= Solution
The real part $A\cos(kx-\omega t+\theta_0)$ represents a monochromatic travelling wave for $k\ne0$. 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 $k\,dx/dt-\omega=0$, so
$$
\boxed{c_p=\frac\omega k.}
$$
On the positive-frequency, positive-$k$ branch,
$$
\boxed{c_p=\sqrt{c^2+\frac a{k^2}}.}
$$
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.
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