This page gives the mathematical derivation of the speed of sound equation beginning with the conservation of mass and momentum and using information from the isentropic flow slide. Information is presented in text only. A graphical version is also available. In this text version, * denotes multiplication, / denotes division, and ^2 means quantity squared. r is the density, p is the pressure, T is the temperature and g is the gas constant. u is the velocity, s is the speed of sound, A is the area, m dot is the mass flow rate, gam is the ratio of specific heats, and d something denotes differential change.
Equation 1. The mass flow rate equation. m dot = r * u * A
Equation 2. Momentum equation. -dp = r * u * du
Let u = s = the speed of sound and assume that the area and the mass flow rate are constant.Equation 3. Then: r * u = r * s = (r + dr) * ( s + du)
Equation 4. Multiply out the right side. r * s = r * s + r * du + s * dr + dr * du
The last term (dr * du) is very small so we can set it to zero.
Equation 5. Simplify equation 4. r * du = - s * dr
Equation 6. Substitute equation 5 into equation 2. dp = s ^2 * dr
For sound waves, the changes in flow properties are small and nearly reversible.We can determine the change in pressure with change in density from the isentropic relations.
Equation 7. Isentropic relation. dp / p = gam * dr / r
Equation 8. Substitute equation 7 into equation 6. gam * p * dr / r = s ^2 * dr
Equation 9. The equation of state is. p = r * G * T
Equation 10. Substitute equation 9 into equation 8. s ^2 = gam * p / r = gam * G * T
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