Freq response of forced sinusoidal motion:

[H_man]

The book I am currently reading derives the response of an electron to an applied sinusoidal field as:

1/[1 + 2i (w - w_a)/g] = 1 / (1 + i delta)

where w and w_a are the induced and resonance response frequencies and g the damping constant.

And delta = 2 (w - w_a)/g

Up to this point there is no problem. But the book then states that the amplitude squared or power response has the form:

1/[1 + {2 (w - w_a)/g}² ] = 1 / (1 + delta²)

And here is the problem... if you square the first equations listed you don't get what they give as the power response. I think the answer should be:

1 / (1 + 2 i delta - delta²)

Is there some assumption I am missing, its not just the imaginary term is missing the sign is wrong for the squared term??

Thanks for any help!

 

[Redbelly98]

For complex amplitudes, one squares the absolute value of the amplitude to get the power response.

 

[H_man]

Thanks!