Mach Number and Compressible Flow

At high speeds, compressibility effects become important. The Mach number characterises how fast a flow moves relative to the local speed of sound.

Speed of Sound

In an ideal gas, the speed of sound depends only on temperature:

$c = \sqrt{\gamma R T}$

where:

• $\gamma = 1.4$ — ratio of specific heats for air (diatomic ideal gas)
• $R = 287\,\text{J/(kg·K)}$ — specific gas constant for air
• $T$ — absolute temperature in kelvin (K)

At 20 °C (293.15 K), $c \approx 343$ m/s. Sound travels faster in warmer air because higher temperature means faster molecular motion.

Mach Number

The Mach number $M$ is the ratio of flow speed $v$ to the local speed of sound $c$:

$M = \frac{v}{c}$

Mach Cone

When an object moves at supersonic speed ($M > 1$), it outruns the pressure waves it creates. These waves pile up into a Mach cone (shock wave). The half-angle $\alpha$ of the cone satisfies:

$\sin \alpha = \frac{1}{M} \quad (M > 1)$

$\alpha = \arcsin\!\left(\frac{1}{M}\right)$

At $M = 2$, $\alpha = 30°$. Higher Mach numbers produce narrower cones.

Your Task

Implement:

• speed_of_sound(T) — speed of sound in m/s for air at temperature $T$ (K). Use $\gamma = 1.4$, $R = 287$ J/(kg·K).
• mach_number(v, T) — Mach number for flow speed $v$ (m/s) at temperature $T$ (K).
• mach_angle_deg(M) — Mach cone half-angle in degrees for $M > 1$; return None if $M \leq 1$.