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In the textbook, it said a wave in the form $y(x, t) = A\cos(\omega t + \beta x + \varphi)$ propagates along negative $x$ direction and $y(x, t) = A\cos(\omega t - \beta x + \varphi)$ propagates along positive $x$ direction. This statement looks really confusing because when it says the wave is propagating along $\pm$ x direction, to my understanding, we can drop the time term and ignore the initial phase $\varphi$ while analyzing the direction, i.e.$y(x, 0) = A\cos(\pm\beta x)$, however, because of the symmetry of the cosine function, $\cos(\beta x)\equiv \cos(-\beta x)$, so how can we determine the direction of propagation from that?

I know my reasoning must be incorrect but I don't know how to determine the direction. So if we don't go over the math, how to figure out the direction of propagation from the physical point of view? Why $-\beta x$ corresponding to the propagation on positive x direction but not the opposite? Thanks.

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up vote 7 down vote accepted

For a particular section of the wave which is moving in any direction, the phase must be constant. So, if the equation says $y(x,t) = A\cos(\omega t + \beta x + \phi)$, the term inside the cosine must be constant. Hence, if time increases, $x$ must decrease to make that happen. That makes the location of the section of wave in consideration and the wave move in negative direction.

Opposite of above happens when the equation says $y(x,t) = A\cos(\omega t - \beta x + \phi)$. If t increase, $x$ must increase to make up for it. That makes a wave moving in positive direction.

The basic idea:For a moving wave, you consider a particular part of it, it moves. This means that the same $y$ would be found at other $x$ for other $t$, and if you change $t$, you need to change $x$ accordingly.

Hope that helps!

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thanks, if "term inside the cosine must be constant" holds, then I understand the reason now. But why physics requires that the phase is constant? Sorry for that, I just feel very confusing because 'phase' is not as straightforward as something like frequency or period etc. – user1285419 Mar 9 '13 at 6:01
$y$ is what is to be constant, that's why phase has to be constant. – Cheeku Mar 9 '13 at 6:16
I still don't understand $y$ is a function of $x$ and $t$, why it is constant? – user1285419 Mar 9 '13 at 6:21
Wish I could show you a diagram. But,lemme try without it. At some $x$ and some $t$, you have a particular $y$, right? So, if you wish to make the wave travel, you need to create the same $y$ at some other $x$. That's what is meant by "wave has travelled". The same $y$ is created at some other $x$ and other $t$. Just like, the same salesman is in other town on other day, but the salesman is the same. Better? – Cheeku Mar 9 '13 at 6:28
oh, I think I get the point now :) thanks Cheeku, it helps :) – user1285419 Mar 9 '13 at 6:34

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