Wave Superposition - Complex Exponential

[djfermion]

Hi guys, I lurk here often for general advice, but now I need help with a specific concept.

Ok, so I started a classical and quantum waves class this semester. We are beginning with classical waves and using Vibrations and Waves by A P French as the text. So in the second chapter he discusses wave superposition and describing the motion of two waves added together with a single equation.

In the book, he explains most things geometrical, using the complex vector and complex exponential notation for the wave. He draws both waves and uses law of sines and law of cosines to determine the combined amplitude/frequency/phase angle.

One of the homework problems is then: Express z=sinwt + coswt in the form z=Re[Ae^i(wt+a)]

I was able to accomplish this geometrically with little difficulty and correctly got the answer to be A=root 2 and a=-pi/4. However, my professor said that I should not necessarily rely on the geometry and should be able to get the answer mathematically using the complex exponential form.

I have tried it this way and do not really understand how to go about it. Do I represent sinwt as -iAe^i(wt+a) or possibly as Ae^i(pi/2-wt-a).

Honestly, that particular problem is not that important. I just want to gain insight on how to find the superposition of waves by manipulating different complex exponentials.

 

[djfermion]

Ok, so after sitting for a second and reconsidering what my professor had said, I have made progress (half way there!). My professor had said the best way to do them is treat the complex exponentials as vectors and add components as you normally would. So that's what I did and I figured out the amplitude:

z=sinwt+coswt
z=Re[-ie^wt+e^wt]

Let the stuff in the bracket equal Ae^i(wt+a) then,

Ae^i(wt+a)=-ie^wt+e^wt
Ae^i(wt+a)=(-icoswt+sinwt)+(coswt+isinwt)
Ae^i(wt+a)=(sinwt+coswt)+i(sinwt-coswt)

A=root(Re^2+Im^2)
A=root[(sinwt+coswt)^2+i(sinwt-coswt)^2]
A=root(2)

That is indeed what the amplitude should be. Now I just need the phase angle, which seems like it is also so close.

 

[djfermion]

Well I figured it out. If I just expand the left side and equate the real and imaginary parts then I can solve for alpha.

Even though I didn't get any help, this served as a good way to exercise my brain and figure it out for myself, so thank you physicsforum and I'm sure you will see me again soon.