Expressing waves in complex form

[inner08]

Question:
Two waves have the same amplitude, speed, frequency moving in the same region of space. The resultant wave can be expressed like the sum of two waves: psy(y,t) = Asin(ky+wt) + Asin(ky-wt+pi).

Express each wave individually using the complex representation. Demonstrate, using this representation, that the resultant wave can be expressed such as: psy(y,t)=2Acos(ky)sin(wt).

What is the amplitude at y=0, y=pi/k?


Attempt to answer:

I've tried breaking this problem into pieces. I'm having quite a bit of difficulty converting the wave equations into complex form.

I started by splitting the equation in two.

psi1(y,t) = (ky + wt)
psi2(y,t) = (ky - wt + pi)

From there, I'm not sure what to do. I know that I have to convert it into a form of Ae^i(wt-kx+epsilon).

Would simply be: psy1(y,t) = Ae^i(wt+ky) and psy2(y,t) = Ae^i(-wt+ky) ?

I'm pretty confused about this. I don't know where to go from here or even if what I did has sense to it. Hopefully someone can help me out!

 

[inner08]

Anyone?

 

[robphy]

Can you write $$\sin(\theta)$$ in terms of the exponential function?

 

[inner08]

I think (not sure) that its: 1/2i(e^i$$(\theta)$$ - e^(-i$$(\theta)$$))

 

[inner08]

Would anyone know any websites that could explain the idea behind this because my book is limited when describing this concept?