Electromarnetisem/ Magnetohydronamics

[leoflindall]

Homework Statement

Consider plane-parrallel, non magnetic single fluid flow in the z direction. Assume that the realtionship between pressure, velocity and density is constrained by equation (3), shown below.

Show that

$$\frac{P_{1}}{\rho_{1}}$$ = $$\frac{5P_{0}}{3\rho_{0}}$$

Where

P$$_{0}$$ and $$\rho_{0}$$

are the pressure and density of the uniform background on which a wave with small pressure and density fluctiations of

P$$_{1}$$ and $$\rho_{1}$$ propagates in the z direction.

Homework Equations

(1) $$\frac{d\rho}{dt}$$ + $$\frac{d}{dz}$$ ($$\rho$$ v) = 0

(Mass Conversation)

(2) v = vz^

(3) $$\frac{d}{dt}$$ [ $$\frac{1}{2}$$ $$\rho$$v$$^{2}$$ + $$\frac{3}{2}$$P ] + $$\frac{d}{dz}$$ [v($$\frac{1}{2}$$ $$\rho$$ v$$^{2}$$ + $$\frac{5}{2}$$ P )] = 0

(The Constraining equation)
Please note where the notation $$\frac{d}{dt/dz}$$ has been used to indicate the partial differential (I'm not the best at Latex!)

The Attempt at a Solution

I'm not looking for an answer to this problem. This a set problem that I have been set, and am at a loss at how to tackle it. I can the see there is a realtionship between the conservation of mass equation and the constraining equation, however I can't see how to tackle it to show what is required.

Any thoughts of hints inthe right direction would be greatly appreciated.

Many Thanks

Leo

 

[leoflindall]

Solved.

Thank you for viewing.