Wave Physics PHYS 2023 Tim Freegarde

Transcription

1 Wave Physics PHYS 2023 Tim Freegarde

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

3 Wave Physics 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 speed of propagation is derived static dynamic particles (Lagrange) equilibrium SHM fields (Euler) eg Poisson s equation WAVES 3

4 Electromagnetic waves vertical component of force 4

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

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

7 Gravitational waves m m t G a t r c 1 2 vertical coalescing component binary stars: of force neutron stars, ~1.4 solar mass 3 separation few tens 4 of 0km r delay due to propagation speed several of rotations force per second stars coalesce after minutes gravitational field = force per unit mass (m 2 ) detector is laser interferometer several km in size centre of mass motion quadrupole radiation F 7

8 Wave Physics 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 speed of propagation is derived static dynamic particles (Lagrange) equilibrium SHM fields (Euler) eg Poisson s equation 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 8

9 Wave Physics 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 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 where rest position separation displacement pressure elasticity density 9

10 Wave equations 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 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 10

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

12 Wave equations 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 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 12

13 Waves on long strings 13

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

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

16 General solutions wave equation is linear i.e. if are solutions to the wave equation, then so is note that two solutions to our example: arbitrary constants 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 16

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

18 Plucked guitar string x 18

19 Plucked guitar string? L? x 19

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