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Wave Physics

Wave Physics. PHYS 2023. Tim Freegarde. today’s lecture:. local and macroscopic definitions of a wave. Coming up in Wave Physics. transverse waves on a string:. wave equation. travelling wave solutions. other wave systems:. electromagnetic waves in coaxial cables.

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Wave Physics

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  1. Wave Physics PHYS 2023 Tim Freegarde

  2. today’s lecture: • local and macroscopic definitions of a wave Coming up in Wave Physics... • transverse waves on a string: • wave equation • travelling wave solutions • other wave systems: • electromagnetic waves in coaxial cables • shallow-water gravity waves • sinusoidal and complex exponential waveforms

  3. Local/microscopic definition: • a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points Wave Physics • speed of propagation is derived particles (Lagrange) fields (Euler) static equilibrium eg Poisson’s equation dynamic SHM WAVES

  4. Electromagnetic waves • vertical component of force

  5. Electromagnetic waves • vertical component of force • delay may be due to propagation speed of force (retarded potentials) • electric field = force per unit charge (q2)

  6. Gravitational waves • vertical component of force • vertical component of force • delay due to propagation speed of force • delay may be due to propagation speed of force (retarded potentials) • gravitational field = force per unit mass (m2) • electric field = force per unit charge (q2) • centre of mass motion  quadrupole radiation

  7. vertical component of force • delay due to propagation speed of force • gravitational field = force per unit mass (m2) • centre of mass motion  quadrupole radiation Gravitational waves • coalescing binary stars: • neutron stars, ~1.4 solar mass • separation few tens of km • several rotations per second • stars coalesce after minutes • detector is laser interferometer several km in size

  8. Local/microscopic definition: • a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points Wave Physics • speed of propagation is derived particles (Lagrange) fields (Euler) static equilibrium eg Poisson’s equation dynamic SHM WAVES Macroscopic definition: • a time-dependent feature in the field of an interacting body, due to the finite speed of propagation of a causal effect • speed of propagation is assumed

  9. Local/microscopic definition: • a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points Wave Physics • speed of propagation is derived • What is the net force on the penguin? • rest position • separation • displacement • For an elastic penguin, Hooke’s law gives • pressure • elasticity • density • If the penguin has mass , Newton’s law gives • where

  10. use physics/mechanics to write partial differential wave equation for system • waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points Wave equations • propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position insert generic trial form of solution e.g. find parameter values for which trial form is a solution • but note that not all wave equations are of the same form

  11. Plucked guitar string • displace string as shown • at time t = 0, release it from rest • …What happens next?

  12. use physics/mechanics to write partial differential wave equation for system • waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points Wave equations • propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position insert generic trial form of solution e.g. find parameter values for which trial form is a solution • but note that not all wave equations are of the same form

  13. Waves on long strings

  14. use physics/mechanics to write partial differential wave equation for system • shallow waves on a long thin flexible string Solving the wave equation • travelling wave insert generic trial form of solution • wave velocity find parameter values for which trial form is a solution

  15. use physics/mechanics to write partial differential wave equation for system • consider a wave shape at • which is merely translated with time Travelling wave solutions where insert generic trial form of solution • use chain rule for derivatives find parameter values for which trial form is a solution

  16. arbitrary constants use physics/mechanics to write partial differential wave equation for system • wave equation is linear – i.e. if General solutions are solutions to the wave equation, then so is insert generic trial form of solution find parameter values for which trial form is a solution • note that two solutions to our example:

  17. x use physics/mechanics to write partial differential wave equation for system • fit general solution to particular constraints – e.g. Particular solutions insert generic trial form of solution find parameter values for which trial form is a solution

  18. x Plucked guitar string

  19. x L Plucked guitar string ? ?

  20. x L x x L-x L+x Plucked guitar string

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