Phase difference in sine waves

Question

Hey so I know that the phase difference is simply the difference in phase between two points on either the same wave or different waves.

Phase is represented by the letter phi in simple harmonic motion. Now I would like to ask, where can you measure the phase angle from, or is it simply mathametically derived? I am asking because I am confused about the way we plot waves in maths vs physics. In mathematics we plot angle in radians against the output (usually called y), whereas in physics we can have distance/displacement or even time/displacement for a wave. I used to have this misunderstanding that the phase difference was the wavelength "elaspsed" between two points, but this is not truly correct as phase difference is an angle and not a distance. Thanks in advance.

Answer 1

Periodic motions, like that of the moon, go through "phases" that repeat in exact order. In a pendulum we can say "pendulum at center moving left" (or $\phi=\phi_0$), "pendulum at left" (or $\phi=\pi/4$), "pendulum at center moving right" ($\phi=\phi_0+\pi$) and so on. One complete period amounts to a phase $2\pi$ from initial state so that half a period to $\pi$, a quarter period to $\pi/2$ and so on.

I used to have this misunderstanding that the phase difference was the wavelength "elapsed" between two points, but this is not truly correct as phase difference is an angle and not a distance.

Phase and distance are related but as you say the dimensions are different. The correct relationship is (see the answer by AccidentalTaylorExpansion for the math equations):

phase difference is the distance elapsed between two points divided by the wavelength of a complete period and multiplied by $2\pi$

Answer 2

In physics radians are almost always considered as unitless.

A proper "physical" sine wave looks as follows

$$u(x,t)=A\sin\left(kx-\omega t\right).$$ where $k=\frac{2\pi}{\lambda}$ and $\omega=2\pi f=\frac{2\pi}{T}$. Here $\lambda, f$ and $T$ are the wavelength, frequency and period respectively. Multiplied with $x$ and $t$, which have units of length and time, these combine to form a unitless quantity.

You could call the quantity inside $\sin$, i.e. $kx-\omega t$ the phase of this wave. Let's say you have to points $(x_1,t_1)$ and $(x_2,t_2)$. The phase difference between those points would be $$\Delta\phi=(kx_2-\omega t_2)-(kx_1-\omega t_1).$$ This quantity is in radians (it is actually unitless) and if you divide it by $2\pi$ you get the number of completed wave cycles that occur between those two points.