d9102782-57e0-4528-9dc4-56e89a89762c 26 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Mach => Iterate on canvas

timing 1 min * ≡

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases :

x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases :

x Subsonic x Supersonic k 1 + 20 black 1 5 mat 3 1 sys

supersonic

Supersonic β 21.7077 233.903 20.0537 4.31361 0.165479 4276.76 491.007 16.7228 0.559674 91 mat :

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

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subsonic

Supersonic Mach segment [J_Fraction] x β Mach β 1 el β 6 el x β 2 el + β 7 el x β 3 el + β 8 el x β 4 el + β 9 el x β 5 el + / - / - / - / - :

Mach x k 15 k 0.25 + (range : i 1 x rows range m i el x i el β Mach : for x m augment 31 line :

speed of sound at sea level = 340.29 m/s

Supersonic Mach Number ... x = qc/P [Dynamic/Static

Mach k 1 + 15 red 1 5 mat 2 1 sys

NOTES: Lot of work was done outsides this Smath document ... Mathcad 11. Either Thiele continued fraction or Padé rational fraction, both can be converted in "J_Fraction". Originally, J_Fraction was a way to reduce the length of Thiele. To my knowledge, only CAS Maple companions have this conversion [Mathcad 11 & earlier, Smath ... other ?]. Before the Padé library of approximation of functions [Hart et al.], computers were running Thiele in limited quantity.

Global fit from GenfitMatrix ... Subsonic => Supersonic
m(x) is the J_Fraction of f(x) x m 12.476502 805.629714 x 76.665245 + 46.132095 x 5.94495 + 1.120185 x 5.968869 - 74.483764 x 11.468397 - / - / - / - / - :

β 4.69292878185 2.485645088668 0.005049502793 - 0.008399478307 - 5.19337552684 0.698811411628 0.043875966805 - 0.000673223801 - 81 mat :

Overall Mach rational segment x f x x β 3 el x * β 2 el + β 4 el x 2^* + (* β 1 el + (* 1 x x β 7 el x * β 6 el + β 8 el x 2^* + (* β 5 el + (* + / :

0.125 Subsonic 0.125 f - 0.25 Subsonic 0.25 f - 0.5 Subsonic 0.5 f - 0.75 Subsonic 0.75 f - k Subsonic k f - 51 mat 0.037 0.006 0.004 - 0.001 - 051 mat

k Supersonic k f - 2 Supersonic 2 f - 3 Supersonic 3 f - 5 Supersonic 5 f - 7 Supersonic 7 f - 10 Supersonic 10 f - 15 Supersonic 15 f - 71 mat 000000 0.001 71 mat

Integrate on canvas

x F 0 x ≤ (x 15 ≤ (& x f outside fit range if :

z Int u f u 0 x int :

Sanity check x f x 015 int 35.509

x Int 0 x ≤ (x 15 ≤ (& z Int 0utside fit range if :

x F k 1 + 15 red 1 5 mat 2 1 sys x Int 15 15 Int + 15 red 1 5 mat 2 1 sys

Integrals accuracy: 100 15 Int 35.5087796

f a b Romberg w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line :

Technically: Romberg + accurate f 015 Romberg 35.5088694

Mach Number

The Cauchy parameter, after transformation becomes the 'Mach number' subsonic for 'Air' as ideal gas is derived from Bernoulli equation.

M 2 γ 1 - / (qc p / 1 + (γ 1 - γ /^1 - (* sqrt ≡

In supersonic condition, compressible fluids and past the "choc wave", Mach number is derived from the Rayleigh-Pitot equation with 'qc' = impact pressure past the "choc wave".

M 0.88128485 qc P / 1 + (117 M 2^* / - (2.5^* sqrt * ≡