https://top-secret-documents.com/en/nasa/index.htmlNASA – Atmospheric oscillations
“Eckart [l960] went through a second derivation in which the effect of
the earth's rotation was included. The other conditions are the same as the first case discussed. He again found that certain frequencies are not allowed and the acoustic and
gravity waves are similar to the ones already described. Figures 2 and 3 show these allowed frequencies versus wave number in the cases of a nonrotating and a rotating earth.”
“A more general approach uses
spherical coordinates. The basic mathematical equations in his book, and ramifications were developed in papers by Wilkes [1951] and by White [1955, 1956, 1960a, 1960bl. The basic equations of motion are taken to be, in the linearized, perturbation form, - au -2wvc0se = at -- ia - (t +n) a ae - 1 a (2 +n) a sin 8 P, av +2wvcose =- at * Eckart used the term sound where
a is the radius of the earth, co is the angular velocity of the earth, 8 is the latitude, 4 is the longitude, z is the height above the earth's surface, u is the southward component of air velocity at (z, 8, 4), v is the eastward component, w is the vertically upward component, c is the velocity of sound at height z, and R is the tide producing potential, gravitational in origin.
In the above the earth is considered to be spherical, and the variation of radius vector, gravity, and an with height are neglected. Also, the vertical acceleration is considered to be negligible.”
The following is an explanation for the term of "flat non rotating earth" used in those NASA papers:
"For the purposes of the equations governing the aerodynamic properties of aeroplanes, a whole lot of things can be ignored because they don't make the slightest bit of difference. Such as the curvature of the earth, its rotation, the earth's orbit around the sun, the sun's orbit around the galactic center etc.
For much the same reason, if I want to calculate the path taken by a ball I drop in my lounge, I don't need to worry about those things either...
A key skill in physics is in understanding what factors need to be taken into account and what can be ignored, at the level of precision you care about.
As far as aerodynamic models of planes go, the curvature of the earth only matter to the extent that the earth deviates from being flat over the size of the aircraft. Likewise the rotation: if the rotation speed of the earth varies significantly over the span of the aircraft it might matter. But since aeroplanes aren't tens of kilometers in size, it really makes no difference, because we are not doing calculations precise enough for those things to matter..."