Engineering
Supersonic Mach Cone Angle & Subsonic Prandtl-Glauert Rule
Calculate the Mach-wave half-angle for supersonic flow and a separate Prandtl-Glauert pressure-coefficient approximation. Distinguish weak waves, finite-strength shocks and the sonic limit.
Two different flow regimes
The Mach angle describes infinitesimal disturbances in supersonic flow. It is not generally the shock angle around a finite wedge or aircraft nose; those require additional flow and geometry information. At Mach 2 the weak-wave half-angle is 30 degrees.
The Prandtl-Glauert relation is a linearized, small-disturbance subsonic approximation. It is not a formula for transonic shocks. Its validity can fail before Mach 1 when local flow reaches sonic speed; critical Mach depends on the configuration.
Mach-wave half-angle in degrees
B2 is the nonnegative Mach number for the wave calculation. Values above 1 return DEGREES(ASIN(1/M)). At exactly 1 the formula has a 90-degree limiting value, reported here as a sonic-limit message rather than calling it supersonic flight.
=IF(NOT(ISNUMBER(B2)),"Enter numeric Mach",IF(B2<0,"Mach must be nonnegative",IF(B2<1,"Subsonic: no Mach cone",IF(B2=1,"Sonic limit: 90 degrees",DEGREES(ASIN(1/B2))))))Prandtl-Glauert pressure coefficient
B3 is a separate Mach input and B4 is the incompressible pressure coefficient. For 0.6 and -0.5, the corrected coefficient is -0.625. This changes the dimensionless coefficient; it is not a pressure in pascals.
This exercise restricts evaluation to 0 <= M < 0.8 to avoid illustrating the divergent near-sonic expression as a usable prediction. This range is not a guarantee of validity: the flow must also satisfy the small-disturbance and subcritical assumptions for its geometry.
At Mach zero the original coefficient is recovered. Positive, negative and zero coefficients retain their sign. The wave-angle and subsonic-correction inputs are independent; do not use them as a single model spanning the sound barrier.
=IF(COUNT(B3:B4)<>2,"Enter numeric Mach and coefficient",IF(B3<0,"Mach must be nonnegative",IF(B3>=0.8,"Outside this subsonic example range",B4/SQRT(1-B3^2))))Try the example
Use the original values or change them to test your understanding.