jw Fundamentals of Physics 1 Chapter 14 Waves - I 1.Waves & Particles 2.Types of Waves 3.Transverse...

jw Fundamentals of Physics 1

Fundamentals of Physics

Chapter 14 Waves - I

1. Waves & Particles2. Types of Waves3. Transverse & Longitudinal Waves4. Wavelength & Frequency5. Speed of a Traveling Wave6. Wave Speed on a Stretched String7. Energy & Power in a Traveling String Wave8. The Wave Equation9. The Principle of Superposition for Waves10. Interference of Waves11. Phasors12. Standing Waves13. Standing Waves & Resonance


Waves & Particles

Particles - a material object moves from one place to another.

Waves - information and energy move from one point to another, but no material object makes that journey.

– Mechanical waves

– Newton’s Laws rule!

– Requires a material medium

e.g. water, sound, seismic, etc.

– Electromagnetic waves

– Maxwell’s Equations & 3.0 x 108 m/s

– No material medium required

– Matter waves

– Quantum Mechanics - ~10-13 m

– Particles have a wave length - De Broglie (1924)


A Simple Mechanical Wave

A single up-down motion applied to a taut string generates a pulse.

The pulse then travels along the string at velocity v.

Assumptions in this chapter:

No friction-like forces within the string to dissipate wave motion.

Strings are very long - no need to consider reflected waves from the far end.

He moves his hand once.


Traveling Waves

Transverse Wave:The displacement (and velocity) of every point along the medium carrying the wave is perpendicular to the direction of the wave.

Longitudinal Wave:The displacement (and velocity) of the element of the medium carrying the wave is parallel to the direction of the wave.

e.g. a vibrating string e.g. a sound wave

They oscillate their hand in SHM.

Longitudinal, Transverse and Mixed Type Waves

http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/long_wave.html

http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/long_wave.html


Transverse Wave

Displacement versus position

not versus time

Each point along the string just moves up and down.

transverse wave applet


• amplitude - maximum displacement

• wavelength - distance between repetitions of the shape of the wave.

• angular wave number

• period - one full oscillation

• frequency - oscillations per unit time

Wave Length & Frequency

2T

2k

21 Tf


The phase, kx – wt , changes linearly with x and t, which causes the sine function to oscillate between +1 and –1.

Wave Length & Frequency

timespace


Each point on the wave, e.g. Point A, retains its displacement, y;

hence:

Speed of a Traveling Wave

Consider a wave traveling in the positive x direction; the entire wave pattern moves a distance x in time t:

vx

t

y x t y kx tm( , ) sin constant

kx t constant Note: both x and t are changing!

dxdt k

v • Differentiating:


Direction of the Wave

y = f (x + v t) traveling towards -x

All traveling waves are functions of (kx + t) = k(x + vt) .

Consider an unchanging pulse traveling along positive x axis.

traveling towards +x y = f (x’) = f (x - v t)


Descriptions of the phase of a Traveling Wave

( , ) sinmy x t y kx t

v fk T

1

2f

T

2 vvT

k f

. .x

kx t k x vt t etc etcv

( , ) sin 2m

x ty x t y

T


Wave Speed on a Stretched String

A single symmetrical pulse moving along a string at speed vwave.

stringwave F

v

In general, the speed of a wave is determined by the properties of the medium through which it travels.


Wave Speed on a Stretched String

Consider a single symmetrical pulse moving along a string at speed v:

Speed of a wave along a stretched string depends only on the tension and the linear density of the string and not on the frequency of the wave.

= tension in the string = the string’s linear density

Moving along with the pulse on the string.

2 sin 2r

lF

R

m l

2va

R

2

2

netF m a

l vl

R R

v

String moving

to the left.

(Roughly a circular arc)


Energy of a Traveling String Wave

Energy

Driving force imparts energy to a string, stretching it.

The wave transports the energy along the string.


• Driving force imparts energy to a stretched string.

• The wave transports energy along the string.

– Kinetic energy - transverse velocity of string mass element, m = x– Potential energy - the string element x stretches as the wave passes.

Energy & Power of a Traveling String Wave

dK dmdy

dt

dK dx y kx t

dK

dtv y

dU

dt

v y

m

avg

m

avg

m

12

2

12

2

14

2 2

12

2 2

cos

Average Power

sinmy y kx t

(See Section 14-3)


The Principle of Superposition for Waves

ytotal(x,t) = y1(x,t) + y2(x,t)

Overlapping waves algebraically add to produce a resultant wave:

y1(x,t) y2(x,t)Add the amplitudes:

Overlapping waves do not alter the travel of each other!


The Principle of Superposition for Waves

Interference of waves traveling in opposite directions.

