# Standing waves - Wave Equation

## Main Question or Discussion Point

I don't completely understand how equation 4.4.4 was derived and determined. I understand the derivation behind the basic wave equation 4.3.4 but not what happened in 4.4.4. Why is there a need for all the negative signs ? Would a simple phase change suffice ?

Please do be a bit detailed in your explanation.... Pretend I'm an idiot. Thank you ! Related Classical Physics News on Phys.org
Redbelly98
Staff Emeritus
Homework Helper
Since the reflected wave travels in the opposite direction as the incident wave, you need to flip the sign on either the kx or the ωt term***. The author chose to reverse the sign of kx to make it -kx. There was already a - sign with -ωt. For the phase term, the choice of sign is arbitrary.

Just changing the phase would express a wave traveling in the same direction as the incident wave, but shifted in phase.

Hope that helps.

*** This is because a rightward-traveling wave has the form f(kx-ωt) or f(-kx+ωt). A leftward-traveling wave has the form f(kx+ωt) or f(-kx-ωt).

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So I could equally have written kx+wt ? Why is there a need for the phase change ∅ ?

Redbelly98
Staff Emeritus
Homework Helper
So I could equally have written kx+wt ?
Yes; see the edit added to my earlier post.
Why is there a need for the phase change ∅ ?
As the book says, the type of boundary will determine ∅. Usually the boundary is either fixed or has a maximum amplitude. Also, the location of the boundary plays a role in what ∅ is.

Eg., for a fixed end located at x=x0:

$$\cos(kx_0 + \phi /2) = 0$$

You'd solve that for ∅, given k and x0. Your book is taking the fixed end to be at x=0, so that simplifies things somewhat.