dxt78,
By use of the term "characteristic impedance" I assume that you mean "transmission line characteristic impedance". For simplicity ignore the 2R in your circuit because when connected the impedance is not sqrt(L/C) Im going to ignore it and show how a lossless t-line will result in the sqrt(L/C). For this tline picture that you place a port at the left hand side L and another connected to the node that ties the L and C together.
For an ideal transmission line with ideal source/load impedance Zout=Zin = Z0
Vin/(sL+Z0||1/sC) = Vout/(sL+Z0)
Iin*((1/sC)||Z0)=Iout*((1/sC)||(sL+Zo))
Vin/In = Vout/iout = Z0
Solve for Z0 and it will equal sqrt(L/C)
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