3.4 Normal and oblique shock waves

Properties of normal shock waves

A normal shock wave is a razor-thin region (on the order of a few mean free paths thick) where supersonic flow abruptly transitions to subsonic flow. Across this region, pressure, temperature, and density all jump sharply upward, while velocity drops. These shocks form when supersonic flow encounters an obstruction or must adjust to a downstream pressure condition it can't meet gradually.

The key thing to internalize: a normal shock is perpendicular to the flow direction. Every property change happens across that perpendicular plane, and the flow downstream is always subsonic.

Pressure ratio across the shock

The static pressure jumps significantly across a normal shock. For a calorically perfect gas with ratio of specific heats $\gamma$, the pressure ratio is:

$\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1}(M_1^2 - 1)$

• Higher upstream Mach numbers produce larger pressure ratios. At $M_1 = 2$ in air ($\gamma = 1.4$), the pressure ratio is about 4.5.
• At $M_1 = 1$ (the weakest possible shock), the ratio equals 1, meaning no shock at all.

Temperature ratio across the shock

Temperature also rises across the shock because kinetic energy of the bulk flow converts into random thermal motion of the molecules. The temperature ratio is:

$\frac{T_2}{T_1} = \frac{p_2}{p_1} \cdot \frac{\rho_1}{\rho_2}$

or equivalently, expressed directly in terms of $M_1$:

$\frac{T_2}{T_1} = \left[1 + \frac{2\gamma}{\gamma+1}(M_1^2 - 1)\right] \cdot \left[\frac{2 + (\gamma-1)M_1^2}{(\gamma+1)M_1^2}\right]$

The temperature increase grows with upstream Mach number, just like the pressure ratio.

Density ratio across the shock

Density increases to satisfy conservation of mass. Since velocity drops across the shock, density must rise so that $\rho u$ remains constant. The density ratio is:

$\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{2 + (\gamma-1)M_1^2}$

Unlike pressure and temperature, the density ratio has an upper limit. As $M_1 \to \infty$, the density ratio approaches $\frac{\gamma+1}{\gamma-1}$, which equals 6 for air ($\gamma = 1.4$).

Mach number change

• Upstream Mach number is always supersonic: $M_1 > 1$
• Downstream Mach number is always subsonic: $M_2 < 1$

The downstream Mach number is given by:

$M_2^2 = \frac{1 + \frac{\gamma-1}{2}M_1^2}{\gamma M_1^2 - \frac{\gamma-1}{2}}$

The velocity drop happens because both the bulk flow speed decreases and the local speed of sound increases (since temperature rises). Both effects push $M_2$ below 1.

Entropy increase

Entropy always increases across a normal shock. This is what makes shocks irreversible. The entropy change per unit mass is:

$\Delta s = c_v \ln\left(\frac{p_2}{p_1}\right) - c_p \ln\left(\frac{\rho_2}{\rho_1}\right)$

• Stronger shocks (higher $M_1$) produce larger entropy increases.
• The second law of thermodynamics requires $\Delta s > 0$, which is why expansion shocks (which would decrease entropy) are physically impossible.
• This entropy rise corresponds directly to a loss in stagnation (total) pressure.

Rankine-Hugoniot relations

The Rankine-Hugoniot relations are the governing equations across a shock wave, derived from the three fundamental conservation laws applied to a control volume straddling the shock. They hold for any gas, not just ideal gases.

Conservation of mass

Mass flow rate per unit area is the same on both sides:

$\rho_1 u_1 = \rho_2 u_2$

Since velocity decreases across the shock, density must increase proportionally.

Conservation of momentum

The sum of pressure and momentum flux is conserved:

$p_1 + \rho_1 u_1^2 = p_2 + \rho_2 u_2^2$

The pressure rise across the shock exactly compensates for the drop in momentum flux ($\rho u^2$). This is Newton's second law applied to the fluid passing through the shock.

Conservation of energy

Total enthalpy is constant across the shock (no heat addition or work):

$h_1 + \frac{1}{2}u_1^2 = h_2 + \frac{1}{2}u_2^2$

Kinetic energy lost by the decelerating flow reappears as increased static enthalpy (higher temperature). Note that stagnation temperature is conserved across the shock, but stagnation pressure is not.

Normal shock in ideal gas

For a calorically perfect ideal gas ($p = \rho R T$, constant $\gamma$), the Rankine-Hugoniot relations yield closed-form expressions for all property ratios as functions of $M_1$ and $\gamma$ alone. This is what makes the normal shock tables you'll find in textbooks possible.

