My summer project is pretty much done. I haven't done much carpentry in the past, and found the construction process surprisingly educational. Constructing the flooring, framing, siding, roof, and putting in the window and door all had their own unique challenges in measurement, cutting, joining, etc. I still have to build some shelving and bins and put up some hooks and other devices to hold our tools - can't do that yet though as the kids have installed a couch and turned it into a play house for now. Another couple of weeks though and our new shed should be fully operational!

Continuing from the previous post, suppose we want to look at solutions where the (s) box is close to the short-time-constant (τ_{-}) solution, and the (o) box then close to the long-time-constant (τ_{+}) solution. That suggests setting the corresponding inverse time constants perturbatively close to τ_{-} and τ_{+}, respectively. Define dimensionless small numbers ε^{s} and ε^{o} as follows:

Eq. 27: α^{s} = (1 - ε^{s})/τ_{-}

Eq. 28: α^{o} = (1 - ε^{o})/τ_{+}

Then from the definitions of ν_{+} and ν_{-} (unnumbered equation between 22 and 23 in the previous post) we find:

This is essentially a continuation of the math in the previous post. The same warnings apply!

The previous analysis indicated we have 3 free variables to play with. Let them be α^{s}, α^{o} and C^{s} (the initial inverse time constants of the two boxes, and the heat capacity of the "surface" box).

Equations 9 and 12 of the previous post show a relationship between w_{+}^{s} and w_{+}^{o} depending on the α's, γ's and τ_{+}, and equations 10 and 13 show the same relationship for w_{-}^{s} and w_{-}^{o} with τ_{-} instead of τ_{+}.

This will probably be painful for anybody who hasn't already been following this (which I'm sure is all but 1 or 2 readers, if that many). So skip this post unless you're really into solving systems of equations...

Ok, this time I'm going to start with the graph, and explain what's going on after. Seems to work for other folks... :

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