ABSTRACT
Discrete time independent equations may arise in different contexts. Our next example is concerned with the loaded vibrating net. Let us be given a rectangular net of weightless cords loaded at each point of inter section Pij = (Xi , Yj ) with a particle of mass m. We shall further assume vibrations small and perpendicular to the plane of the net when the net is at rest . We thus assume the tension in each string constant through out its length. Denote the tension in the string over (Xi , Yj-l ) , (Xi , Yj ) , (Xi , Yj+l ) by Ti and in the string over (Xi-I , Yj ) , (Xi , Yj ) , (XHI . Yj ) by Tj . We assume Ti and Tj independent of t. We also assume vibrations so small that we can replace sin () by tan () where e is an angle of the type indicated. Denote the vertical coordinate of Pij by Uij (t) . Under the circumstances, as outlined, the motion of Pij is determined by the following differential equation
T ui ,Hl - Uij T Uij - Ui,j-l + i - i , I S; i S; m, 1 S; j S; n . YHI - Yj Yj - Yj-l For convenience in writing, we let
t'.. [3 . . -
J tJ ( ) ' m XHI - Xi
Then the above equation can be written as
where Ll1Vij = VHl ,j - Vij and L::.2Vij = vi,Hl - Vij . We proceed by letting Uij (t) = ewtvij (t) . Substituting into the above equation and divide by eWt , we obtain the partial difference equation
where 1 S; i S; m and 1 S; j S; n. It is expected only for certain values of w, there will be solutions to the above equations. Therefore, we are now dealing with eigenvalue problems.
