 Solution

The real part $A\cos (kx-\omega t+{\theta }_0)$ represents a monochromatic travelling wave for $k\ne 0$. The magnitude $|A|$ is its displacement or field amplitude, and $\theta =kx-\omega t+{\theta }_0$ is its phase, with ${\theta }_0$ an initial phase offset. The complex exponential is a convenient representation of that real oscillation.

Following a constant-phase point gives $kdx/dt-\omega =0$, soOn the positive-frequency, positive-$k$ branch,For negative $k$, $\omega /k$ is negative on that frequency branch and encodes the reversed direction. At $k=0$, the mode is spatially uniform with frequency $\sqrt{a}$, so a propagation phase speed is not defined.