Stretched Membrane

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Stretched Membrane
(xdx,zdz)
(x,zdz)
dz
dx
(x dx,z)
(x,z)
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(No Transcript)
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Laplacian in Cartesian and Polar Coordinates
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2-D Helmholtz Equation
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Separation of Variables
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The Full Solution
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Boundary Conditions Fixed Edges
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Frequency
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Circular Membrane
SOV
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Azmuthal (Longitudinal) Solution
Radial Solution
Bessels Equation
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Bessel Functions Append A4/5
Bessel Functions of the First Kind
Order 0 Black 1 red 2 green
What is a Hankel Function?
Bessel Functions of the Second Kind
a.k.a Webers or Neumanns function
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Radial Solution
BC
Table pg 516
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Complete Solution
m determines the number of radial nodal lines n
determines the number of nodal circles g
determines the orientation of the radial nodal
lines and is determined by the initial
excitation of the membrane
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Symmetric Modes m 0