PX275 - I0 - introduction to waves

a stationary plane wave:

e^{\pm i k x}

where, $k = 2 π / λ$

\exp (\pm i (k x - ω t))

where, $ω = k x$

considering 1D plane waves:

u (x, t) = A \exp (\pm i (k x - ω t)) = A \exp (\pm i k (x - c t))

and 3D plane waves:

u (\vec{r}, t) = C \exp (\pm) i (\vec{k} \cdot \vec{r} \pm) ω t

where, $C$ is the complex amplitude

the dispersion relation is given by:

ω = | \vec{k} | c

FT of a plane wave

the FT:

\tilde{u} (k, ω) = \iint_{- \infty}^{\infty} \exp (- i (k x + ω t)) u (x, t) d x d t

also, the inverse FT:

u (x, t) = {(\frac{1}{2 π})}^{2} \iint_{- \infty}^{\infty} \exp (i (k x + ω t)) \tilde{u} (k, ω) d k d ω

let $u (x, t) = A \exp (k^{'} x - ω^{'} t)$
taking its FT:

\begin{aligned} \tilde{u} (k, ω) & = \iint_{- \infty}^{\infty} \exp (- i (k x + ω t)) A \exp (k^{'} x - ω^{'} t) d x d t \\ = A \int_{- \infty}^{\infty} \exp (i (k^{'} - k)) d x \int_{- \infty}^{\infty} \exp (- i (ω^{'} + ω) t) d t \\ ⟹ \tilde{u} (k, ω) & \propto A δ (k - k^{'}) δ (ω + ω^{'}) \end{aligned}

so, a pure plane wave with wave number, $k^{'}$ , and frequency, $ω^{'}$ , maps to the product of delta functions in reciprocal space
physical phenomena with wave-like properties:
- surface waves on a fluid
- pressure waves in a compressible fluid
- optics: propagation, interference and diffraction of light waves
- quantum mechanics wavefunction
- wave-like properties of particles