For the stretched sheet, constant, while constant far from the clamped boundaries, so that constant. At the clamped boundaries the boundary conditions are written as,

(18.7)

Substituting in equation (18.7), a periodic solution of the form , where and is the number of wrinkles, yields a Sturm-Liouville-like problem,

,

(18.8)

Writing, , equation 18.8 simplifies to . We can use solutions of the form for , Since the solution with least bending energy corresponds to , so that , rearranging, we have . Then the solution of is written as, . Plugging this expression in 18.4, yields , relating the wave number and amplitude, so that we can write total energy as . Minimizing , i.e. from we obtain an expression for the wavelength:

(18.9)

Substituting this result in the condition of inextensibility along ,

(18.10)

We obtain an expression for the amplitude

(18.11)

Putting the expressions for flexural rigidity of the sheet, , tensile stress, and transverse displacement, and substituting these expressions in equation 18.9, we obtain in terms of measurable quantities as,

and

(18.12)

The above expressions have been verified from experiments.