THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

6.3 Rectangular Membrane with Fixed Edges 113We can recast the wave equation (6.2) in the more general Laplacian format:∇ 2 z = 1 ∂ 2 z(6.3)c 2 ∂t 2Equation (6.2) is suitable for treatment of rectangular membranes, but Equation(6.3) should be expressed in terms of polar coordinates to facilitate mathematicaltreatment of circular membranes:∂ 2 z∂r + 1 ∂z2 r ∂r + 1 ∂ 2 zr 2 ∂θ = 1 ∂ 2 z(6.4)2 r 2 ∂t 2For normal vibrational modes it is the standard mathematical procedure to assumethat the solution to Equation (6.3) consists of a spatially dependent functionΨ and a strictly time-dependent function e iωt (we dispense with the other functione −iωt as being superfluous for the current physical applications):z = Ψe iωt (6.5)Inserting the above expression into Equation (6.3) and setting k = ω/c yields thetime-independent Helmholtz equation:∇ 2 Ψ + k 2 Ψ = 0 (6.6)whose solutions upon insertion into Equation (6.5) yield normal modes of vibrationsin a membrane of a given geometry and boundary conditions.6.3 Rectangular Membrane with Fixed EdgesConsider a stretched rectangular membrane that is fixed at its four edges x = 0,x = L x , y = 0, and y = L y . The boundary conditions may be expressed asz(0, y, t) = z(L x , y, t) = z(x, 0, t) = z(x, L y , t) = 0 (6.7)In the Cartesian format the solution z(x, y, t) = Ψ(x, y)e iωt to Equation (6.2) mustderive from the Helmholtz equation given below for Cartesian coordinates:∂ 2 Ψ∂x + ∂2 Ψ2 ∂y + 2 k2 Ψ = 0 (6.8)But Ψ(x, y) can be stated as the product of two singly dimensioned functions X(x)and Y (y) so thatand then Equation (6.8) transforms toΨ(x, y) = X(x)Y(y)1 ∂ 2 XX ∂x + 1 ∂ 2 Y2 Y ∂y + 2 k2 = 0 (6.9)