Difference Between Spatial and Temporal Frequency Decomposition of Waves?

[Niles]

Homework Statement

Hi

In Griffiths it is stated that any possible wave can be written as a sum of plane waves

$$f(z,t)=\int_{-\infty}^{\infty}{A(k)e^{i(kz-\omega t)}dk}$$

This is a sum over spatial frequencies. In another book I have, they write the H-field as

$$H(r, t) = \sum_n{C_n H_n(r)\exp (-i\omega_n(t-t_0)}$$

where C_n are the amplitudes. This is a sum over temporal frequencies (i.e. ω, not k) of monochromatic waves. What is the difference between expressing an arbitrary wave in terms of frequency and spatial components? Is the answer that there is no difference, since (because of dispersion), when I vary k I also automatically vary ω and vice versa?Niles.

 

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Homework Equationsf(z,t)=\int_{-\infty}^{\infty}{A(k)e^{i(kz-\omega t)}dk}H(r, t) = \sum_n{C_n H_n(r)\exp (-i\omega_n(t-t_0)}The Attempt at a SolutionNo attempt at a solution has been made.