# Phases and sinusoidal waves

When we're talking about a wave, just a singular sinusoidal wave, what exactly is a 'phase'?

I came across a question that gave values of frequency ($550$Hz), and speed ($330$m/s). The question then asked to find how far apart two points are that differ in phase by $\frac{\pi}{3}$ rad. The answer came out to be $0.1$m. I was a little confused by the result, because if the two points differ by a phase of $\frac{\pi}{3}$ rad, the distance between them should also be $\frac{\pi}{3}$ rad, right?

• I wrote an answer describing how to define the group velocity of a wave that also describes the phase of a wave at: http://physics.stackexchange.com/a/143717/59023. The answer to your question involves the $\mathbf{k} \cdot \mathbf{x}$ term in the phase expression. – honeste_vivere May 3 '16 at 14:01
• I'm not familiar with treating waves how you did in your answer. Is it possible to reword your answer in simpler terms? – Aaron May 3 '16 at 14:08
• Sorry, I do not have time to rewrite the entire answer but I added some links to Wiki articles that may be easier to digest. – honeste_vivere May 3 '16 at 16:30

The clue here is " how far apart". The question is asking for distance which must be in terms of the wave's wavelength. Phase measures fractions of wavelength. And you are given information of the wave's speed and the periodic time in which it propagates (frequency).

The fundamental "distance = rate * time" applies in terms of the wave speed, wavelength and period, but period is the reciprocal of frequency.

$$\lambda=\frac{c}{f}$$ where $\lambda$ is the wavelength, $c$ is the wave's velocity , and $f$ the wave's frequency. Plugging the numbers in you get one wavelength of the wave equal to 0.6 meters. But you don't want to know a full wavelength ($2\pi$ radians), but rather the fraction $\frac{pi}{3}$ radians. So just apply a simple ratio $$\frac{0.6}{2\pi}=\frac{x}{\frac{pi}{3}}$$ x is equal to 0.1 m

Phase is the argument of the wave.

This is the definition written in my book and quite hard for beginners (like me).

So the question "What is phase?"

Phase is the quantity which tells us the status of the wave.

In normal x-y grid like x-axis tells us the distance from the origin, in similar way you can think a phase is along x-axis and it gives us distance from the origin of a particular point.

Confusion:

In normal grid, we measure distance of a point from origin in "meters" but phase is unit less. I told you to think phase as a distance from origin but this is for getting idea only, phase is not like distance between a particular point rather phase is difference in status of two waves .

And you can define that status as $\omega t+\phi$

In your question the phase difference is gives as $\pi /3$ this means there status differ by $\pi /3$. Now you define status difference =$\omega t=\pi /3$ and after this it is eating work!

Phase means the stage in its cycle which a wave has reached. Peak and trough are phases of the cycle, but it is more useful mathematically to describe phase in radians, as though the wave is a point moving round a circular track - the phase is the angle which the current radius makes with the starting radius.