Relfected wave equation for free and fixed end

[kapitan90]

Homework Statement

Hello,
I have problems with expressing a reflected wave mathematically.
In my printed notes I found the following formulas for reflected waves:
a) For a fixed end: incoming wave: y_1(x,t)=e^{-i(kx+ωt)}
reflected wave: y_2(x,t)=re^{i(kx-ωt)} where r is the reflection coefficient.
I understand the - in front of ωt is because the wave has changed its direction, and the - in front of the whole power (before i) is because the wave is reflected.
Now for a free end: y_1(x,t)=e^{i(kx-ωt)} it is said that y_2(x,t)=re^{i(-kx-ωt)}

Homework Equations

Shouldn't the reflected wave for a free end in this case be:
y_2(x,t)=re^{i(kx+ωt)} because the wave is not inverted, but it changes its direction?

Any help appreciated!

 

[kapitan90]

Could anyone please have a look at this problem? I thought it was quite straightforward :(