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Adding together waves using the Method of
Phasors. Lets say that you want to add two
waves together. In this example, they will be
the electric field part of an EM wave. Well say
the wave is harmonic and plane polarized. At
some point P, we have that the waves are E1
(4 N/C) sin (w1t p/6) and E2 (3 N/C) sin
(w2t p/3). If the (angular) frequencies of the
waves are the same (ie. if w1 w2), then you can
use the METHOD OF PHASORS.
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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 Atot 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 ftot. Step 5 The resultant wave from the
superposition of the individual waves will be
Atot sin (wt ftot)
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We were given E1 (4 N/C) sin (w1t p/6) and
E2 (3 N/C) sin (w2t p/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. Weve just checked Step 1. Now, well
use Step 2 and represent the waves by phasors
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.)
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Atot ? A1 A2
ftot ? f1 f2
Must do VECTOR addition
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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.)
Atot,x A1x A2x
A1 cos f1 A2 cos f2
Atot,y A1y A2y
A1 sin f1 A2 sin f2
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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 Df f2 f1. If there are multiple
waves, you continue this way, always rotating the
next phasor with respect to the previous phasor.
Df f2 f1 p/3 p/6 p/6
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Atot,x A1 cos 0 A2 cos Df
Atot,y A1 sin 0 A2 sin Df
EXACT same amplitude as before!!! NOTE Can NOT
get correct phase shift with this picture
THE END