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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