Newton's Second Law Gives the POE of the Model. By Newton's second law (see Sec. 2.4)

Chapter 12, Problem 12.1.120

(choose chapter or problem)

Newton's Second Law Gives the POE of the Model. By Newton's second law (see Sec. 2.4) the sum of the forces given by (I) and (2) is equal to the mass p~A of that small SEC. 12.8 Rectangular Membrane. Double Fourier Series 571 portion times the acceleration a2 u/at2 ; here p is the mass of the undeflected membrane per unit area, and .lA = Llx Lly is the area of that portion when it is undeflected. Thus a2 u p Llx~)' at2 = T Lly [ux(x + .lX .'"1) - lIx(X, )'2)] + T ~.\ [lIiXI' )' + .1y) - Uy (X2, y)] where the derivative on the left is evaluated at some suitable point (x, Y) corresponding to that portion. Division by p .lx LlY gives a 2 u = T [lIx (X + .lx. )'1) - II"J', }'2) + lIy(x., y + ~)') - lIiX2. Y) ] .