Adding same-frequency complex waves using phasors and Euler's form to rewrite r1sin(ax+α1)+r2sin(ax+α2) as a single sine rsin(ax+α), with resultant amplitude r=∣∣r1eiα1+r2eiα2∣∣ and phase α=arg(r1eiα1+r2eiα2).
Adding two waves with the same frequency produces a wave with the same frequency:
Notice that the produced wave's amplitude never exceeds the sum of the two amplitudes we started with, but it can be smaller than both.
When the waves we add together have near opposite phases (the max of one is the min of the other), they cancel each other out partially.
For any two complex waves f(x)=r1sin(ax+α1) and g(x)=r2sin(ax+α2) with the same frequency but different phase, we can use Euler's form of complex numbers to express h(x) as a single sine function:
So we can write
where
This can be done on the calculator, but you can also just graph the waves instead.
The y-coordinate of the maxima tells us the amplitude is a≈2.83.
The x-coordinate of the first positive root tells us the phase shift: 2.60.
So the resulting wave is
Nice work completing Complex Phasors, here's a quick recap of what we covered:
Exercises checked off
Adding same-frequency complex waves using phasors and Euler's form to rewrite r1sin(ax+α1)+r2sin(ax+α2) as a single sine rsin(ax+α), with resultant amplitude r=∣∣r1eiα1+r2eiα2∣∣ and phase α=arg(r1eiα1+r2eiα2).
Adding two waves with the same frequency produces a wave with the same frequency:
Notice that the produced wave's amplitude never exceeds the sum of the two amplitudes we started with, but it can be smaller than both.
When the waves we add together have near opposite phases (the max of one is the min of the other), they cancel each other out partially.
For any two complex waves f(x)=r1sin(ax+α1) and g(x)=r2sin(ax+α2) with the same frequency but different phase, we can use Euler's form of complex numbers to express h(x) as a single sine function:
So we can write
where
This can be done on the calculator, but you can also just graph the waves instead.
The y-coordinate of the maxima tells us the amplitude is a≈2.83.
The x-coordinate of the first positive root tells us the phase shift: 2.60.
So the resulting wave is
Nice work completing Complex Phasors, here's a quick recap of what we covered:
Exercises checked off