Confused about phase lead and lag of a wave.

A sine wave traveling in +x direction can be represented by [itex]\cos(\omega {t}- kx+\phi)[/itex].

As shown in the drawing attached where [itex] \phi=\frac{\pi}{2}[/itex]:

a) Is a plot that holds t=0 and two waves along +x direction.
[tex]t=0\Rightarrow\; \cos(\omega {t}- kx+\phi)\;=\;\cos(- kx+\frac{\pi}{2})[/tex]
This implies maximum at [itex]-kx+\frac{\pi}{2}\;=\;0[/itex]
[tex]k=\frac{2\pi}{\lambda}\;\Rightarrow\; \frac {2\pi}{\lambda}x=\frac {\pi}{2}\;\Rightarrow\;x=\frac{\lambda}{4}[/tex]
This gives the wave form in RED that LAGs the [itex]\cos(\omega{t}-kx)[/itex].





b) Is a plot of the waveform at x=0 and the two waves vary with time t.
[tex] x=0\;\Rightarrow\; \cos(\omega {t}- kx+\phi)\;=\;\cos(\omega{t}+\frac{\pi}{2})[/tex]
This implies maximum at [itex]\omega{t}=-\frac{\pi}{2}[/itex]
[tex]\omega{t}\;=\;\frac{2\pi}{T}t\;=\;-\frac{\pi}{2}\;\Rightarrow\;t=-\frac{T}{4}[/tex]
This means waveform in RED LEADs the original wave.


So the two cases give opposite result. Can anyone explain this to me?
Thanks

Alan

Attached Images

Phase.png (126.2 KB)