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ω =κI

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ω =gL

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⇒ T = 2π Lg

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ω =mghrot−com

I rot ⇒ T = 2π I rot

mghrot−com

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α (t) = −ω 2θ (t)

Chapter16:Waves

WAVES Pulse - single wave Periodic wave - sinusoidal waves

Particle - Displacement - Velocity - Acceleration

Waves and particles Vibration = waves - Sound - medium vibrates - Surface ocean waves - no net water is displaced - Mechanical waves - Newton’s equations with medium - Electro-magnetic waves - NO MEDIUM light (photons)

MechanicalWaves

TransversevsLongitudinalWaves

Transverse: Displacement of particle is perpendicular to the direction of wave propagation

Longitudinal: Displacement (vibration) of particles is along same direction as motion of wave

-Sound (fluids…) -Ocean currents

- top vs bottom

Traveling Waves - they travel from one point to another - Nodes move Standing Waves - they look like they’re standing still - Nodes do not move

TransverseWaves

In a transverse wave the motion of the particles of the medium is perpendicular to the direction of the wave’s travel

LongitudinalWaves

A longitudinal pulse travels along the medium but does not involve the transport of matter … just energy

Here are periodic longitudinal waves – pick a single particle and follow its motion as the wave goes by

AWaveontheWater

A water wave is a combination of a longitudinal and a transverse wave … notice how the blue dots make a circular motion:

How do we describe the such waves?

Descrip<onoftransversetravelingwave

Displacement (y) versus position (x)

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y(x,0) = ymax sin kx( )

t = 0

t = δ

t = 2δ

t = 3δ

t = 4δ

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k =2πλ

Spatially Periodic ( repeats ) : kλ = 2π

Temporally Periodic ( repeats ) : ωΤ = 2π

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y(0,t) = ymax sin −ωt( )

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ω =2πT

Displacement versus time does not show “shape”

Wave number Wavelength

Descrip<onoftravelingwave:mathema<cal

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k =2πλ

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ω =2πT

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phase : kx ±ωt

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kx +ωt⇒

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kx −ωt⇒ Wave traveling in + x direction

Wave traveling in - x direction

What is the velocity at which the wave crests move?

Wavespeed

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phase : kx −ωt = const.

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⇒ k dxdt−ω = 0

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vwave =dxdt

=ωk

=λT

= λf

Velocity at which crests move = wave velocity or phase velocity

A wave crest travels a distance of one wavelength, λ, in one period, T

Descrip<onoftransversewave

- To describe a wave (particle) on a “string” , the transverse displacement (y) depends on both the position (x) along the string and the time (t)

Displacement Y versus position X

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vT = λ

Spatially Periodic ( it repeats )

Displacement Y versus time t

Temporally Periodic ( it repeats )

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f =1T

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v =λT

= fλ

Example:WavelengthsofRadioSta<onsWaves like radio, light, x-rays etc. are part of the electromagnetic spectrum. They travel with a velocity:

What is the wavelength of talk radio WJBO am 1150?

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f =1150 kHz =1150 ×103Hz=1.15 ×106Hz

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v = c = speed of light= 3×108m /s

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λ =cf

=3×108m /s1.15 ×106 /s

= 261 m

What is the wavelength of KLSU fm 91.1?

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f = 91.1 MHz = 91.1×106Hz= 9.11×107Hz

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λ =cf

=3×108m /s9.11×107 /s

= 3.29 m

A wave traveling along a string is described by: in which the numerical constants are in SI units (0.00327 m, 72.1 rad/m, and 2.72 rad/s).

a)  Which direction are the waves traveling?

b)  What is the amplitude of the waves?

c)  What is the wavelength?

d)  What is the period?

e)  What is the frequency?

f)  What is the velocity of the wave (vw)?

g)  What is the displacement y at x = 22.5 cm and t=18.9 s?

h)  What is u, (or vt) the transverse velocity, at x = 22.5 cm and t=18.9 s?

i)  What is at, the transverse acceleration, at x = 22.5 cm and t=18.9 s?

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y(x,t) = 0.00327sin 72.1x − 2.72t( )

Positive x-direction

ymax = 0.00327 m = 3.27 mm

k = 72.1 rad/m → λ = 0.0871 m = 87.1 mm

ω= 2.72 rad/s → Τ = 2.31 s

Τ = 2.31 s → f = 0.433 Hz

vw = ω / k = λf =38 mm/s

y = 1.92 mm

u=-ωymcos(kx-ωt)

at = -ω2y(x,t)

make sure your calculator is in radians

Sample problem 16-1 & -2

Wavespeedonstretchedstring

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vwave =λT

=ωk

= λfWavelength and period are NOT independent:

Wave speed depends on physical properties: on properties of the medium in which it travels

If particles of medium move, then velocity must be a function of elasticity and mass (inertia) {potential energy and kinetic energy}

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τΔlR

= (µΔl) v2

R

VelocityinString

Conceptually 1)  the speed which the waves moves to right depends on how quickly one particle is accelerated

upward in response to net pulling force by neighbors.

2)  Newton’s 2nd law - stronger force → greater upward acceleration (a = F/m) -> leads to a faster moving wave

3)  Ability to “pull on neighbors” depends on how tight the string is -> TENSION greater tension → greater force → greater speed

4) Inertia → F = ma: travel faster with small mass ( or linear mass density - mass/length)

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vwave =τµ

= λf For transverse wave in physical medium

shows a plot of the displacement as a function of position at time t=0. The scale

Problem16‐21:wavespeedinstretchedstring

Find (a) amplitude (b) wavelength (c) wave speed (d) period of the wave

shows a plot of the displacement as a function of position at time t=0. The scale

Problem16‐21:wavespeedinstretchedstring

Find

(e) maximum speed of a particle in the string

(f) If the wave form is determine all variables

(g) The correct choice of sign in front of ω