= Solution
Write $Z=\tilde z$ and introduce an arbitrary constant streamfunction scale $\Psi_0$, with $\hat\psi=\Psi_0\tilde\psi$. The displacement jump condition acquires only a constant prefactor $2\Psi_0/\Delta U$, which can be divided out:
$$
\boxed{\left[\frac{\tilde\psi}{\tilde U-\tilde c}\right]=0.}
$$
For the second of the <jump conditions for stratified inviscid shear flow>, the two derivative terms have common scale $\Delta U\Psi_0/h$. Dividing by it gives
$$
\left[(\tilde U-\tilde c)\tilde\psi_Z-\tilde\psi\tilde U_Z
-\frac{2gh}{\Delta U^2}\frac{\tilde\psi}{\tilde U-\tilde c}
-J\tilde\rho\frac{\tilde\psi}{\tilde U-\tilde c}\right]=0.
$$
The term proportional to the reference density has zero jump by the first condition. Thus
$$
\boxed{\left[(\tilde U-\tilde c)\tilde\psi_Z-\tilde\psi\tilde U_Z
-J\tilde\rho\frac{\tilde\psi}{\tilde U-\tilde c}\right]=0,\qquad
J=\frac{g\Delta\rho h}{\rho_0\Delta U^2}.}
$$
No physical reference-density force has been discarded inside either layer; its continuous interface contribution has canceled. The layer equation becomes $\tilde\psi_{ZZ}=\alpha^2\tilde\psi$. This explains why only the density anomaly appears in the nondimensional jump condition.
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