Finding the real component of a two dimensional wave

[rmjmu507]

Homework Statement

Show that if the displacement of the waves on a membrane of width b is given by the superposition

$z=A_1\exp^{i(\omega t-(k_1x+k_2y))}+A_2\exp^{i(\omega t-(k_1x-k_2y))}$

with $z=0$ when $y=0$ and $y=b$ then the real part of z is

$z=2A_1sin(k_2)sin(\omega t-k_1x)$ where $k_2=\frac{n\pi}{b}$

Homework Equations

The Attempt at a Solution

So I've found that $A_1=-A_2$ and $k_2=\frac{n\pi}{b}$, but I don't quite see how I can show that the real part of z is $z=2A_1sin(k_2)sin(\omega t-k_1x)$. Can someone please provide some guidance?

 

[BvU]

How do you find the real part of a complex $z$ ?