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Wave Physics PHYS 2023 Tim Freegarde 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 • shallow-water gravity waves • sinusoidal and complex exponential waveforms 2 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 particles (Lagrange) fields (Euler) static equilibrium eg Poisson’s equation dynamic SHM WAVES 3 Electromagnetic waves • vertical component of force 4 Electromagnetic waves • vertical component of force • delay may be due to propagation speed of force (retarded potentials) • electric field = force per unit charge (q2) 5 Gravitational waves a • vertical component of force • delay due may to bepropagation due to propagation speed ofspeed force of force (retarded potentials) • electric field = force per unit (q2)(m2) gravitational field = force percharge unit mass • centre of mass motion quadrupole radiation 6 Gravitational waves m1m2 • coalescing binary stars: • neutron G~1.4 solar 3mass vertical component of force F tstars, at r c 4 r • separation few tens of 0km several per second • delay due to propagation•speed of rotations force • stars coalesce after minutes • gravitational field = force per unit mass (m2) • detector is laser interferometer several km in size • centre of mass motion quadrupole radiation 7 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 particles (Lagrange) fields (Euler) static equilibrium eg Poisson’s equation dynamic SHM 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 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? • rest position • For an elastic penguin, Hooke’s law gives • separation • displacement • If the penguin has mass , Newton’s law gives • pressure • elasticity • where • density 9 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 use physics/mechanics to write partial differential wave equation for system insert generic trial form of solution e.g. • but note that not all wave equations are of the same form find parameter values for which trial form is a solution 10 Plucked guitar string • displace string as shown • at time t = 0, release it from rest • …What happens next? 11 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 use physics/mechanics to write partial differential wave equation for system insert generic trial form of solution e.g. • but note that not all wave equations are of the same form find parameter values for which trial form is a solution 12 Waves on long strings 13 Solving the wave equation • shallow waves on a long thin flexible string • travelling wave • wave velocity 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 14 Travelling wave solutions • consider a wave shape at which is merely translated with time where use physics/mechanics to write partial differential wave equation for system insert generic trial form of solution • use chain rule for derivatives find parameter values for which trial form is a solution 15 General solutions use physics/mechanics to write partial differential wave equation for system • wave equation is linear – i.e. if are solutions to the wave equation, then so is insert generic trial form of solution arbitrary constants • note that two solutions to our example: find parameter values for which trial form is a solution 16 Particular solutions • fit general solution to particular constraints – e.g. x 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 17 Plucked guitar string x 18 Plucked guitar string yx,0t L ? ? x 19 Plucked guitar string yx, t y x x y L x y x x L+x L L-x x y L x 20