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The fundamental Child–Langmuir limit on the maximum current density in a vacuum between two infinite parallel electrodes is one of the most well known and often applied rules of plasma physics. We develop a simple model using vacuum capacitance, conservation of energy, and conservation of charge to derive the Child–Langmuir space-charge-limited emission. This capacitive model provides physical insight into the origins of the well known (voltage)3/2/(gap distance)2 scaling of the classical current density and does not require the solution of the nonlinear differential equation normally associated with the Child–Langmuir formulation. In addition, the full spacecharge-limited solution is reproduced without imposing the condition that the electric field be driven to zero at the cathode surface.

Bi-orthogonality relationships an~ establislwd for a vertically lwterogeneous porous media in contact with a fluid, a solid, a sr~cond porous medium, and a free surface. Fraser's bi-orthogonality relation for propagation of Rayleigh-Lamb modes in a plate with traction free surfaces is shown to b e a special case of the bi-orthogonality relations derived herein in which the medium can be thought of as a porous slab with z:ero porosity.

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