10.3: Fourier Transforms for Time-Domain Functions
So far, we have been dealing with functions of a spatial coordinate \(x\) . Of course, mathematical relations don’t care about what kind of physical variable we are dealing with, so the same equations could be applied to functions of time \(t\) . However, there is a important difference in convention . When dealing with functions of the time coordinate \(t\) , it is customary to use a different sign convention in the Fourier relations!
The Fourier relations for a function of time, \(f(t)\) , are:
Definition: Fourier relations
\[\begin{align}\left\{\;\,\begin{aligned}F(\omega) &= \;\int_{-\infty}^\infty dt\; e^{i\omega t}\, f(t) \\ f(t) &= \int_{-\infty}^\infty \frac{d\omega}{2\pi}\; e^{-i\omega t}\, F(\omega).\end{aligned}\;\,\right.\end{align}\]
Compared to the Fourier relations previously given in Eq. (10.2.9), the signs of the \(\pm i \omega t\) exponents are flipped.
There’s a good reason for this difference in sign convention: it arises from the need to describe propagating waves, which vary with both space and time. As discussed in Chapter 5, a propagating plane wave can be described by a wavefunction of the form \[f(x,t) = A e^{i(kx - \omega t)},\] where \(k\) is the wave-number and \(\omega\) is the angular frequency. We write the plane wave function this way so that positive \(k\) indicates forward propagation in space (i.e., in the \(+x\) direction), and positive \(\omega\) indicates forward propagation in time (i.e., in the \(+t\) direction). This requires the \(kx\) and \(\omega t\) terms in the exponent to have opposite signs. Thus, when \(t\) increases by some amount, a corresponding increase in \(x\) leaves the exponent unchanged.
As we have seen, the inverse Fourier transform relation describes how a wave-form is broken up into a superposition of elementary waves. For a wavefunction \(f(x,t)\) , the superposition is given in terms of plane waves: \[f(x,t) = \int_{-\infty}^\infty \frac{dk}{2\pi} \int_{-\infty}^\infty \frac{d\omega}{2\pi}\;\; e^{i(kx-\omega t)}\, F(k,\omega).\] To be consistent with this, we need to treat space and time variables with oppositely-signed exponents: \[\begin{align} f(x) &= \int_{-\infty}^\infty \frac{dk}{2\pi}\; e^{ikx}\, F(k) \\ f(t) &= \int_{-\infty}^\infty \frac{d\omega}{2\pi}\; e^{-i\omega t}\, F(\omega).\end{align}\] The other equations follow similarly.