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A graphic showing the physics of the Mach angle.

As an object moves through a gas, the gas molecules are deflected around the object. If the speed of the object is much less than the speed of sound of the gas, the density of the gas remains constant and the flow of gas can be described by conserving momentum, and energy in the flow. As the speed of the object increases towards the speed of sound, we must consider compressibility effects on the gas. The density of the gas varies locally as the gas is compressed by the object. Near and beyond the speed of sound (about 330 m/s or 700 mph on Earth at sea level), small disturbances in the flow are transmitted to other locations isentropically (with constant entropy) as sound waves.

For supersonic and hypersonic flows, small disturbances are transmitted downstream within a cone. The edge of the cone is depicted two-dimensionally by the blue lines on the figure at the top of this page. The sound waves strike the edge of the cone at a right angle and the speed of the sound wave is denoted by the letter a. The flow is moving at velocity v which is greater than a. From trigonometry, the sine of the cone angle mu is equal to the ratio of a and v:

sin(mu) = a / v

But the ratio of v to a is the Mach number of the flow.

M = v / a

With a little algebra, we can determine that the cone angle mu is equal to the inverse sin of one over the Mach number.

sin(mu) = 1 / M

mu = asin(1 / M)

where asin is the trigonometric inverse sine function. It is also written as shown on the slide sin^-1. Mu is an angle which depends only on the Mach number and is therefore called the Mach angle of the flow.

We are interested in determining the Mach angle because small disturbances in a supersonic flow are confined to the cone formed by the Mach angle. There is no upstream influence in a supersonic flow; disturbances are only transmitted downstream within the cone.

Here's a JavaScript program which solves for the Mach angle .

Isentropic Flow Calculator

Mach Angle
P-M Ang

You select an input variable by using the choice button labeled Input Variable. Next to the selection, you then type in the value of the selected variable. When you hit the red COMPUTE button, the output values change. The default input variable is the Mach number, and by varying Mach number you can see the effect on Mach angle. You can also select Mach angle as an input, and see its effect on the other flow variables.

If you are an experienced user of this calculator, you can use a sleek version of the program which loads faster on your computer and does not include these instructions. You can also download your own copy of the program to run off-line by clicking on the yellow button. Look for the Isentropic Flow Calculator.

Button to Download a Copy of the Program


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Editor: Nancy Hall
NASA Official: Nancy Hall
Last Updated: May 13 2021

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