Changes between Version 27 and Version 28 of u/ehansen/RT

Timestamp:

09/14/11 13:10:27 (13 years ago)

Author:

ehansen

Comment:

—

u/ehansen/RT

@@ -25 +25 @@
 
 For small perturbations, a linear analysis is used to show that the position of the interface at x = 0 rises exponentially as:
-[[latex($y(t)=y_o e^{\gamma t}$)]].
+[[latex($y(t)=y_o e^{\lambda t}$)]].
 
-Taking the log of both sides yields: [[latex($\log{y(t)}=\log{y_o}+\lambda t$)]].  
+Taking the log of both sides yields: [[latex($\log y(t) = \log y_o + \lambda t$)]].  
 
-Therefore, the slope of a [[latex($\log{y(t)}$)]] vs. [[latex($t$)]] plot would yield the growth rate [[latex($\lambda$)]].
+Therefore, the slope of a [[latex($\log y(t)$)]] vs. [[latex($t$)]] plot would yield the growth rate [[latex($\lambda$)]].
 Said plot was created and used to determine which time period would be most appropriate for calculating the growth rate.
 * [attachment:snapshot1.png Log(interface_position) Plot]
@@ -35 +35 @@
 
 The plots were used to get a rough idea of when to calculate the growth rate.  Bear2fix was then used to actually do the calculation directly from the data.  The data does not actually need to be plotted to find the slope of a best fit line.  Essentially, this slope (the growth rate) was calculated using the method of least squares. 
-== Results ==
+== The Analytic Growth Rate and Results ==
 It can be shown that the analytic growth rate [[latex($\lambda_a=\sqrt{A k g}$)]]
 Where [[latex($A$)]] is the Atwood number, [[latex($k$)]] is the spatial wave number, and [[latex($g$)]] is the acceleration due to gravity.
 The Atwood number is defined as [[latex($A=\frac{\rho_h-\rho_l}{\rho_h+\rho_l}$)]] where [[latex($\rho_h$)]] and [[latex($\rho_l$)]] are the densities of the heavy fluid and light fluid respectively.
-For the initial conditions used here [[latex($\lambda_a=0.6472$)]]
+For the initial conditions used here [[latex($\lambda_a~0.6472$)]]