What the term 'phase' in a wave actually means? [closed]

I actually want to clear my conception about the phase. I have used it while dealing with wave equations. But could not get the actual significance of it. I have learned that, 'phase is a quantity by which state of motion is represented'. But how? If somebody wants can give suggestion of books.

P.S. (I don't think it is a duplicate question. I have read some other questions related to this, but did not get my answer)

• Phase is an offset in angle of a periodic function. It simply tells you where a period starts. – CuriousOne Jan 14 '16 at 16:29
• Its like that teen wave who went emo. She was going to fail every subject because she was always late for class. Her parents weren't worried though, they knew it was just a phase... – AccidentalFourierTransform Jan 14 '16 at 16:29
• First: I don't understand that quote. There are two closely related meanings. One is the spatial or temporal offset of one wave relative to another. The other is the argument of a sinusoidal wave. That is, in $\cos(\omega t -kx)$ the quantity $(\omega t - kx)$ is sometimes called 'the phase of the wave'. If I squint my eyes, I can make your quote correspond to the second definition. – garyp Jan 14 '16 at 16:34
• Explain how the other answer does not answer your question and how your question is different. That will help keep your question alive. – Muze Jan 14 '16 at 16:57

It's best to go back to basics here. We know that a wave is a Harmonic Oscillation (HO) travelling at a constant speed. Now let's consider that HO:

The point $P$ travels on a circle of radius $A$ with constant angular speed $\omega$. At each point in time the the projection of the point $P$ on the $y$-axis is:

$$y=A\sin\theta$$

At $t=0$, then $\theta=\phi$ and with the angular speed we get:

$$\theta=\omega t+\phi$$

Substituting we get the full description of the HO:

$$y=A\sin(\omega t+\phi)$$

Where $\phi$ is the phase angle.

It's now clear that two such travelling HO's (waves), otherwise identical in all respects ($A$, $\omega$ and wave speed), can differ in phase $\phi$.