Wave Physics PHYS 2023 Tim Freegarde

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 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

4 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

5 Waves on long strings

6 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

7 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

8 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:

9 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

10 Plucked guitar string x

11 Plucked guitar string ? x L ?

12 Plucked guitar string x L x xL-x L+x

13 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

14 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.

Electromagnetic waves Radio Gamma radiation Light aliexpress.com NASA/DOE/Fermi LAT Collaboration

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

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

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

Water waves Severn bore Kelvin ship wake Ocean waves theguardian.com © Jason Hawkes / Getty Images Tsunami © Reuters / Mainichi Shimbun

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

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

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

23 Complex wave functions φ simple harmonic motion circular motion

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

25 Kelvin ship waves

26 Hull speed