Help with Gas dynamics problem

[cotts123]

Homework Statement

for a weak shock wave (p2-p1)/p1=epsilon
show that (s2-s1)/cv=(gamma^2-1)epsilon^3/(12*gamma^2)+higher order terms


HINT: it is helpful to subtract this equation [p]/<p>+gamma[v]/<v>=0
from the equation [ln p(v)^gamma]=/cv
before expanding in terms of epsilon
[x]=x2-x1
<x>=(x1+x2)/2

Homework Equations

The Attempt at a Solution

I subtracted the two equations given expanded the natural log terms according to ln ab=lna+lnb
and taylor series series expansion and I am stuck at a massive equation with square terms of p2 and p1 which is no where near the answer. any help!

 

[KataKoniK]

First, let's start by writing out the two equations given:

1) (p2-p1)/p1 = epsilon

2) [ln p(v)^gamma] = /cv

We can rewrite equation 2) as:

[ln (p2^gamma/p1^gamma)] = (s2-s1)/cv

Next, we can use the hint given and subtract equation 1) from equation 2):

[ln (p2^gamma/p1^gamma)] - (p2-p1)/p1 = (s2-s1)/cv - epsilon

Using the property of logarithms, we can rewrite the left side as:

[ln (p2/p1)^gamma] - (p2-p1)/p1 = (s2-s1)/cv - epsilon

Expanding the natural log using the Taylor series expansion, we get:

gamma[ln(p2/p1)] + (gamma^2/2)[ln(p2/p1)]^2 + (gamma^3/3)[ln(p2/p1)]^3 + ... - (p2-p1)/p1 = (s2-s1)/cv - epsilon

We can then use the fact that ln(p2/p1) = ln(p2) - ln(p1) to rewrite the equation as:

gamma[ln(p2)-ln(p1)] + (gamma^2/2)[ln(p2)-ln(p1)]^2 + (gamma^3/3)[ln(p2)-ln(p1)]^3 + ... - (p2-p1)/p1 = (s2-s1)/cv - epsilon

Using the Taylor series expansion for ln(p2) and ln(p1), we get:

gamma[(p2-p1)/p1 + (p2-p1)^2/(2p1^2) + (p2-p1)^3/(3p1^3) + ...] + (gamma^2/2)[(p2-p1)/p1 + (p2-p1)^2/(2p1^2) + (p2-p1)^3/(3p1^3) + ...]^2 + (gamma^3/3)[(p2-p1)/p1 + (p2-p1)^2/(2p1^2) + (p2-p1)^3/(3p1^3) + ...]^