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Wave PhysicsPHYS 2023
Tim Freegarde
2
Wave Physics
WAVE EQUATIONS & SINUSOIDAL SOLUTIONS
wave equations, derivations and solution
sinusoidal wave motions
complex wave functions
WAVE PROPAGATION
Huygens’ model of wave propagation
interference
general wave phenomena
Fraunhofer diffraction
longitudinal waves
BEHAVIOUR AT INTERFACES
continuity conditions
boundary conditions
SUPERPOSITIONSlinearity and superpositions
Fourier series and transforms
FURTHER TOPICSwaves from moving sources
operators for waves and oscillations
waves in three dimensions
http://www.avcanada.ca/albums/displayimage.php?album=topn&cat=3&pos=7
further phenomena and implications
3
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• Huygens reflection, refraction and diffraction• reflection and transmission at interfaces• string motion from initial conditions
4
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
5
Plucked guitar string
x
6
Plucked guitar string
? xL
?
• boundary conditions:
7
Plucked guitar string
xL
txy ,
x x L-xL+x
xy
xy
xLy
xLy
• boundary conditions:
Acoustic waves
Ophélie Gaillard, Bach, Prelude, Cello suite n. 1
9
Spectrum
• periodic waveform can be expressed as a superposition of harmonics
amplitude or intensity
• commonly represented by spectrum
10
Spectrum
• periodic waveform can be expressed as a superposition of harmonics
• commonly represented by spectrum• often plotted on log-log or lin-log scales
11
Spectrum
• periodic waveform can be expressed as a superposition of harmonics
• commonly represented by spectrum• often plotted on log-log or lin-log scales• may be plotted with frequency axis upwards...• ... with marks for integer ... • ... and time on the horizontal axis
12
Boundary conditions• for a boundary at :
• transverse waves on a string: fixed point
• sound waves in organ pipe:
• water waves:
• thermal waves:
• electromagnetic waves:
wall
since
lagged end
conductor
open
stopped
•
thermal reservoir
13
Open organ pipe• first six harmonics lie on major chord
J S Bach Toccata & Fugue in D minor BWV565
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Stopped organ pipe• even harmonics are absent
•
• register key raises by a twelfthCLARINET:
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Oboes and saxophones• even harmonics are present
•
• register key raises by a twelfthCLARINET:
• register key raises by an octaveSAXOPHONE:
Wind instruments
•
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The bell
• cyclic boundary conditions