Engineering
Oblique Shock Theta-Beta-Mach Wedge Deflection & Normal Mach Formula
Calculate supersonic wedge deflection angle theta (degrees) from upstream Mach M_1 and shock wave angle beta using the exact Theta-Beta-Mach relation, plus normal Mach component M_n1 = M_1 * SIN(beta).
Analyze supersonic intake ramps and wedge shock waves via Theta-Beta-Mach!
When supersonic flow (M_1 > 1) strikes a compressive ramp or wedge, an attached Oblique Shock Wave forms at wave angle beta (measured from upstream flow) between the Mach wave limit mu = DEGREES(ASIN(1 / M_1)) and a 90-degree normal shock! Only the velocity component perpendicular to the shock wave—Normal Mach M_n1 = M_1 * SIN(RADIANS(beta)) >= 1—undergoes Rankine-Hugoniot compression, while tangential velocity is preserved!
Combining mass, momentum, and energy conservation yields the exact Theta-Beta-Mach Equation: TAN(theta) = 2 * (1 / TAN(beta)) * ((M_1^2 * SIN(beta)^2 - 1) / (M_1^2 * (gamma + COS(2 * beta)) + 2))! At Mach 2.5 in air (gamma = 1.4, Mach angle mu = 23.58 deg), a 35 deg oblique shock corresponds to a 13.17 deg ramp with M_n1 = 1.4339!
| Shock Wave Angle beta (Air M_1 = 2.5, gamma = 1.4, mu = 23.58 deg) | Normal Mach M_n1 = M_1*SIN(beta) | Wedge Deflection theta (Deg) | Shock Regime |
|---|---|---|---|
| Mach Angle Limit (beta = 23.58 deg) | 1.0000 | 0.00 deg | Infinitesimal Mach wave |
| Attached Weak Shock (beta = 35.0 deg) | 1.4339 | 13.17 deg | Supersonic downstream |
| Steeper Ramp Shock (beta = 45.0 deg) | 1.7678 | 21.57 deg | Stronger compression |
Exact Theta-Beta-Mach Wedge Deflection Angle theta (Degrees)
Validates M_1 > 1 and M_1 * SIN(RADIANS(beta)) >= 1 (so beta >= Mach angle mu), then computes DEGREES(ATAN(tan_theta)).
=IF(B2 <= 1, "Requires Supersonic M1 > 1", IF((B2 * SIN(RADIANS(B3))) < 1, "Wave Angle Beta Below Mach Angle", ROUND(DEGREES(ATAN(2 * (1 / TAN(RADIANS(B3))) * ((((B2 ^ 2) * (SIN(RADIANS(B3)) ^ 2)) - 1) / (((B2 ^ 2) * (B4 + COS(2 * RADIANS(B3)))) + 2)))), 2)))Shock-Normal Upstream Mach Component M_n1 = M_1 * SIN(beta)
Computes the perpendicular Mach component that plugs directly into 1D normal-shock jump tables.
=ROUND(B2 * SIN(RADIANS(B3)), 4)Try the example
Use the original values or change them to test your understanding.