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THE WAVE EQUATION. Alemayehu Adugna Arara Supervisor : Dr. J.H.M. ten Thije Boonkkamp November 04 , 2009. Outline. Occurence of the Wave Equation 1D Waves Spherical Waves Cylinderical Waves Supersonic Flow Past a body Revolution Initial value Problems in Two and Three Dimensions.
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THE WAVE EQUATION Alemayehu Adugna Arara Supervisor : Dr. J.H.M. ten Thije Boonkkamp November 04 , 2009
Outline • Occurence of the Wave Equation • 1D Waves • Spherical Waves • Cylinderical Waves • Supersonic Flow Past a body Revolution • Initial value Problems in Two and Three Dimensions
1. Occurrence of the Wave Equation • Acoustics • Electromagnetism • Elasticity • Hyperpolic wave equation.
Acoustics • Linearized • Small disturbance about an equilibrium state. • Body forces are neglected The initial disturbance has a uniform entropy
Acoustics • From conservation of mass we have • From balance momentum we also have • Then from these equation we have
Acoustics • Introducing we have, Hence
Linearized Supersonic Flow • Disturbance is to remain small, • the motion should have to be very small or • the body must be very slender. • Slender body moving with arbitrary constant velocity relates acoustics with aerodynamics. The velocity components in frame are
Linearized Supersonic Flow Hence the equation wave for acoustics become The velocity
Electromagnetic Waves • Maxwell‘s equation Where B is the magnetic induction and E is the electric field. Therefore And E satisfies the same equation.
2. 1D Wave Equation • The functions f and g follows from initial and boundary conditions For the intial value problem,
3. Spherical Waves • For waves symmetric about the origin This can be written as The general solution will be
3. Spherical Waves • For outgoing waves, the solution is Standard form is to prescribe source is In acoustics, ∂φ/∂R is radial velocity and Q(t) is the flux of volume.
3. Spherical Waves • IVP, “balloon problem“ in acoustics: No source at the origin
3. Spherical Waves t R- a0 t=0 E R- a0 t=R0 R+ a0 t=R0 D R0 /a0 C A B R0 /2a0 R 0 R0 /2 R0
3. Spherical Waves Region A: Region B:
3. Spherical Waves Region C: Region D: Region E:
3. Spherical Waves • Time Evolution of pressure difference p-p0 P p2 p1 R 0 R0-a0t R0 R0+a0t
3. Spherical Waves p-p0 P p2 R 0 R0-a0t R0 R0+a0t p1
3. Spherical Waves p-p0 p2 R 0 R0 R0+a0t
4. Cylinderical Waves • The sources are uniformily distrubeted on the z axis with a uniform strength q(t) per unit length. • Total disturbance is q(t)dz- z R r
4.Cylinderical Waves • Various forms of this solution are valuable. If q‘(t) →0 suficiently fast as t→-∞.
5. Supersonic Flow Past a body of Revolution • Linearization • Body must be slender • R‘(x) is small and Φx, are both small.
5. Supersonic Flow Past a body of Revolution The components of the velocity perturbation are obtained by suitable modification
6. Initial Value Problem in Two and Three Dimensions • We consider We know from the spherical wave solution is
6. Initial Value Problem in Two and Three Dimensions • The initial source strength determining f acts only for an instant,
6. Initial Value Problem in Two and Three Dimensions The general initial value therefore is
6. Initial Value Problem in Two and Three Dimensions Two Dimesional Problem
Conclusion • The wave equations occurs in many problems. • The solution strongly depends on the initial value and boundary conditions. • The wave equations are easier to solve in odd dimensions. • One can easily solve the IVP in two and three dimensions.
