Basic linear stability analysis consists of presuming the existence of sinusoidal disturbances to a basic state (also called a background, initial, or equilibrium state), which is the flow whose stability is being investigated. For example, the velocity field of a basic state involving a flow parallel to the x-axis and varying along the z-axis is \( \mathbf{U} = U(z)\mathbf{e}_x \). On this background flow we superpose a spatially extended disturbance of the form: \[ \underbrace{u(x, y, z, t)}_{\text{Disturbance field}} = \underbrace{\hat{u}(z)}_{\text{Amplitude in } z} \underbrace{e^{i(kx + my + \sigma t)}}_{\text{Planar wave in } x, y \text{ with temporal growth}} = \hat{u}(z) \underbrace{e^{i|\mathbf{K}| (\mathbf{e}_K \cdot \mathbf{x} - ct)}}_{\text{Traveling wave in direction of } \mathbf{K}} \] where \( \hat{u}(z) \) is a complex amplitude, \( i = \sqrt{-1} \) is the imaginary root, \( \mathbf{K} = (k, m, 0) \) is the disturbance wave number, \( \mathbf{e}_K = \mathbf{K}/|\mathbf{K}| \), \( \mathbf{x} = (x, y, z) \), \( \sigma \) is the temporal growth rate, \( c \) is the complex phase speed of the disturbance, and the real part of this equation is taken to obtain physical quantities
For simplicity ignore interfacial (surface) tension, and assume that only small-slope linear waves exist on the interface and that both fluids are infinitely deep, so that only those solutions that decay exponentially from the interface are allowed. Therefore, complex notation will be used to ease the algebraic and trigonometric effort
The two forms of \(u(x, y, z, t) = \hat{u}(z) \exp \{i kx + imy + \sigma t \} = \hat{u}(z) \exp \{i |\mathbf{K}| (\mathbf{e}_K \cdot \mathbf{x} - ct) \}\) are useful when