Linear Hydrodynamic Stability - Periodicity of Perturbations I am currently taking a course in Hydrodynamic Stability, based around Drazin's 'Introduction to Hydrodynamic Stability'. 
Up until this point, we have considered linear stability, where we assume a basic background state of e.g $\mathbf{u, p, \rho}$ etc and then add in perturbations $\mathbf{u', p'}$ etc. These are subbed into the governing equations and terms of e.g $\mathcal O (\mathbf u'^2)$ are discarded.
After this, it is assumed that the perturbations take the form of e.g $\mathbf u' = U(z)e^{ikx+st}$.
The conceptual difficulty I am having is as follows:
Why would these perturbations be spatially periodic? It seems an incredibly unphysical assumption to make. Is it necessary that the solutions to linear equations must take this form?
 A: If a disturbance to a steady base flow $(\mathbf{U}(\mathbf{x}), P(\mathbf{x}))$ (velocity and pressure field) is introduced at time $t= 0$ then the time-dependent altered flow is expressed in terms of disturbance variables as
$$\mathbf{u}(\mathbf{x},t) = \mathbf{U}(\mathbf{x}) + \mathbf{u}'(\mathbf{x},t), \quad p(\mathbf{x},t) = P(\mathbf{x}) + p'(\mathbf{x},t) $$
with some initial condition $\mathbf{u}'_0(\mathbf{x}) = \mathbf{u}'(\mathbf{x},0),\,\, p'_0(\mathbf{x}) = p'(\mathbf{x},0) $.
Hydrodynamic stability analysis seeks to determine conditions under which the disturbance will be amplified (unstable) or decay to zero (stable) so that the altered flow reverts back to the base flow.
Linearization of the governing equations for the disturbance is a typical first step because solution in closed form may be possible. As such it is an approximate analysis in that it is valid in the limit as the magnitude of the disturbance becomes infinitesimally small. Nevertheless, if conditions are found for which the flow is unstable to such small disturbances, then these are likely to be sufficient for instability of disturbances with finite amplitudes.
Solution of the linearized equations is typically facilitated by decomposition of the disturbance as an assumed convergent Fourier series (for bounded domains) or a convergent Fourier integral (for unbounded domains), both with respect to the spatial variables, that is  
$$\mathbf{u}'(\mathbf{x},t) = \sum_{\mathbf{k} \in\mathbb{Z}^3}\hat{\mathbf{u}'}(\mathbf{k},t)e^{i \mathbf{k} \cdot \mathbf{x}}$$
or 
$$\mathbf{u}'(\mathbf{x},t) = \int_{\mathbb{R}^3}\hat{\mathbf{u}'}(\mathbf{k},t)e^{i \mathbf{k} \cdot \mathbf{x}} \, d \mathbf{k}$$
Typically a physical disturbance is spatially localized as it is generated with finite energy which lends credibility to this type of representation. 
Since the governing partial differential equations for the disturbance are linear then the usual approach of separation of variables (for bounded domains) or Fourier transformation (for unbounded domains) may be applied leading to ordinary differential equations for the "amplitudes" $\hat{\mathbf{u}}'({\mathbf{k},t})$ of the disturbance modes.
Frequently, but not always, the simple exponential time behavior $\hat{\mathbf{u}}'({\mathbf{k},t}) = e^{\sigma t}\hat{\mathbf{u}}_0'({\mathbf{k}})$ will be discovered through the solution procedure.
So assuming general disturbances of the form $e^{\sigma t}e^{i \mathbf{k} \cdot \mathbf{x}}$ as an ansatz is not as you say "incredibly unphysical."
