1. Wave Physics PHYS 2023 Tim Freegarde

2. 2 Coming up in Wave Physics... local and macroscopic definitions of a wavetransverse waves on a string: wave equation travelling wave solutionsother wave systems:electromagnetic waves in coaxial cablesshallow-water gravity wavessinusoidal and complex exponential waveforms today’s lecture:

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

4. 4 Electromagnetic waves vertical component of force

5. 5 Electromagnetic waves delay may be due to propagation speed of force (retarded potentials) electric field = force per unit charge ( q 2 ) vertical component of force

6. 6 Gravitational waves vertical component of force delay due to propagation speed of force gravitational field = force per unit mass ( m 2 ) centre of mass motion  quadrupole radiation delay may be due to propagation speed of force (retarded potentials) electric field = force per unit charge ( q 2 ) vertical component of force

7. 7 Gravitational waves vertical component of force delay due to propagation speed of force gravitational field = force per unit mass ( m 2 ) centre of mass motion  quadrupole radiation 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. 8 Wave Physics a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points Local/microscopic definition: a time-dependent feature in the field of an interacting body, due to the finite speed of propagation of a causal effect Macroscopic definition: speed of propagation is derived speed of propagation is assumed static dynamic particles (Lagrange)fields (Euler) equilibrium SHM eg Poisson’s equation WAVES

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

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

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

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

13. 13 Waves on long strings

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

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

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

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

18. 18 Plucked guitar string x

19. 19 Plucked guitar string ? x L ?

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