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pde_notes02

# pde_notes02 - Notes for TUT course MAT-51316 Robert Pich e...

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Notes for TUT course MAT-51316 Robert Pich´e 4.9.2007 2 Wave Equation Models How to derive the PDE for vibrating string vibrating membrane 2.1 Vibrating String Consider the motion of a thin string moving in the xz plane. Assume that points of the string move in the z direction only, and let u ( x, t ) denote the string displace- ment. Longitudinal force balance: Let T ( x, t ) be the tension, assumed to act tan- gentially along the string. Let θ = tan - 1 u x denote the angle between the string tangent and the x axis. The only longitudinal forces acting on the part of the string between x = a and x = b are the x components of the tension force, and because there is no longitudinal motion these forces must be equal, that is, T ( b, t ) cos θ ( b, t ) = T ( a, t ) cos θ ( a, t ) . Because the segment is arbitrary, this implies T cos θ is constant with respect to x , say T ( x, t ) cos θ ( x, t ) = τ ( t ) . Mass conservation: Let ρ ( x, t ) be the string’s mass per unit length; it may vary as the string stretches during the motion. Let ρ 0 ( x )

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pde_notes02 - Notes for TUT course MAT-51316 Robert Pich e...

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