Upstream and downstream states

The upstream state has high Mach number, low static pressure, low static temperature, and low density. The downstream state is the opposite: low Mach number, high pressure, high temperature, and high density. All property ratios ($p_2/p_1$, $T_2/T_1$, $\rho_2/\rho_1$) increase monotonically with $M_1$.

Mach number limits

• A normal shock requires $M_1 > 1$. At $M_1 = 1$, the "shock" has zero strength and nothing happens.
• The downstream Mach number $M_2$ is always less than 1.
• As $M_1 \to \infty$, $M_2$ approaches a finite lower limit: $M_2 \to \sqrt{\frac{\gamma - 1}{2\gamma}}$, which is approximately 0.378 for air.

Stagnation pressure ratio

Stagnation (total) pressure always decreases across a normal shock:

$\frac{p_{02}}{p_{01}} < 1$

This ratio drops more steeply as $M_1$ increases. For example, at $M_1 = 2$ in air, $p_{02}/p_{01} \approx 0.72$, meaning about 28% of total pressure is lost. At $M_1 = 3$, the loss exceeds 67%. This stagnation pressure loss is directly tied to the entropy increase and represents a permanent degradation in the flow's ability to do work.

Maximum entropy increase

The entropy increase across a normal shock grows monotonically with $M_1$ for a calorically perfect gas. There is no local maximum at some intermediate Mach number. The entropy rise is zero at $M_1 = 1$ and increases without bound as $M_1 \to \infty$.

Moving normal shock waves

So far, the analysis has assumed the shock is stationary. In many real situations, the shock itself moves through the fluid. The same physics applies, but you need to be careful about reference frames.

Stationary vs. moving shocks

• A stationary shock is fixed in space; the fluid flows through it.
• A moving shock propagates through fluid that may itself be stationary or moving.

The standard approach is to transform into the shock-fixed reference frame, apply the normal shock relations, and then transform back. All property ratios (pressure, temperature, density) are the same in both frames because they don't depend on the observer's velocity. Velocities, however, must be adjusted for the frame change.

Shock velocity relative to flow

In the shock-fixed frame, the flow approaching the shock must be supersonic. If the shock moves at velocity $V_s$ into a stationary gas, the gas approaches the shock at speed $V_s$ in the shock-fixed frame. The downstream gas velocity (in the lab frame) is then:

$u_{\text{lab}} = V_s - u_2$

where $u_2$ is the downstream velocity in the shock-fixed frame.

Shock propagation in ducts

When a moving shock enters a duct with varying cross-section:

• In a converging duct, the shock strengthens (higher pressure ratio) and accelerates.
• In a diverging duct, the shock weakens and decelerates.

The duct's area change modifies the effective upstream conditions seen by the shock, altering its propagation speed and strength.

Oblique shock waves

Oblique shocks are inclined at an angle to the incoming flow direction. They form when supersonic flow is forced to turn, such as at a compression corner or a wedge. Unlike normal shocks, oblique shocks allow the downstream flow to remain supersonic.

Oblique vs. normal shock waves

A normal shock is actually a special case of an oblique shock where the wave angle $\beta = 90°$.

Shock wave angle

The shock wave angle $\beta$ is measured between the oblique shock and the upstream flow direction. It depends on:

• The upstream Mach number $M_1$
• The flow deflection angle $\theta$
• The specific heat ratio $\gamma$

These are related through the theta-beta-Mach relation:

$\tan\theta = 2\cot\beta \cdot \frac{M_1^2 \sin^2\beta - 1}{M_1^2(\gamma + \cos 2\beta) + 2}$

This equation is implicit in $\beta$, so it's typically solved graphically (using oblique shock charts) or numerically.

Deflection angle

The deflection angle $\theta$ is the angle through which the flow turns as it passes through the oblique shock. It's set by the geometry, such as the angle of a wedge or compression ramp.

For a given $M_1$, there is a maximum deflection angle $\theta_{\max}$. If the required turning exceeds $\theta_{\max}$, no attached oblique shock solution exists, and a detached (bow) shock forms instead. $\theta_{\max}$ increases with $M_1$.

