Presentation Chapter 2: Wave equation

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    29-Nov-2014

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  • 1. Derivation of Wave Equation (Ta3520) Derivation of Wave Equation p. 1/1
  • 2. Derivation wave equationConsider small cube of mass with volume V : p+pz z p+p y y x p+p xDesired: equations in terms of pressure p and particlevelocity v Derivation of Wave Equation p. 2/1
  • 3. Deformation EquationConservation of mass: (t0 )V (t0 ) = (t0 + dt)V (t0 + dt) 0 V (0 + d)(V + dV )or: d dV , 0 Vneglecting term ddV . Derivation of Wave Equation p. 3/1
  • 4. Deformation EquationAssume linear relation between density and pressure p: d dp = 0 Kwhere K is bulk modulus.This is one. Now dV /V . Derivation of Wave Equation p. 4/1
  • 5. 1-Dimensional motion (in x-direction): dV dx V xdx is difference in change of particle displacement dux : dx = (dux )x+x (dux )x (dux ) vx = x = dt x x xwhere vx is xcomponent of particle velocity. Derivation of Wave Equation p. 5/1
  • 6. Substitute d/0 and dV /V : dp vx = dt K xAssuming low-velocity approximation: dp p dt t Gives deformation equation: 1 p vx = K t x Derivation of Wave Equation p. 6/1
  • 7. Equation of MotionNewtons law applied to volume V : dvx Fx = V dtwhere Fx is force in x-direction (1-dimensional motion) Fx = px Sx p p = x + dt Sx x t p V (for dt 0) x Derivation of Wave Equation p. 7/1
  • 8. Assuming low-velocity approximation: dvx vx dt t Then Newtons Law becomes: p vx = 0 x t This is the equation of motion. Derivation of Wave Equation p. 8/1
  • 9. Wave EquationCombine deformation equation and equation of motion.Let operator (/x) work on equation of motion andassume constant: p vx = x x t xSubstitute deformation equation: Derivation of Wave Equation p. 9/1
  • 10. Wave Equation 2 p 0 2 p 2 2 =0 x K tor: 2p 1 2p 2 2 2 =0 x c tThis is the wave equation.c is velocity of sound: c = K/ Derivation of Wave Equation p. 10/1
  • 11. Need to know for examConservation of mass, with some approximations, leadsto deformation equation: 1 p vx = K t xConservation of forces, with some approximations, leadsto equation of motion: p vx = 0 x tFrom then on, everything. Derivation of Wave Equation p. 11/1

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