chapter15-1

Zaza Metreveli - experimental particle physicist

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Required: University Physics with Modern Physics

by Young and Freedman

12th edition, Pearson Addison Wesley

1

12th edition, Pearson Addison Wesley

Lab Manual

Physics 135-3 Problem Manual

by D. Brown (availability TBA)

Final Exam: Monday, March 12, 9-11 am

I wish a great success to all of you.

Mechanical Waves

Chapter 15

University Physics, 12th edition

Hugh D. Young and Roger A. Freedman

Lectures by Zaza Metreveli

2

Goals for Today’s Lecture

� Study waves and their properties

� Define wave functions and wave dynamics

� Calculate the power in a wave� Calculate the power in a wave

� Consider wave superposition

� Study standing waves on a string

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Mechanical Waves - Definition

A mechanical wave is a disturbance that travels through some material or substance called the medium for the wave.

As the wave travels, the particles that make up the medium

undergo displacements of various kind, depending on the nature of the wave. nature of the wave.

Transverse wave – displacements of the medium are perpendicular to the direction of travel of the wave.

Longitudinal wave – displacements of the medium are parallel to thedirection of travel of the wave.

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Types of Mechanical Waves

Transverse

Longitudinal

Both Transverse

And Longitudinal5

Transverse wave

Longitudinal wave

Types of Mechanical Waves - Demonstration

Longitudinal wave

Both transverse and

longitudinal

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General Common Characteristics of Mechanical Waves

Disturbance propagates (travels) with a definite speed through the medium. This speed is called wave speed.

The medium itself does not travel through space. Its individual

particles undergo back-and-forth or up-and-down motions around the equilibrium positions.around the equilibrium positions.

To set any of these systems into motion, we have to put in energy by doing mechanical work on the system.

Wave transports energy, but not matter, from one region to another.

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Periodic Waves - Parameters

A detailed look at periodic transverse

waves will allow us to extract parameters.

A – amplitude [m, cm]

magnitude of change in the

oscillating variable.

λ – wavelength [m, cm]

distance over with the wave’s shape

repeats.

periodic sinusoidal waves

T – period [s]

length of time taken by one cycle.

– frequency [Hertz, 1/s]

event repeats per second.f

ω – angular frequency [rad/s]

rate of change of angular displacement.

repeats.

fπω 2=ω

π21==

fT

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Periodic Waves - Parameters

• Wave travels with constant speed

• In time interval T wave travels distance λλλλ

υ

T

λυ =

f1

=Since

Tf

1=

fλυ =

This fundamental equation holds for

all types of periodic waves.

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Question:

If you double the wavelength λ of a wave on a string, what happens

to the wave speed v and the wave frequency f?

A. v is doubled and f is doubled.

B. v is doubled and f is unchanged.

C. v is unchanged and f is halved.

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D. v is unchanged and f is doubled.

E. v is halved and f is unchanged.

If you double the wavelength λ of a wave on a string, what happens

to the wave speed v and the wave frequency f?

Answer:

A. v is doubled and f is doubled.

B. v is doubled and f is unchanged.


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D. v is unchanged and f is doubled.

E. v is halved and f is unchanged.

fλυ =

Mathematical Description of a Wave

Wave function – determines the position of any particle of wave at any time.

),( txyy =We consider simple harmonic

motion (SHM) – sinusoidal wavesftAtAtxy πω 2coscos),0( ===

)](cos[),(υ

ωx

tAtxy −=

)(2cos)](cos[),( tx

fAtx

Atxy −=−= πω )(2cos)](cos[),( tx

fAtx

Atxy −=−=υ

πυ

ω

)(2cos),(T

txAtxy −=

λπ

Wave number [rad/m]λ

π2=k

kυω =

Graphing the wave function

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)cos(),( tkxAtxy ω−=

Mathematical Description of a Wave

)cos(),( tkxAtxy ω−= - sinusoidal wave moving in +x direction

)cos(),( tkxAtxy ω+= - sinusoidal wave moving in -x direction

for t=0kxAtxy cos)0,( ==

tAtAtxy ωω cos)cos()0,( =−== for x=0tAtAtxy ωω cos)cos()0,( =−== for x=0

tkx ω± is called the phase.

If phase = 0, 2π, 4π, … y=A

If phase = π, 3π, 5π, … y=-A

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Which of the following wave functions describe a wave that moves

in the –x-direction?

A. y(x,t) = A sin (–kx – ωt)

B. y(x,t) = A sin (kx + ωt)

C. y(x,t) = A cos (kx + ωt)

Question:

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D. both B. and C.

E. all of A., B., and C.

Answer:

Which of the following wave functions describe a wave that moves

in the –x-direction?

A. y(x,t) = A sin (–kx – ωt)

B. y(x,t) = A sin (kx + ωt)


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D. both B. and C.

E. all of A., B., and C.

)cos(),( tkxAtxy ω+=

Particle Velocity and Acceleration in a Sinusoidal Wave

� υy - transverse velocity of any particle in a transverse wave, partial derivative

of with respect to time:

)sin(),(

),( tkxAt

txytxy ωωυ −=

∂

∂=

� ay – acceleration of any particle in a transverse wave, second partial derivative

of with respect to time:

),()cos(),(

),(22

2

2

txytkxAt

txytxa y ωωω −=−−=

∂

∂=

),( txy

),( txy

t∂

x

txy

∂

∂ ),(- slope of the string

2

2),(

x

txy

∂

∂- curvature of the string

),()cos(),( 22

2

2

txyktkxAkx

txy−=−−=

∂

∂ω

2

2

2

22

22

/),(

/),(υ

ω==

∂∂

∂∂

kxtxy

ttxy

2

2

22

2),(1),(

t

txy

x

txy

∂

∂=

∂

∂

υWave Equation – very important

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Particle Velocity and Acceleration in a Sinusoidal Wave

t=0

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Speed of a Transverse Wave

F – tension [N]

µ − linear mass density [kg/m]

- Increasing tension increases

the wave speed.

- Increasing mass decreases

the wave speed.F

Fy is constant

1

2

tF υ υ 1⇒=t

t

F

F yy

υ

υtFtF

y

yυ

υ=

m = µ [mass per unit length] υt [length] = µυt

yy mtF υ= (impulse-momentum theorem) y

yttF υµυ

υ

υ=

µυ

F=

equilibrumtoreturntheresistinginertia

equlibrumtosystemreturningforserestoring

_____

_____=υ

18• Wave speed v does not depend on vy, also on amplitude and frequency.

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