The term shock polar is generally used with the graphical representation of the Rankine-Hugoniot equation

Rankine-Hugoniot equation

The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, relate to the behaviour of shock waves traveling normal to the prevailing flow. They are named in recognition of the work carried out by Scottish engineer and physicist William John...

s in either the hodograph

Hodograph

A hodograph is a diagram that gives a vectorial visual representation of the movement of a body or a fluid. It is the locus of one end of a variable vector, with the other end fixed. The position of any plotted data on such a diagram is proportional to the velocity of the moving particle. It is...

 plane or the pressure ratio-flow deflection angle plane. The polar itself is the locus of all possible states after an oblique shock

Oblique shock

An oblique shock wave, unlike a normal shock, is inclined with respect to the incident upstream flow direction. It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and compresses. The upstream streamlines are uniformly deflected after the shock wave...

.

Shock Polar in the Plane

The minimum angle, , which an oblique shock can have is the Mach angle , where is the initial Mach number before the shock and the greatest angle corresponds to a normal shock. The range of shock angles is therefore . To calculate the pressures for this range of angles,the Rankine-Hugoniot equation

Rankine-Hugoniot equation

The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, relate to the behaviour of shock waves traveling normal to the prevailing flow. They are named in recognition of the work carried out by Scottish engineer and physicist William John...

s are solved for pressure:



To calculate the possible flow deflection angles, the relationship between shock angle and is used:



Where is the ratio of specific heats and is the flow deflection angle.

Uses of Shock Polars

One of the primary uses of shock polars is in the field of shock wave reflection. A shock polar is plotted for the conditions before the incident shock, and a second shock polar is plotted for the conditions behind the shock, with its origin located on the first polar, at the angle through which the incident shock wave deflects the flow. Based on the intersections between the incident shock polar and the reflected shock polar, conclusions as to which reflection patterns are possible may be drawn. Often, it is used to graphically determine whether regular shock reflection is possible, or whether Mach reflection occurs.