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When dealing with high-speed aircraft or projectiles that surpass the speed of sound, understanding the different operational modes of air inlets is vital. Supercritical mode is one such configuration where airflow entering the inlet is at supersonic speeds, but the objective is typically to slow down this fast-moving air to subsonic speeds before it enters the engine.
This deceleration is often achieved using a shock wave that forms inside the inlet. A normal shock is one that stands perpendicular to the direction of the airflow. With a flight Mach number above 1 and the presence of a shock within the inlet, as described in the given exercise scenario, we are clearly exploring a supercritical mode of operation.

The Mach number, denoted as M, is a dimensionless quantity used in fluid dynamics that describes the ratio of the speed of an object in a fluid to the speed of sound in that fluid. It's named after the Austrian physicist Ernst Mach.
For an object moving at the speed of sound in air, the Mach number is exactly 1. If the object is moving slower than sound, it's traveling at subsonic speeds and has a Mach number less than 1. Conversely, if it's going faster than sound, it's in supersonic or transonic flight and has a Mach number greater than 1. In the problem given, the flight Mach number is 1.6, indicating supersonic speed.

Normal shock relations are a set of mathematical formulas used to calculate changes in various properties of air crossing a normal shock. The properties include pressure, temperature, density, and Mach number.
These relations are pivotal to understanding and designing supersonic inlets, nozzles, and diffusers. In our example, the Mach number ahead of the shock, M_x, is calculated with these relations. Analyzing the properties of airflow before and after the shock helps in designing inlets that optimize engine performance and efficiency.

In supersonic flight, managing pressures is critical for efficiency and engine performance. Total pressure is the sum of the static pressure and the pressure from the flow's motion, called dynamic pressure.
In the critical mode of an inlet, the aim is to maximize the air pressure recovery to provide the highest pressure possible to the engine's compressor. A total pressure gain might occur if the inlet system is designed to operate more efficiently at a particular speed, leading to an increase in the engine's overall pressure ratio. The percentage total pressure gain, calculated in this exercise, can give engineers an idea of the potential efficiency improvement by switching to critical mode operation from a supercritical mode.

Critical mode operation refers to a condition in air inlet dynamics where the inlet is optimized for a precise Mach number and flow condition.
At this optimized regime, the inlet captures the highest total pressure possible at a given Mach number, without any spillage or unstart conditions that could lead to engine flameouts or other performance issues. Designers look for the best balance that will produce the critical mode for an intended flight regime.