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#### Alan0354

##### Full Member level 4
A sine wave traveling in +x direction can be represented by $\cos(\omega {t}- kx+\phi)$.

As shown in the drawing attached where $\phi=\frac{\pi}{2}$:

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

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

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

Alan

#### Attachments

• Phase.png
126.2 KB · Views: 70

All depends what direction you choose in the plot. If you start with the X origin going LEFT with time for the second wave, the first being reference, then the red in your figure LAGS the black, in the second, it LEADS the black.

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