# Definition of Phase Shifts (Coupled Oscillators)

I was wondering if maybe someone could look at this excerpt from a textbook (Attached). It states that “the displacement of the two masses will be in opposite directions (out of phase by pi)” but I thought that the Definition of Phase was: Phase = theta in the following formula, cos(wt+theta).

However, the two equations in this book excerpt have the exact same expression in parenthesis, and both are nested in the same function, cosine - so how is it that they have different phases? Many thanks.

Here's a simple GIF to complement @garyp's succinct answer:

I've plotted the two functions $$-\cos{(\sqrt{5}t)}$$ (blue) and $$\cos{(\sqrt{5}t + \phi)}$$ (orange), and I've varied $$\phi$$ between 0 and $$2\pi$$. As you can see, they start off being exactly exactly reflected, as you'd expect, and now as I tune $$\phi$$, the orange curve moves "closer" to the blue one until it matches it exactly when $$\phi = \pi/2$$, as one would expect.

• I was expecting an animation when you mentioned "GIF". Oh well... Dec 4, 2020 at 20:07
• @wyphan It's animated for me! Doesn't it work for you? :'( Dec 4, 2020 at 20:17
• Oh, it works now! Must be my ISP messing around with me again... Dec 4, 2020 at 20:18
• @wyphan Ok it turns out that the GIF I made didn't loop, so you basically see it once and then you don't see it again for a while :P Just made one that loops, so it should be better! :) Dec 4, 2020 at 20:22
• Wow thank you so much - what an awesome animation! What programming did you use to do that? Dec 4, 2020 at 21:35

$$-4\cos{(\sqrt{5}t)} = +4\cos{(\sqrt{5}t + \pi)}$$

• That’s interesting and very helpful! But I’m interested to know why this works? If we were to graph both equations (y=4cos(sqrt(5)t) and y=-4cps(sqrt(5)t), we’d find that the negative one was simply reflected over the t axis, yes? So then, why the phase shift of pi? It doesn’t look like a phase shift (as in one has y=0 at origin vs. another one at y=pi)? Dec 4, 2020 at 19:33
• @Yelena being "reflected" is one way to see it, but how about you "shift" the second equation until it exactly overlaps the first: now, see how far you had to shift it in order for this to happen. Dec 4, 2020 at 19:35
• Thanks so much, that is so interesting! Dec 4, 2020 at 19:37