Constructive

Interference

Destructive

Interference


Interference of Waves Traveling in the Same Direction

It is easy to show that:

= “phase difference”

sin sin sin cos 2 12

12

1( , ) sinmy x t y kx t

2 ( , ) sinmy x t y kx t

my

1 2( , ) ( , ) ( , )y x t y x t y x t


Interference of Waves

The magnitude of the resultant wave depends on the relative phases of the combining waves - INTERFERENCE.

Constructive Interference Destructive Interference Partial Interference


Standing Waves

y’(x,t) = y1(x,t) + y2(x,t)

y1(x,t) = ym sin (k x - t)

y2(x,t) = ym sin (k x + t)

Two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a string:

For a standing wave, the amplitude, 2ymsin(kx) , varies with position.

sin sin sin cos 2 12

12

positive x direction

Their interference with each other produces a standing wave:

negative x direction

It is easy to show that:


Standing Wave

The amplitude of a standing wave equals zero for:

minimums @ x = ½ n n = 0, 1, 2, . . . NODES

0sin xk

3,2,1,0nnxk

2

k


Standing Waves

2

k

minimums @ x = ½ n n = 0, 1, 2, . . . NODES

maximums @ x = ½(n + ½) n = 0, 1, 2, . . . ANTINODES


Reflections at a Boundary

Reflected pulse has opposite

sign

Reflected pulse has same sign

Newton’s 3rd Law

“soft” reflection“hard” reflection

Tie the end of the string to the wall

End of the string is free to move

Antinodeat boundary

Nodeat boundary


Reflections at a Boundary

From high speed to low speed (low density to high density)

From high density to low density


Standing Waves & Resonance

Resonance for certain frequencies for a string with both ends fixed.

This can only be true when:

v

f

v is the speed of the traveling waves on the string.

,3,2,12

nL

vnfn

,3,2,12

nnL

Consider a string with both ends fixed; it has nodes at both ends.

Only for these frequencies will the waves reflected back and forth be in phase.



A standing wave is created from two traveling waves, having the same frequency and the same amplitude and traveling in opposite directions in the same medium.

Using superposition, the net displacement of the medium is the sum of the two waves.

When 180° out-of-phase with each other, they cancel (destructive interference).When in-phase with each other, they add together (constructive interference).



The Harmonic Series

both ends fixed

,3,2,1

2 1

n

fnL

vnfn


String Fixed at One End

fixed end

Prenault’s applets

free end

Resonance:

,5,3,1

4

n

nL

,5,3,1

4 1

n

fnL

vnfn

Standing wave applet


Fundamentals of Physics

Waves - II

1. Introduction2. Sound Waves3. The Speed of Sound4. Traveling Sound Waves5. Interference6. Intensity & Sound Level

The Decibel Scale7. Sources of Musical Sound8. Beats9. The Doppler Effect

Detector Moving; Source StationarySource moving; Detector StationaryBat Navigation

10. Supersonic Speeds; Shock Waves


Sound Waves

A sound wave is a longitudinal wave of any frequency passing through a medium (solid, liquid or gas).


Elastic property of the medium– Strings - tension ( in N)– Sound - Bulk Modulus (B in N/m2)

• Inertial property of the medium– Strings - linear mass density ( in kg/m)– Sound - volume mass density ( in kg/m3)

The Speed of Sound

vB

v

p BV

V

property inertial

property restoring velocity

The speed of waves depends on the medium, not on the motion of the source.


The Speed of Sound

For an ideal gas, B/ can be shown to be proportional to absolute temperature; hence, the speed of sound depends on the square root of the absolute temperature.

vB

vR T

M

T = absolute temperature

= 1.4 for O2 and N2 (~air)

R = “universal gas constant” = 8.314 J/mol-K

M = molar mass of the gas = 29 x 10-3 kg/mol (for air)

p B

V

V Equation for the speed of sound:

343 760m mis hv


Traveling Sound Waves

tkxstxs m cos,

Pressure-Variation Function:(pressure change as wave passes x)

Displacement Function:(of the air element about x) SHM

tkxptxp m sin,



As a sound wave moves in time, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source.

Slinky Demo



mm svp

As a sound wave moves in time, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source.


Interference

Consider two sources of waves S1 and S2, which are “in phase”:

L

2

1L

2L

12 LLL

,3,2,,0

,3,2,1,02

L

mm

“Constructive Interference”

2L

is the “phase difference” @ P1

“arrive in phase”


Interference

Two sources of sound waves S1 and S2:

L

2

1L

2L

12 LLL

,,,

,3,2,1,012

25

23

21

L

mm

“Destructive Interference”

,,, 25

23

21 L

arrive “out of phase”


Psychological dimensions of sounds

Pitch

Loudness

1500 Hz

150 Hz

150 Hz with twice the amplitude of 1500 Hz

A healthy young ear can hear sounds between 20 - 20,000 Hz.

