I started from @dharr code but did slightly different operations on the equations.
My point is to finder under chich conditions each term in the final dimensionless equation are indeed dimensionless, and check that these conditions ar mutually compatible.
Doing this I find that f, for instance, x is a length, so is z, and if t is a time, then u and v are velocities
More of this gamma and gamma are frequencies and kappa is surfave (look inside the attached code to see what kappa is).
But there is a problem, already underlined by @dharr, concerning f and f.
Ultimately a true dimensionless equation should contain "Dimensionless Numbers" (such as Reynolds, Mach, ...) build from some combinations of the elementary dimensions present in the initial equation (2 in the present case).
I believe this require some clarifications.