Must a 1D sine wave in a linear homogeneous medium have a constant speed? It is clear that a sinusoidal driver in a linear time-independent homogeneous medium will produce sinusoidal waves. All examples of such non-dissipative waves in one dimension seem to have constant speed. Is that necessarily the case? (I do not assume the wave-equation applies.)
 A: The speed of sound $c(\omega)$ is a property of a medium at specific conditions; if we assume small oscillations in a non-dissipative homogenous medium, the fact that all sound waves of a given frequency propagate at the same constant speed follows from the wave equation, which, although not assumed to be true in the question, can be inferred from Euler's equation and the continuity equation.

Here's a more rigorous justification:
Assume that the medium is of constant density $\rho_0$ and pressure $p_0$; the sound propagates as a fluctuations in pressure and density such that $$p=p_0+\delta p(\mathbf{x},t);\quad \rho=\rho_0+\delta\rho(\mathbf{x},t).$$
Assuming the wave's motion is adiabatic, at a constant entropy $S$ we have $$\delta p=\frac{\partial p}{\partial \rho}\bigg|_S\delta \rho.$$
The continuity equation is $$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0\implies\frac{\partial}{\partial t}\delta \rho+\rho_0\nabla\cdot\mathbf{v}=0,$$
where we have neglected the second order term $\mathbf{v}\cdot\nabla\delta\rho$.
Euler's equation can be linearized as $$\frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}+\frac{\nabla p}{\rho}=0\implies\frac{\partial\mathbf{v}}{\partial t}+\frac{\nabla p}{\rho_0}=0.$$
Combining these three equations and defining the velocity potential $\phi$ such that $\mathbf{v}=\nabla\phi$, we have $$\frac{\partial^2\phi}{\partial t^2}-\frac{\partial p}{\partial \rho}\Bigg|_S\frac{\partial^2\phi}{\partial x^2}=0.$$
This is the wave equation with speed $c=\sqrt{\partial p/\partial\rho|_S}$! Thus, disturbances of this form, which are what you'd get with a sinusoidal driving force, must follow the wave equation, and thus they will propogate with this constant speed.
