An initially unperturbed two-dimensional inviscid jet in $-h<y<h$ has uniform speed $U$ in the $x$ direction, while the surrounding fluid is stationary. The unperturbed velocity field $\mathbf{u}=(u, v)$ is therefore given by

$$\begin{array}{ll} u=0 & \text { in } \quad y>h \\ u=U & \text { in } \quad-h<y<h \\ u=0 & \text { in } \quad y<-h \end{array}$$

Consider separately disturbances in which the layer occupies $-h-\eta<y<h+\eta($ varicose disturbances) and disturbances in which the layer occupies $-h+\eta<y<h+\eta($ sinuous disturbances $)$, where $\eta(x, t)=\hat{\eta} e^{i k x+\sigma t}$, and determine the dispersion relation $\sigma(k)$ in each case.

Find asymptotic expressions for the real part $\sigma_{R}$ of $\sigma$ in the limits $k \rightarrow 0$ and $k \rightarrow \infty$ and draw sketches of $\sigma_{R}(k)$ in each case.

Compare the rates of growth of the two types of disturbance.