Interpreting Imaginary Component Amplitudes in Fourier Series

[da_willem]

If you express a wave as a Fourier series like:

z(x,t)= \sum _{n=1} ^{ inf.} A_n cos(nk_0 x - \omega (n) t )

Then what is the physical interpretation of a non-zero imaginary part of a component amplitude?

 

[marlon]

If you express a wave as a Fourier series like:

z(x,t)= \sum _{n=1} ^{ inf.} A_n cos(nk_0 x - \omega (n) t )

Then what is the physical interpretation of a non-zero imaginary part of a component amplitude?

Hi Willem

Somebody wants to prevent me from helping you out here so i have written this answer for you


regards
marlon

 

[da_willem]

I read your word document, and would like to thank you very much for that. But maybe I should have specified my question some more, it is still not very clear to me...

If you want to describe a real wave in a formula you give it's height as a function of position and time. A height cannot be imaginary. So when an imaginary component amplitude appears in you Fourier representation of that wave there must be somethig wrong. This component cannot be canceled by another component can it? Is the appearance of an imaginary component in the sum a mathematical curiosity, or does it simply never appear for a real signal, or...?

 

[marlon]

I read your word document, and would like to thank you very much for that. But maybe I should have specified my question some more, it is still not very clear to me...

If you want to describe a real wave in a formula you give it's height as a function of position and time. A height cannot be imaginary. So when an imaginary component amplitude appears in you Fourier representation of that wave there must be somethig wrong. This component cannot be canceled by another component can it? Is the appearance of an imaginary component in the sum a mathematical curiosity, or does it simply never appear for a real signal, or...?

The imaginary factor does not have a real and direct physical meaning. When you take the real and imaginary parts (Re and Im) of this complex term you get physical useful info, just as explained in my word-doc.

As an example : these complex factors in the wave-equation often arise from the differential-equations that describe the physical system or from a process called harmonization that is used in order to set up the MHD-equations that describe the classical plasma-physics


regards
marlin