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

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 dynamic equilibrium eg Poisson’s equation SHM WAVES

Electromagnetic waves vertical component of force

Electromagnetic waves vertical component of force delay may be due to propagation speed of force (retarded potentials) electric field = force per unit charge (q2)

Gravitational waves vertical component of force delay due to propagation speed of force gravitational field = force per unit mass (m2) centre of mass motion  quadrupole radiation delay may be due to propagation speed of force (retarded potentials) electric field = force per unit charge (q2) vertical component of force

Gravitational waves vertical component of force delay due to propagation speed of force gravitational field = force per unit mass (m2) centre of mass motion  quadrupole radiation coalescing binary stars: neutron stars, ~1.4 solar mass separation few tens of km several rotations per second stars coalesce after minutes detector is laser interferometer several km in size

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 dynamic equilibrium eg Poisson’s equation 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

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 density where

Wave equations waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points use physics/mechanics to write partial differential wave equation for system propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position insert generic trial form of solution e.g. find parameter values for which trial form is a solution but note that not all wave equations are of the same form

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

Wave equations waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points use physics/mechanics to write partial differential wave equation for system propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position insert generic trial form of solution e.g. find parameter values for which trial form is a solution but note that not all wave equations are of the same form

Waves on long strings

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

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 insert generic trial form of solution use chain rule for derivatives find parameter values for which trial form is a solution

General solutions wave equation is linear – i.e. if use physics/mechanics to write partial differential wave equation for system are solutions to the wave equation, then so is insert generic trial form of solution arbitrary constants find parameter values for which trial form is a solution note that two solutions to our example:

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

Plucked guitar string x

Plucked guitar string x L ? ?

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