3. A simple element analysis (cont'd)

If we define \(\lambda^2 = \sigma \omega^2 / T\) and \(g = p/T\), then it can be shown (Forsythe & Wasow, 1960, Chap. 3) that the energy functional corresponding to this system is

$$F(w)=\frac{1}{2}\iint {\left|\nabla w\right|}^2\text{d}S+\frac{1}{2}\iint {\lambda }^2{w}^2\text{d}S-\iint wg\text{d}S$$

(Eqn. 4)

where the integrals are over the entire region of interest, in this case the triangular element.

The three integral terms on the right-hand side represent energy due to membrane tension, inertia, and the applied pressure, respectively.

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