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Coming up in Wave Physics...
• 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
• today’s lecture:
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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
static
dynamic
particles (Lagrange)
fields (Euler)
equilibrium
SHM
eg Poisson’s equation
WAVES
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Electromagnetic waves
• delay may be due to propagation speed of force (retarded potentials)
• electric field = force per unit charge (q2)
• vertical component of force
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Gravitational waves
a
• 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
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Gravitational waves
• vertical component of force
• delay due to propagation speed of force
crtar
mmGtF 3
0
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• 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
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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
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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
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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.
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Plucked guitar string
• displace string as shown
• at time t = 0, release it from rest• …What happens next?
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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.
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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
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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
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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:
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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
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