How to determine the direction of a wave propagation?

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?

Answer

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!