Standing waves - Wave Equation

[elemis]

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 !

 

[Redbelly98]

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.

EDIT added:
*** 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).

 

[elemis]

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

 

[Redbelly98]

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.