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Homework Statement
[tex] \rho_0, c_0 [/tex] is the mean density, the mean speed of sound in the ideal gas.
Is the following correct?
[tex] c(\rho)=c_0\left(\frac{\rho}{\rho_0}\right)^{\frac{\kappa-1}{2}} [/tex]
Homework Equations
[tex] p = const * \rho^\kappa, c=\sqrt{\frac{\partial p}{\partial \rho}} [/tex]
The Attempt at a Solution
[tex] c=\sqrt{\frac{\partial p}{\partial \rho}} = \sqrt{const*\kappa*\rho^{\kappa-1}}=const*\rho^{\frac{\kappa-1}{2}} [/tex]
With c(\rho_0)=c_0, I get:
[tex] c(\rho)=c_0\left(\frac{\rho}{\rho_0}\right)^{\frac{\kappa-1}{2}} [/tex]
Can I then say, that the refractive index is:
[tex] n(\rho)=\frac{c_0}{c(\rho)}=\left(\frac{\rho}{\rho_0}\right)^{\frac{1-\kappa}{2}} [/tex]
Hence, the ratio of 2 refractive indexes, like it is needed in the refraction law, is independent of [tex] \rho_0 [/tex]?
Is there a mistake in the reasoning? Thank you very much for your help!