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Unformatted text preview: of
the membrane located at (x, y ) with sides of length dx and dy is given by the
expression
∂2u
F = ρ 2 (x, y )dxdy k.
∂t
Suppose the force displaces the membrane from a given position u(x, y ) to a
new position
u(x, y ) + η (x, y ),
where η (x, y ) and its derivatives are very small. Then the total work performed
by the force F will be
F · η (x, y )k = η (x, y )ρ ∂2u
(x, y )dxdy.
∂t2 Integrating over the membrane yields an expression for the total work performed
when the membrane moves through the displacement η :
Work = η (x, y )ρ
D ∂2u
(x, y )dxdy.
∂t2 (5.14) On the other hand, the potential energy stored in the membrane is proportional to the extent to which the membrane is stretched. Just as in the case of
the vibrating string, this stretching is approximated by the integral
Potential energy = T
2 D 2 ∂u
∂x + 2 ∂u
∂y dxdy, where T is a constant, called the tension in the membrane. Replacing u by u + η
in this integral yields
New potential energy = T
2 D 129 ∂u
∂x 2 + ∂u
∂y 2 dxdy ∂u
∂x +T
D + T
2 ∂η...
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 Winter '14
 Equations

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