ch15. mechanical waves - university of california,...
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Ch15. Mechanical Waves
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15-1. Introduction
Source: disturbance + cohesive force between adjacent piecesA wave is a disturbance that propagates through spaceMechanical wave: needs a medium to propagate
Wave pulse
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DistinctionsWave velocity vs. particle velocity
Wave can travel & Medium has only limited motion
Waves are moving oscillations not carrying matter along
What do they carry/transport?Disturbance & Energy
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Types of WavesTransverse wave
Longitudinal wave
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More ExamplesTransverse
Longitudinal
Transverse + Longitudinal
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Sound Wave: Longitudinal
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15-2. Periodic Waves
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Continuous / Periodic Wave
Caused by continuous/periodic disturbance: oscillations
Characteristics of a single-frequency continuous wave
Crest / peak
Trough
Wavelength: distance between two successive crestsor any two successive identical points on the wave
Frequency f: # of complete cycles that pass a given point per unit time
Period T: 1/f
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Wave Velocity
v=λ/T =λf
Different from particle velocity
Depends on the medium in which the wave travels
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15-3. Mathematical Description
Approach:
extrapolate motion of a single point to all points
from displacement, derive velocity, acceleration, energy…
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Recall SHM: Position of Oscillator
cosθ=x/A θ=ωtω - angular frequency
(radians / s)= 2π/Τ= 2πf
x =Acosθ=Αcos ωt=Acos(2πt/Τ)
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Motion of One Point Over Time
y =Αcos ωt=Acos(2πt/Τ)
y
Rewriting y as the particle displacement:
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Motion of Any Point at Any Time: Wave Function
)(cos),(vxtAtxy −= ω
For wave moving in +x direction
)(2cos)(2cos)(2cos),(TtxAx
TtA
vxtfAtxy −=−=−=
λπ
λππ
Wave numberλπ2
=k
vk=ω
)cos(),( tkxAtxy ω−=
Phase: kx±ωtFor wave moving in -x direction
Motion at x trails x=0 by a time of x/v
)cos(),( tkxAtxy ω+=
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Example