: Section 2.6 Example 1 correction
Post by: Jingxuan Zhang on February 06, 2018, 07:28:59 AM
I noticed this too. I seems to me a problem as well.
Title: Re: Section 2.6 Example 1 correction
Post by: Ioana Nedelcu on February 06, 2018, 02:48:20 PM
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
Title: Re: Section 2.6 Example 1 correction
Post by: Victor Ivrii on February 06, 2018, 07:19:11 PM
Adam, thanks. Fixed