Problem #1:
Kelvin-Helmholtz Instability

Two layer unbounded system with density contrast and shear. See Kundu,  page 375.

For z>0, the basic state is u=U1 and ρ=ρ1.  For z<0, the basic state is u=U2 and ρ=ρ2.

Your tasks:
 ☺ For disturbances of form (e^(i(kx-ωt))) with k>0, find ω(k).
 ☺ For the special case of case1 , plot Im[ω(k)].  Your
 graph might look like: 
 
 case1 could correspond to a layer of fresh water over salt water with 1 m/s shear across the interface.
 ☺ Do something else or make some intelligent comments about the problem
 you have solved.  Make a nice notebook. Put it on your web page.

case1={U1->0,U2->1,ρ1->1,ρ2->1.04,g->9.8}

exampleofuse = (g*(-ρ1^2 + ρ2^2))/((U1 - U2)^2*ρ1*ρ2)/.case1