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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:
• energy in wave motions
• standing wave solutions
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Sinusoidal waves
• conveniently manipulated in differential equations• allow standing wave solutions• only real travelling wave solutions in dispersive systems• useful terminology for dissipative systems• complete set of orthogonal solutions in linear systems
• occur when source is rotating or performing SHM• detected by resonant detectors
• convenient for energy/power calculations
44
Sinusoidal waves
0tt
0tt at ,
• wavenumber2
k
x
y
000 sin, tkxytxy
• spectroscopists’
1~ wavenumber
00~2sin txy
00
2sin txy • wavelength
kxtytxy sin, 0
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Sinusoidal waves
0xx
0xx at ,
• angular frequency 2
t
y
000 sin, kxtytxy
• frequency
1 00 2sin kxty
00
2sin kxty • period
kxtytxy sin, 0
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Complex wave functions
φ
tkxtxy sin, tkxi expRe txi ,expRe
• simple harmonic motion • circular motion
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Wave propagation
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
• transverse motion of taut string
• travelling wave:
• e-m waves along coaxial cable• shallow-water waves• flexure waves• string with friction
• general form• sinusoidal• complex exponential
• standing wave• damped
• soliton
• speed of propagation• dispersion relation• string motion from initial conditions
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Velocities of waves on a string
xx
• phase velocity• (group velocity)• transverse string velocity
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Sumatra-Andaman earthquake 2004
• magnitude 9.15; 275,000 perished
Tsunami Inundation Mapping EffortsNOAA/PMEL - UW/JISAO
• 1200 km along India/Burma plate subduction zone• slip of 15 m sideways, several metres vertically• formed ridges 1.5 km high, trench kms wide
26 Dec 2004
• 30 km3 water displaced
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Sumatra-Andaman earthquake 2004
• magnitude 9.15; 275,000 perished
NOAA
• 1200 km along India/Burma plate subduction zone• slip of 15 m sideways, several metres vertically• formed ridges 1.5 km high, trench kms wide• 30 km3 water displaced
• ocean waves ~ 60 cm• 25 m high near shore
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Group velocity
• 2 sinusoidal components:• 10 sinusoidal components:• spreading of wavepacket
• this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation
• generally: the group velocity = speed of energy propagation= speed of information propagation
•
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Dispersion and group velocity
• time-dependent Schrödinger equation(in 1-D)
• wavefunction of atomic electron
• flexural waves (Kirchhoff plate theory)
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Wave propagation
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
• transverse motion of taut string
• travelling wave:
• e-m waves along coaxial cable• shallow-water waves• flexure waves• string with friction
• general form• sinusoidal• complex exponential
• standing wave• damped
• soliton
• speed of propagation• dispersion relation• string motion from initial conditions
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