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 $z$ ?