Phasors are just another way to represent waves

We can represent waves using equations:

We can represent waves using pictures of wiggling lines:

Today, we will come up with a new way to represent waves: by little clocks.

More specifically, we will use the position of the hand on the clock to tell us about the amplitude and phase of a wave at some given time. Let me illustrate with an example: watch as one particular wave propagates to the right:

Now, watch again as I put one of the little phasor clocks next to the graph.

The hand on the little clock rotates as the wave progresses -- the position of the hand says something about the amplitude of front of the wave, or something. In fact, there are several ways to interpret the position of the hand.

• 

  The position of the hand tells us how far the wave has moved since it was at its maximum positive amplitude.

• 

  The position of the hand tells us the angular position of the wave -- in other words, the value of all the stuff inside the sine or cosine function.

  

  Of course, we can choose to think of this angular position in terms of degrees, or in terms of radians.

  

In summary, a phasor uses the hand of a little clock to represent the wave.

• the position of the hand corresponds to the location of the wave within its cycle of oscillation
• the length of the hand corresponds to the amplitude of the wave

Different ways to add waves

So, suppose someone gives us two waves and asks, "What happens when these waves meet? How do they interfere?"

One way to answer the question is to write down equations and use trigonometric identities and algebra ...

Another way to answer the question is to draw diagrams showing each wave function

then read the values for each function from the graph, add them, and plot the result:

But perhaps the quickest and easiest way to do this is to draw little phasor clocks for each of the waves:

and then use the clocks themselves to do the addition. There is a very simple way to find the sum of two waves using phasors: just treat the little hands as vectors and add them:

 

  Q:  Call the amplitude of the black wave A.
      What is the amplitude of the wave
      which is the sum of the red and black waves?

  
  Q:  What is the phase of this sum relative to the
      black wave?

Let's check your answers by looking at the wave pictures:

A problem which is easy if you use phasors

Joe sets up three tuning forks on the table in front of him. Each tuning fork has exactly the same frequency: 440 Hz. However, the forks are different sizes:

1. tiny fork produces a wave with amplitude 2 units
2. middle fork produces a wave with amplitude 6 units
3. big fork produces a wave with amplitude 10 units

Joe strikes the tuning forks with his rubber mallet. Even though he tries to strike them all at once, there is a small offset in the production of each wave.

1. tiny fork has phase phi = 0
2. middle fork has phase phi = 90 degrees
3. big fork has phase phi = 180 degrees

       Q:  What is the amplitude of the combined wave produced
              by all three forks?  


       Q:  What is the phase of the combined wave produced
              by all three forks, relative to the 
              wave of the tiny fork?

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.