Can Any Wave Be Expressed as a Complex Fourier Series?

[Niles]

Homework Statement

Hi

In Griffiths (chapter 9) he states that any wave can be expressed as a linear combination of sinusoidal waves,

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

Is it correct to say that this in principle is a complex Fourier series?

 

[vela]

Yes.

 

[Niles]

Thanks.

 

[vela]

Actually, it's not the series; it's the Fourier transform. It's the same basic idea though.

 

[Niles]

If its the transform and not the series, I don't see why Griffiths believes we can write *any* wave like that. I agree that if
$$f(z,t)=A(r)e^{-i\omega t}$$
then we can always write
$$f(z,t)=\int_{-\infty}^{\infty}{A(k)e^{i(kz-\omega t)}dk}$$
But that is not the same as saying that we can express *any* wave like this.

 

[vela]

You could look at it this way. At t=0, for any wave, you can write $$f(z,0) = \int_{-\infty}^\infty A(k)e^{ikz}\,dk.$$ You're taking a snapshot of the wave at an instant in time and expressing it in terms of its spatial frequency components. Each component then propagates independently, so at time t, the kth component is given by $A(k)e^{i(kz-\omega t)}$. Then by superposition, you simply sum over all the components to find the total wave, which gives you the expression Griffith's has.

Keep in mind that the waves satisfy the dispersion relation $\omega=|k|v$. When you integrate over k, you're not simply varying k but $\omega$ as well.

 

[Niles]

You could look at it this way. At t=0, for any wave, you can write $$f(z,0) = \int_{-\infty}^\infty A(k)e^{ikz}\,dk.$$ You're taking a snapshot of the wave at an instant in time and expressing it in terms of its spatial frequency components. Each component then propagates independently, so at time t, the kth component is given by $A(k)e^{i(kz-\omega t)}$. Then by superposition, you simply sum over all the components to find the total wave, which gives you the expression Griffith's has.

Keep in mind that the waves satisfy the dispersion relation $\omega=|k|v$. When you integrate over k, you're not simply varying k but $\omega$ as well.

Thanks, that is a good explanation, but one thing remains unclear to me: Why can I always decompose a wave at some time t=0 as

$$f(z,0) = \int_{-\infty}^\infty{A(kt)e^{ikz}dk}$$

There is a final thing, and it is perhaps a little relevant to the above. If you know the answer, I would very much appreciate it: In a book I have, they state that:" If we assume that we are dealing with harmonic modes, then we can write

$$E(r,t)=E(r)e^{-i\omega t}$$

My question is why we can factor the spatial and temporal parts in this case?Niles.