deletions | additions       diff --git a/data and analysis c4.tex b/data and analysis c4.tex  index 38ed0b1..a836c70 100644  --- a/data and analysis c4.tex  +++ b/data and analysis c4.tex 
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Fourth sound is now calculated in the same fashion as above by the equation \[c_4=f2L\] for n=1 and L=.0495 meters as before. This time though there is a scattering factor,n, that is caused by the powder in the fourth sound cell that results in a lower measured velocity for fourth sound. This scattering can be determined by the equations \[ c_4=nc_{4 exp}=\sqrt{\frac{\rho_s}{\rho}}c_1 \] coupled with the equations \[ \frac{\rho_s}{\rho_n}=\frac{\frac{\rho_s}{\rho}}{1-\frac{\rho_s}{\rho}} \] the scattering factor can be solved for. The maynard tables give a value for $\frac{\rho_n}{\rho}$ at for our lowest temperature. The lowest temperature measured was 1.34 K for fourth sound so ,rounding up to 1.35 K , the value for $\frac{\rho_n}{\rho}=.0598$. solving for $\frac{\rho_s}{\rho}$ and utilizing the value for $\frac{\rho_n}{\rho}$ , it is found that $\frac{\rho_s}{\rho}=1-.0598=0.9402$. Then to find n, use the values for $c_1$ and $c_4$ at 1.34 K.\[n=\sqrt{0.9408}\left(\frac{236.511 K}{185.229 K} \right)=1.2385\] Then our true $c_4$ is found by multiplying n by our experimental $c_4$ values. \\  In comparing the corrected value of $c_4$ with the theoretical plots of figure 3, it is seen that the measured $c_4$ does agreee with theory in that it decays as the temperature approaches $T_{\lambda}$ Although the theory plot drops rapidly after 2 K, the measured $c_4$ decays appropriately but "stalls" right before $T_{\lambda}$.   \subsection{Error in $c_4$ and Temperaure}  Error in $c_4$ was found similarly as other sounds by the equation equations  \[dc_{4exp}=\sqrt{(2f)^2 dL^2+(2L)^2 (df)^2}\\ (df)^2}\] \[  dc_{4 correction}=n\sqrt{(2f)^2 dL^2+(2L)^2 (df)^2}\]\] (df)^2}\]  The same errors in f,L, and temperature were used here.