1. Wave Physics Tim Freegarde School of Physics & Astronomy University of Southampton

2. 2 Mexican wave equation response to action of neighbour delayed reaction comparing the differentials with respect to time and position, we see (without proof) that the Mexican wave equation could be e.g.

3. 3 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 e.g. but note that not all wave equations are of the same form

4. 4 Waves on long strings xxx+δxx-δx y W W δxδx

5. 5 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

6. 6 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 consider a wave shape at which is merely translated with time where

7. 7 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:

8. 8 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

9. 9 Plucked guitar string x

10. 10 Plucked guitar string ? x L ?

11. 11 Plucked guitar string x L x xL-x L+x

12. 12 Waves along a coaxial cable b xx+δx r -δQ-δQ I δQδQ I x x a V(x)V(x+δx)

13. 13 Waves along a coaxial cable b xx+δx x x a -δQ-δQ r δQδQ

14. 14 Waves along a coaxial cable b xx+δx x x a r I I

15. 15 Deep water waves x xx+δxx-δx h(x) δxδx v2v2 v1v1 volume = h(x) (δx+ε 2 -ε 1 ) δy ε2ε2 ε1ε1

16. 16 Deep water waves x xx+δxx-δx h(x) δxδx v1v1 volume = h(x) (δx+ε 2 -ε 1 ) δy

17. 17 Sumatra-Andaman earthquake 2004 magnitude 9.15; 275,000 perished Tsunami Inundation Mapping Efforts NOAA/PMEL - UW/JISAO 1200 km along India/Burma plate subduction zone slip of 15 m sideways, several metres vertically formed ridges 1.5 km high, trench kms wide 26 Dec 2004 30 km 3 water displaced

18. 18 Sumatra-Andaman earthquake 2004 magnitude 9.15; 275,000 perished NOAA 1200 km along India/Burma plate subduction zone slip of 15 m sideways, several metres vertically formed ridges 1.5 km high, trench kms wide 30 km 3 water displaced ocean waves ~ 60 cm 25 m high near shore

19. 19 Energy of waves on a string x x h(x) δxδx v δyδy

20. 20 Complex wave functions φ simple harmonic motion circular motion

21. 21 Dispersion in dissipative systems xxx+δxx-δx y W W δxδx

22. 22 Phasors