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Unformatted text preview: 12.7 2D Wave Equation: Vibrating Membrane In a similar way as for the vibrating string (Section 12.2 ) we now want to derive a mathematical model of small transverse vibrations of an elastic membrane, such as a drumhead. The membrane at rest covers a bounded domain R 2 and is fixed on the boundary . The deflection of the membrane at point ( x,y ) and time t > 0 is given by u ( x,y,t ) [m]. For a fixed t > 0, the function u ( ,t ) describes the shape of the membrane at time t , whereas for a fixed ( x,y ) , the function u ( x,y, ) describes the vertical motion of this point on the membrane over time. We make the following simplifying physical assumptions (idealization!): 1. The mass per unit area of the membrane, [kgm 2 ] is constant (ho- mogeneous membrane). The membrane is perfectly flexible and offers no resistance to bending. 2. The membrane is stretched and then fixed along its entire boundary . The tension per unit length T [Nm 1 ] caused by stretching the membrane is the same at all points and in all directions and does not change during the motion. The tension is so large that the action of the gravitational force on the membrane can be neglected. 3. The deflection u ( x,y,t ) of the membrane during the motion is small compared to the size of the membrane, and all angles of inclination are small. To derive the model, we consider the forces acting on a small portion of the membrane (an area element [ x,x + x ] [ y,y + y ] of area x y , with < x, y 1). Because of the model assumptions, these will be tensile forces, i. e. tangential to the membrane. We consider the four forces acting on the edges of the area element, F 1 , 2 x , F 1 , 2 y . With the assumption of small deflections we conclude that the magni- tudes of these forces are given by | F 1 , 2 x | = T y, | F 1 , 2 y | = T x. (237) We write F 1 , 2 y in polar coordinates in the yz plane: F 1 y = T x cos T x sin...
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- Spring '08