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Adding together waves using the Method of Phasors. Let’s say that you want to add two waves together. In this example, they will be the electric field part of an EM wave. We’ll say the wave is harmonic and plane polarized. At some point P, we have that the waves are E 1 = (4 N/C) sin (   t +  /6) and E 2 = (3 N/C) sin (   t +  /3). If the (angular) frequencies of the waves are the same (ie. if   =   ), then you can use the METHOD OF PHASORS. Click for next slide There are some animations in this example. If you’d like to see something again, simply hit the up arrow followed by the down arrow. When you are ready to move on, click the Play button shown on the right.

METHOD OF PHASORS. Step 1: Can you use the Method of Phasors? If the (angular) frequencies are the same, then YES. (Of course, the waves must overlap in space and time, and be the same kind of wave!) Step 2: Represent each wave by a VECTOR, whose MAGNITUDE is the AMPLITUDE of the wave. The DIRECTION of the vector is given by the PHASE of the wave. The PHASE is the angle of the VECTOR as measured COUNTER- CLOCWISE with respect to the POSITIVE x-axis. These vectors are called PHASORS. Step 3: Add the PHASORS together. Remember they are VECTORS! Step 4: Determine the amplitude A tot of the resultant vector (phasor) you obtain by adding the individual phasors together. Determine the angle of the resultant phasor (measured counter-clockwise from the + x-axis); this will be  tot. Step 5: The resultant wave from the superposition of the individual waves will be: A tot sin (  t +  tot ) Click for next slide

We were given E 1 = (4 N/C) sin (   t +  /6) and E 2 = (3 N/C) sin (   t +  /3) at the point P. Since they are the same kind of wave, with the same angular frequency, and we see they overlap in space, we can use the Method of Phasors. We’ve just checked Step 1. Now, we’ll use Step 2 and represent the waves by phasors:  /6 4 N/C  /3 3 N/C 22 A2A2 11 A1A1 Click for next slide The MAGNITUDE of the phasor is the AMPLITUDE of the wave. The DIRECTION of the phasor is given by the PHASE of the wave. (For convenience, the numerical values will be represented symbolically.)

11 A1A1 A2A2 22 A tot ≠ A 1 + A 2  tot ≠  1 +  2  tot A tot 11 A1A1 A 1x A 1y 22 A2A2 A 2x A 2y Must do VECTOR addition A 1x A 2x A 1y A 2y A 1x A 2x A 1y A 2y A tot  tot Click for next slide

A 1x A 2x A 1y A 2y A tot  tot A tot,x = A 1x + A 2x A tot,y = A 1y + A 2y Since all the angles are measured counter- clockwise from the +x axis, then can use our standard trig definitions to relate magnitudes and components. Note that with this convention, the SIGN of the component will automatically be taken care of (still should check that it makes sense, though.) = A 1 cos  1 + A 2 cos  2 = A 1 sin  1 + A 2 sin  2 A tot,x = (4 N/C) cos  /6 + (3N/C) cos  /3 = N/C A tot,y = (4 N/C) sin  /6 + (3N/C) sin  /3 = N/C N/C Click for next slide

A2A2 22 11 A1A1 A tot  2 –  1 =  /3 –  /6 =  /6 Sometimes, when adding waves together, you only care about the amplitude. Then, the only thing that matters is the phase difference between one wave and the next. You lay the first phasor along the horizontal, then draw the second phasor with an angle (counterclockwise from the +x axis) that is the difference in the phases of the two waves:  =  2 –  1. If there are multiple waves, you continue this way, always rotating the next phasor with respect to the previous phasor.  Click for next slide

 2 –  1 =  /3 –  /6 =  /6  A tot,x = A 1 cos  + A 2 cos  A tot,x = (4 N/C) cos  + (3N/C) cos  /6 = N/C A tot,y = (4 N/C) sin  + (3N/C) sin  /6 = N/C N/C A tot,y = A 1 sin  + A 2 sin  EXACT same amplitude as before!!! NOTE: Can NOT get correct phase shift with this picture THE END