waves - What is a phase of a waveform?

1. 
   

   Can please someone give a simple explanation of the phase of a waveform, particularly the sine function?

   
   


2. 
   

   Also, what is the angular frequency, and how does it differ from the frequency?

   
   

Answer

Phase means how far an oscillation has gotten through its cycle. There are various ways of expressing this. You could use a fraction - eg it is 1/4 way through its cycle.

Cycles suggest moving in a circle so it is convenient to compare different kinds of oscillations with motion in a circle with uniform speed, even when the oscillation is actually in a straight line, or when there is no "motion" at all (eg an oscillation of potential difference in an electric circuit). Referring to the circle, 1/4 way through the cycle is $90^{\circ}$. Using radians has a number of advantages over using degrees mathematically, so radians is preferred.

Image courtesy of SlideShare

The connection between circular motion and linear harmonic oscillation is that if you project the motion of an object P rotating around a circle of radius $A$ onto the horizontal or vertical axis, then the motion of the object's "shadow" Q along the axis is described by $y=A\sin(\omega t)$ or $x=A\cos(\omega t)$. Here $t$ is time and $\omega t=\theta$ is the "phase" or reference angle, ie the instantaneous angle between the x axis and the radius from the origin to the object P.

$\omega$ is the angular frequency. Frequency $f$ is how many times the object rotates round the circle every second, or how many times an oscillation goes through a full cycle every second. Angular frequency is how many times the object rotates through 1 radian every second, or how many times the oscillation goes through a fraction of $\frac{1}{2\pi}$ of a complete cycle every second. A full circle is $2\pi$ radians so angular frequency is related to frequency by $\omega=2\pi f$.