User:Tohline/Appendix/Ramblings/SphericalWaveEquation
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Playing With Spherical Wave Equation
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The traditional presentation of the (spherically symmetric) adiabatic wave equation focuses on fractional radial displacements, , of spherical mass shells. After studying in depth various stability analyses of PapaloizouPringle tori, I have begun to wonder whether the wave equation for spherical polytropes might look simpler if we focus, instead, on fluctuations in the fluid entropy.
Assembling the Key Relations
In the traditional approach, the following three linearized equations describe the physical relationship between the three dimensionless perturbation amplitudes , and , for various characteristic eigenfrequencies, :
Linearized Linearized Linearized 
First Effort
Let's switch from the perturbation variable, , to an enthalpyrelated variable,



The second expression then becomes,









Taking the derivative of this expression with respect to gives,












Hence, the linearized equation of continuity becomes,









Second Effort
Direct Approach
Let's switch from the perturbation variable, , to an enthalpyrelated variable,



where,
Note, as well, that,












The second expression then becomes,












Taking the derivative of this expression with respect to gives,









Hence, the linearized continuity equation gives,















Playing Around
Multiply thru by :



Now,









Also,


















where,



Let,
Then we have,









Therefore, it must also be the case that,



See Also
© 2014  2020 by Joel E. Tohline 