Why Is the Wave Equation Dimensionally Incorrect?

[Sorgen]

Alright so I'm clueless. I've read the chapter and the concept of a one dimensional box is never mentioned before this problem. I'm thinking i have to integrate some stuff but i have no idea where to begin.

Any help?

 

[tms]

Don't worry about the box. You could just as well think of it as a vibrating string with the ends fixed.

 

[collinsmark]

Hello Sorgen,

I'm not sure what kind of class this is for, but if it is a class that does not require knowledge of differential equations as a prerequisite (or is not a differential equations class itself), the desired solution just might be taking one equation and plugging it into the other, and making sure everything is consistent with the third. That's my speculation anyway.

That said, something is awry with the problem statement.

$$\frac{1}{c} \frac{\partial^2 \Psi}{\partial t^2} - \frac{\partial^2 \Psi}{\partial x^2} = 0$$
is not dimensionally correct.

Are you sure it's not a mistake in the coursework, and the c shouldn't be squared, making it something like,

$$\frac{1}{c^2} \frac{\partial^2 \Psi}{\partial t^2} - \frac{\partial^2 \Psi}{\partial x^2} = 0?$$

[Edit: Hint. Although I phrased that last part as a question, treat it rhetorically. I'm pretty certain that the 1/c should be 1/c². Use $\frac{1}{c^2} \frac{\partial^2 \Psi}{\partial t^2} - \frac{\partial^2 \Psi}{\partial x^2} = 0$ as the wave equation. Plug $\Psi(x,t) = a(t) \sin \left( \frac{n \pi x}{L} \right)$ into that and see what you get, and check that it matches up with the rest of the problem. Also, inform your instructor of the 1/c vs. 1/c² error.]