Weak vs. strong solutions

For any combination of $M_1$ and $\theta < \theta_{\max}$, the theta-beta-Mach relation gives two solutions:

• Weak shock: smaller $\beta$, weaker compression, downstream flow is usually supersonic ($M_2 > 1$)
• Strong shock: larger $\beta$, stronger compression, downstream flow is subsonic ($M_2 < 1$)

In practice, the weak solution almost always occurs in external flows. The strong solution typically appears only when a high downstream pressure forces it, such as near the centerline of a detached bow shock.

Oblique shock relations

The property ratios across an oblique shock use the same normal shock formulas, but applied only to the normal component of the Mach number: $M_{n1} = M_1 \sin\beta$. The tangential velocity component is unchanged across the shock.

Pressure ratio across oblique shock

$\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_{n1}^2 - 1)$

This is identical to the normal shock formula with $M_1$ replaced by $M_{n1} = M_1\sin\beta$. Since $M_{n1} < M_1$ for any oblique shock ($\beta < 90°$), the pressure ratio is always less than for a normal shock at the same freestream Mach number.

Density ratio across oblique shock

$\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_{n1}^2}{2 + (\gamma-1)M_{n1}^2}$

Again, substitute $M_{n1}$ into the normal shock density ratio. The same upper limit of $(\gamma+1)/(\gamma-1)$ applies.

Temperature ratio across oblique shock

$\frac{T_2}{T_1} = \frac{p_2}{p_1} \cdot \frac{\rho_1}{\rho_2}$

The temperature ratio follows from the ideal gas law once you know the pressure and density ratios.

Downstream Mach number

The normal component of the downstream Mach number is:

$M_{n2}^2 = \frac{1 + \frac{\gamma-1}{2}M_{n1}^2}{\gamma M_{n1}^2 - \frac{\gamma-1}{2}}$

The full downstream Mach number is then recovered using the deflection angle:

$M_2 = \frac{M_{n2}}{\sin(\beta - \theta)}$

For the weak shock solution, $M_2 > 1$ in most cases. The downstream flow is only subsonic for weak shocks very close to $\theta_{\max}$ or for the strong shock solution.

Supersonic flow over wedges

A symmetric wedge in supersonic flow is one of the cleanest examples of oblique shock generation. The half-angle of the wedge equals the deflection angle $\theta$, and the resulting flow is uniform between the shock and the wedge surface.

Attached vs. detached shocks

• Attached shock: Forms when the wedge half-angle $\theta$ is less than $\theta_{\max}$ for the given $M_1$. The shock is straight and attached to the wedge apex.
• Detached shock: Forms when $\theta > \theta_{\max}$. The shock stands off from the wedge, curves around it, and includes a region of subsonic flow near the nose. The detached shock is locally normal at the centerline and becomes increasingly oblique farther from the axis.

Wedge angle for attached shock

The maximum wedge half-angle for an attached shock depends on $M_1$ and $\gamma$. For air at $M_1 = 2$, $\theta_{\max} \approx 23°$. At $M_1 = 3$, it increases to about $34°$. You find $\theta_{\max}$ by locating the peak of the $\theta$-$\beta$ curve for a given $M_1$ in the theta-beta-Mach relation.

Maximum deflection angle

The maximum deflection angle increases with Mach number but approaches a finite limit as $M_1 \to \infty$. For $\gamma = 1.4$, this limiting value is about $45.6°$. Designing a supersonic vehicle means keeping compression surfaces below $\theta_{\max}$ to avoid detached shocks, which produce much higher drag.

Reflection of oblique shocks

When an oblique shock hits a solid wall, it reflects to turn the flow back parallel to the wall. The reflected shock must cancel the deflection imposed by the incident shock. Shock reflections also occur when two shocks from opposite walls intersect.

Regular vs. Mach reflection

• Regular reflection: The incident shock hits the wall and a single reflected shock redirects the flow parallel to the surface. Both shocks meet at the wall. This occurs when the required deflection for the reflected shock is below $\theta_{\max}$ for the post-incident-shock Mach number.
• Mach reflection: When the reflected shock would need to turn the flow beyond $\theta_{\max}$, regular reflection is impossible. Instead, a nearly normal shock segment called a Mach stem forms at the wall, and a triple point appears where the incident shock, reflected shock, and Mach stem meet. A slip line (contact discontinuity) trails downstream from the triple point.

Mach reflection is more common with stronger incident shocks and larger initial deflection angles.

