Sources:
Technology of Tanks by Richard M. Ogorkiewicz
A Jet Penetration Model Incorporating Effects of Compressibility and Target Strength by William J. Flis

This equation is derived from Birkhoff's Hydrodynamic (incompressible) Bernoulli formula, and is known variously as:

• Density Law
• Hill-Mott-Pack Equation
• Hydrodynamic Limit

Depending on who you ask.

To find for Penetration:

OR

(both equations are the same mathematically; they're just different ways of showing a square root)

To find for Jet Length:

Where:

T = Thickness of Armor Penetrated (millimeters)
L = Length of Shaped Charge Jet (millimeters)
D_J = Density of Jet Material (kg/m3)
D_T = Density of Target Material (kg/m3)

Validity of the Hydrodynamic Formula

The formula works really well when both the target and the shaped charge liner are made of the same thing, i.e. Steel Liner vs Rolled Homogenous Armor (RHA), and thus have unity (or near 1) ratios in densities. However, as target densities wildly diverge, so does penetration.

Another factor (found out by Eichelberger in Experimental Test of the Theory of Penetration by Metallic Jets [J. Appl. Phys. 27, 63–68 (1956)]) was that as the shape charge jet loses velocity, the compressive yield strength of both target and jet material comes into play. Effectively, at short standoffs and high jet velocities, the Hydrodynamic Limit formula works well, but breaks down as the jet slows down while penetrating extremely thick targets.

This is why if you put the exact same shaped charge against mild steel and Rolled Homogenous Armor (RHA), you'll get a totally different result, due to the difference in compressive/yield strengths of the two steels.