I am wondering what the notion Fourier mode means.
Here is what I think it means:
Suppose, we have a function $u=u(x,t)$ which we can express as Fourier integral with respect to $x$: $$ u(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}\tilde{u}(k,t)\, dk, $$ where $\tilde{u}(k,t)$ is the Fourier transform. Now, for $\tilde{u}(k,t)$ we make the product ansatz $$ \tilde{u}(k,t)=e^{-i\omega t}\tilde{u}(k). $$ That is, we are trying to write $u(x,t)$ as $$ u(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(kx-\omega t)}\tilde{u}(k)\, dk. $$
Now, as far as I am informed, the functions $$ e^{i(kx-\omega t)} $$ are called Fourier modes.
So, Fourier modes are complex travelling waves?
How can they be vizualized?