Prandtl's Condition

It can be easily observed that the temperature from both sides of the shock wave is discontinuous. Therefore, the speed of sound is different in these adjoining mediums. It is therefore convenient to define the star Mach number that will be independent of the specific Mach number (independent of the temperature).

$$M^{*} = { U \over c^{*} } = {c \over c^{*} } {U \over c} = {c \over c^{*} } M$$ (5.32)

The jump condition across the shock must satisfy the constant energy.

$${c^2 \over k-1} + {U^2 \over 2 } = {{c^{*}}^2 \over k-1} + {{c^{*}}^2 \over 2 } = { k+ 1 \over 2( k -1) } {c^{*}}^2$$ (5.33)

Dividing the mass equation by the momentum equation and combining it with the perfect gas model yields

$${{c_1}^2 \over k U_1 } + U_1 = {{c_2}^2 \over k U_2 } + U_2$$ (5.34)

Combining equation (5.33) and (5.34) results in

$${1 \over kU_1} \left[ {k+1 \over 2 } {c^{*}}^2 - {k-1 \over 2 } U... ... \over kU_2} \left[ {k+1 \over 2 } {c^{*}}^2 - {k-1 \over 2 } U_2 \right] + U_2$$ (5.35)

After rearranging and diving equation (5.35) the following can be obtained:

$$U_1U_2 = {c^{*}}^2$$ (5.36)

or in a dimensionless form

$${M^{*}}_1 {M^{*}}_2 = {c^{*}}^2$$ (5.37)