- age reduces our hearing acuity for high frequencies.

300-Hz sound

500-Hz sound

2.3soundvm

f

0.23m


Intensity of a Sound Wave:

The power of the wave is time rate of energy transfer.

The area of the surface intercepting the sound.

Intensity & Sound Level

Area

PowerI

24 R

PI source

All the sound energy from the source spreads out radially and must pass through the surface of a sphere:

In terms of the parameters of the source and of the medium carrying the sound, the sound intensity can be shown to be as follows:

2221

msvI

2

1~

RI

2~ msI

2221 AvP



The Decibel Scale: - Sound Level

Mammals hear over an enormous range:

where I0 is the approximate threshold of human hearing.

(pain) 110 :humans 2212

mW

mW

Sound level (or loudness) is a sensation in the consciousness of a human being. The psychological sensation of loudness varies approximately logarithmically; to produce a sound that seems twice as loud requires about ten times the intensity.

0

log10I

I (decibel)

010 212

0 m

WI

Alexander Graham Bell



Every 10dB is a factor 10 change in intensity; 20 dB is a factor 100 change in intensity

Human Perception of Sound

0

log10I

I

010 212

0 m

WI

~3dB is a factor 2 change in intensity

See Table 17-2.


Sources of Musical Sound

Closed End– Molecules cannot move

• Displacement node

• Pressure Antinode

Open End– Molecules free to move

• Displacement Antinode

• Pressure Node

Both ends closed 2 nodes with at least one antinode in between.

Both ends open 2 antinodes with at least one node in between.

One end closed 1 node and one antinode.

Standing Waves in a Pipe



Fundamental Frequency

“1st Harmonic”

“Fundamental mode”

nodes or antinodes at the ends of the resonant structure



Both Ends Open One End Open

Harmonic Number

L

vnvf

nn

L

2

,3,2,12

1n

L

vnvf

nn

L

4

,5,3,14



length of an instrument fundamental frequency



Fundamental & Overtones

Overtones


Beats

2 waves with slightly different frequencies are traveling to the right.

ttss m coscos2

2121

2121

The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency.

The superposition of the 2 waves travels in the same direction and with the same speed.


Beats

tss

tss

m

m

22

11

cos

cos

21

ttss

ttss

sss

m

m

2121

2121

21

21

coscos2

coscos

Consider two similar sound waves:

Superimpose them:

2121

2121

“Beat Frequency”:

2121 2 beatandthenIf

21 fffbeat

ttss m coscos2


Interference:

Standing Wave created from two traveling waves:

2 waves with slightly different frequencies are traveling in the same direction. The superposition is a traveling wave, oscillating with the average frequency with its amplitude envelope varying according to the difference frequency.

As the two waves pass through each other, the net result alternates between zero and some maximum amplitude. However, this pattern simply oscillates; it does not travel to the right or the left; it stands still.

2 sinusoidal waves having the same frequency (wavelength) and the same amplitude are traveling in opposite directions in the same medium.

[one dot at an antinode and one at a node]

Beats created from two traveling waves:

Beats demo


The Doppler Effect

Doppler Effect

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/#sound_waves

Applet: Doppler Effect


The Doppler Effect

S - Wave Source D - Detector (ear)

Sv

Dv

Case 0: both are stationary 0 DS vv

frequency detected: ff

v

f

sound theoffrequency

sound theof wavelength

f

v

mediumthe in sound of speed


The Doppler Effect

Case 1: Detector is moving towards the source

0Dv

frequency detected: ff

0Sv


The Doppler Effect

vD t v t


Number of wavefronts intercepted

tvtv D

“Rate of Interceptions”

v

vvf

t

tvtvf DD

f

v

vf

A higher frequency is detected

See section 14.8


The Doppler Effect


0Dv


0Sv

v

vvff D

Case 2: Detector is moving away from the source v

vvff D

The detected frequency is less than the source frequency.


The Doppler Effect

Case 3: Source is moving towards the detector

Svv

vff

S@ emissions between time the isT

D@ h wavelengtdetected the is

Tf

1

The detected frequency is greater than the source frequency.

Case 4: Source is moving away from the detector

Svv

vff

TvTv

vvf

S


Supersonic Speeds



Supersonic Speeds & Shock Waves

No waves in front of the source.

Waves pile up behind the source to form a shock wave.

The “Mach Cone” narrows as vS goes up.

sound

vMach Number

v

sv

vsin


Supersonic Speeds

sv

vsin

o33

vS = Mach 1.8

You won’t hear it coming!


Supersonic

sin sound

source

v

v

jw Fundamentals of Physics 1 Chapter 14 Waves - I 1.Waves & Particles 2.Types of Waves 3.Transverse...

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