PX275 - G8 - complex exponential form of solution in 3D

1D case

the wave equation in 1D:

\frac{\partial^{2} u}{\partial t^{2}} = c^{2} \frac{\partial^{2} u}{\partial x^{2}} ⟹ (A \cos k x + B \sin k x) (C \cos ω t + D \sin ω t)

A \cos k x + B \sin k x = α \cos (k x + ϕ) = β \sin (k x + ϕ)

where, $α, β$ are the amplitudes, and $ϕ$ is the phase

a compact alternative is to use complex exponentials

C e^{i k x} = C (\cos k x + i \sin k x) = | C | e^{i (k x + ϕ)}

where, $C = | C | e^{i ϕ}$

note: $R e (| C | e^{i (k x + ϕ)}) = | C | \cos (k x + ϕ)$
same applies to the time part:

u (x, t) = C e^{i (k x - ω t)} = | C | e^{i (k x - ω t + ϕ)}

physical solutions are represented by the real part
note: $C e^{i (k x + ω t)}$ also works!
therefore, the full solution:

u (x, t) = C_{1} e^{i (k x - ω t)} + C_{2} e^{i (k x + ω t)}

there are four degrees of freedom: $R e (C_{1})$ , $I m (C_{1})$ , $R e (C_{2})$ , $I m (C_{2})$

generalizing to 3D

the position vector:

\vec{r} = (x, y, z)

the wave equation:

\frac{\partial^{2} u (\vec{r}, t)}{\partial t^{2}} = c^{2} \nabla^{2} u (\vec{r}, t)

u (\vec{r}, t) = C_{1} \exp (i (k_{x} x + k_{y} y + k_{z} z - ω t)) + C_{2} \exp (i (k_{x} x + k_{y} y + k_{z} z + ω t))

defining the wavevector:

\vec{k} = (k_{x}, k_{y}, k_{z})

⟹ u (\vec{r}, t) = C_{1} \exp (i (\vec{k} \cdot \vec{r} - ω t)) + C_{2} \exp (i (\vec{k} \cdot \vec{r} + ω t))

where, $ω^{2} = c^{2} | \vec{k} |^{2}$