16. Wave_equation_in_2D_and_3D

16. Wave_equation_in_2D_and_3D - 4.3 Wave Equation in 2-D...

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Unformatted text preview: 4.3 Wave Equation in 2-D and 3-D Guitar string derivation – one spatial coordinate (x). Derivation led to: 1-D wave eqn. for s(x,t) Two Dimensions Rectangular flat plate • • Volume: ∆V= h∆x∆y s Use Cartesian coordinates to describe the shape Similar derivation for (h∆x∆y) element leads to 2-D wave eqn. for s(x,y,t) y thickness, h x Circular flat plate • • s thickness, h Use polar coordinates to describe the shape Similar derivation in polar coordinates for an element ∆V= hr∆r∆θ leads to 1 1 ∆V r∆θ ∆r 2-D wave eqn. for s(r,θ,t) thickness, h Similar ideas in 3-D. General statement of the wave equation: s where · is the Laplacian operator. Expand out the Laplacian operator to get the wave eqn. in 1-D, 2-D, or 3-D in any coordinate system. 1-D ∂ ∂x Cartesian 3-D Cylindrical Spherical Cartesian 2-D Polar ∂ ∂x ∂ ∂y ∂ ∂z ∂ ∂x ∂ ∂r ∂ ∂y 1∂ r ∂r see next page For irregular shapes – can use Cartesian coordinates and a computer 1∂ r ∂θ 4.3 Summary of Coordinate Systems Cartesian Coordinates z Cylindrical Coordinates z P Spherical Coordinates P P θ r y θ r φ x ,, ∆∆∆ ∆ ,, ∆∆∆ ∆ ,, sin ∆ ∆ ∆ ∆ (Some books interchange θ and definitions) Polar in 2-D = Cylindrical with no z term (a) Gradient Cartesian Φ Φ Φ Φ Φ Cylindrical Φ Φ Φ Φ Spherical Φ Φ 1Φ Φ Φ Φ some function 1Φ 1 sin Φ Φ Φ (b) Divergence Cartesian: Cylindrical: ·A Spherical: ·A A ·A 1 some vector 1 1 1 sin 1 sin sin (c) Laplacian ∂Φ ∂x 1∂ Cylindrical: Φ r ∂r 1∂ Spherical: Φ r ∂r ∂Φ ∂r (d) The Biharmonic Operator, Cartesian: Φ ∂Φ ∂y ∂Φ r ∂r ∂Φ r ∂r 2 ∂Φ r ∂r Cartesian: Cylindrical: Spherical: ∂Φ ∂z 1 ∂ Φ ∂ Φ ∂ Φ 1 ∂Φ 1 ∂ Φ ∂ Φ r ∂θ ∂z ∂r r ∂r r ∂θ ∂z ∂ ∂Φ ∂Φ 1 1 sin θ ∂θ r sin θ ∂θ r sin θ ∂ ∂Φ 1 ∂ Φ cot θ ∂Φ 1 r ∂θ r sin θ ∂ r ∂θ 2 1∂ ∂ r r ∂r ∂r 1∂ ∂ r r ∂r ∂r 1 · r 2 2 1∂ 1∂ ∂ ∂ 1∂ ∂ · r ∂z r ∂r ∂r ∂z r ∂θ r ∂θ 1 ∂ ∂ 1 ∂ sin θ r sin θ ∂θ ∂θ r sin θ ∂ ∂ ∂ 1 ∂ ∂ 1 ∂ r sin θ ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂ ...
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