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 Waves on long strings 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

4. 4 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

5. 5 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

6. 6 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:

7. 7 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.

8. Water waves Severn bore Kelvin ship wake Ocean waves www.fluidconcept.co.uk/Images/Uploads/capetown1-400-279.jpg theguardian.com © Jason Hawkes / Getty Images Tsunami © Reuters / Mainichi Shimbun

9. 9 Deep water waves x xx+δxx-δx h(x) δxδx v2v2 v1v1 volume = h(x) (δx+ε 2 -ε 1 ) δy ε2ε2 ε1ε1

10. 10 Deep water waves x xx+δxx-δx h(x) δxδx v1v1 volume = h(x) (δx+ε 2 -ε 1 ) δy

11. 11 Energy of waves on a string x x h(x) δxδx v δyδy

12. 12 Complex wave functions φ simple harmonic motion circular motion

13. 13 Dispersion in dissipative systems xxx+δxx-δx y W W δxδx

14. 14 Kelvin ship waves

15. 15 Hull speed