Polar Form Confusion: Understand Damped & Driven Wave Amplitude

[m1ke_]

Hello All,

I'm reviewing some notes for a course and am confused by one step that they do. They are deriving an equation for the ampiltude of a wave that is being damped and driven by a force. I understand it all except for one step in which they state that:

(\omega_{o}^{2} - \omega^{2}) - i(2\beta \omega) = \sqrt{(\omega_{o}^{2} - \omega^{2})^{2} + (2\beta\omega)^{2}}e^{-i\delta}

Can anyone explain this? Or do I need to post more material in order for it to be explained?

 

[spamiam]

As I think you know, they're just writing a complex number in its polar form. Any nonzero complex number can be written

<br /> a + ib = re^{i\theta}<br />

where r = \sqrt{a^2 + b^2} is the modulus and theta is a real number (called the argument).

If you're not familiar with complex numbers, just think of it as a vector (a,b) in \mathbb{R}^2 with x component a and y component b. Then r is the magnitude of the vector and theta is the angle it makes with the x-axis.