Shock-shock interaction

When two oblique shocks intersect (for example, from opposite walls of a duct), the interaction produces:

• A triple point where three shocks meet
• Transmitted shocks downstream of the intersection
• A slip line (contact discontinuity) separating regions with the same pressure but different entropy, density, and velocity magnitude

The flow pattern depends on whether the two incident shocks have equal or unequal strengths.

Shock polars

A shock polar is a plot (typically in the pressure-deflection plane or the hodograph plane) that shows all possible downstream states for a given upstream Mach number. Each point on the polar corresponds to a different shock wave angle $\beta$.

• The upper branch represents strong shocks, the lower branch represents weak shocks.
• The maximum deflection angle appears at the tip of the polar.
• For shock-shock interactions, you overlay the polars for each region. The intersection points of the polars give the compatible downstream conditions.

Shock polars are a powerful graphical tool for solving complex shock interaction problems where algebraic solutions become unwieldy.

Shock wave-boundary layer interaction

Real flows have boundary layers, and when a shock wave impinges on or forms within a boundary layer, the interaction can dramatically alter the flow. This is one of the most challenging problems in compressible aerodynamics.

Shock-induced separation

The sharp pressure rise across a shock creates a severe adverse pressure gradient. If the boundary layer doesn't have enough momentum to push through this pressure rise, it separates from the surface.

• Turbulent boundary layers resist separation better than laminar ones because they carry more momentum near the wall.
• Separation creates a recirculation bubble that thickens the effective body shape, altering the shock position and strength.
• The resulting flow is often unsteady, with the shock oscillating back and forth, a phenomenon called shock buffet on transonic airfoils.

Lambda shock structure

When a shock interacts with a boundary layer, the subsonic portion of the boundary layer "communicates" the pressure rise upstream. This causes the shock to split near the wall into a characteristic $\lambda$-shaped pattern:

1. A leading oblique shock (the forward leg of the $\lambda$) forms where the thickening boundary layer first deflects the outer flow.
2. A trailing, nearly normal shock (the rear leg) completes the compression.
3. The two legs merge above the boundary layer into the main shock.

Between the legs, a separation bubble often exists. This structure appears on transonic airfoils, in supersonic inlets, and wherever shocks meet viscous layers.

Shock train in supersonic flow

In confined supersonic flows (like ducts and isolators), a single normal shock may not be stable. Instead, a series of shocks forms, called a shock train.

• Each successive shock is weaker than the one before it.
• The boundary layer thickens after each shock, and the interaction between the growing boundary layer and the confined flow produces the next shock.
• The spacing between shocks decreases in the downstream direction.
• Shock trains are common in scramjet isolators and supersonic diffusers, where they serve to decelerate the flow over a longer distance with lower total pressure loss than a single normal shock would produce.

Applications of shock waves

Supersonic inlets

Supersonic inlets must decelerate the incoming flow to subsonic speeds before it enters the engine compressor. The goal is to achieve this with the highest possible total pressure recovery (minimizing entropy production).

• Pitot inlets use a single normal shock, which is simple but produces large total pressure losses at high Mach numbers.
• External compression inlets use a series of oblique shocks followed by a weak terminal normal shock. Each oblique shock produces less entropy than a single strong normal shock would, so the overall pressure recovery is much better.
• Mixed compression inlets place some oblique shocks outside and some inside the duct, offering high performance but requiring careful control to prevent inlet unstart (shock expulsion).

Shock tubes and tunnels

A shock tube is a long tube divided by a diaphragm into a high-pressure driver section and a low-pressure driven section.

1. The diaphragm bursts.
2. A normal shock wave propagates into the low-pressure gas, compressing and heating it.
3. An expansion fan propagates back into the driver gas.
4. The region between the shock and the contact surface contains gas at known, uniform high-temperature conditions.

Shock tunnels extend this concept: the shock reflects off the end wall, further heating and compressing the test gas, which then expands through a nozzle to simulate hypersonic flight conditions for brief test times (typically milliseconds).

Shock wave lithotripsy

Shock waves have a medical application in extracorporeal shock wave lithotripsy (ESWL), used to break up kidney stones without surgery.

• A shock wave source (electrohydraulic, electromagnetic, or piezoelectric) generates a focused shock outside the body.
• The shock propagates through soft tissue (which has acoustic properties similar to water) and converges on the stone.
• Repeated shock impacts fracture the stone into fragments small enough to pass naturally.

This is a direct application of shock focusing and propagation principles from compressible flow theory.