applied natural sciences - porous media · 2019-10-01 · 1 p ave f a • when a duck swims, it...

Applied Natural Sciences

Leo Pele‐mail: 3nab0@tue.nlhttp://tiny.cc/3NAB0

Het basisvak Toegepaste Natuurwetenschappen

http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html

Copyright © 2012 Pearson Education Inc.

PowerPoint® Lectures forUniversity Physics, Thirteenth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by Wayne Anderson

Chapter 15

Mechanical Waves

LEARNING GOALS

• What is meant by a mechanical wave, and the different varieties of mechanical waves

• How to use the relationship among speed, frequency, and wavelength for a

periodic wave.

• How to interpret and use the mathematical expression for a sinusoidal periodic wave.

• How to calculate the speed of waves on a rope or string.

• How to calculate the rate at which a mechanical wave transports energy.

• What happens when mechanical waves overlap and interfere.

• The properties of standing waves on a string, and how to analyze these waves.

• How stringed instruments produce sounds of specific frequencies.

4

5

Simple Harmonic oscillator

5

Harmonic oscillator and waves

6

Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:• Transverse – if the direction of displacement is perpendicular to the direction of propagation

Types of waves

Earth quake wavesAnimation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 7

Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:• Longitudinal – if the direction of displacement is parallel to the direction of propagation

Types of waves

Sound

Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 8

But also a combination of Transverse and Longitudinal

Types of waves

Water waves

Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 9

Transverse and longitudinal waves

Sound (part II)air

Mechanical (part I)materials

10

Waves have in common:

1. In each case the disturbance travels or propagateswith a 

definite speed through the medium:  the wave speed.

2. The medium itself does not travel through space.

3. To set any of these systems into motion, we have to put in 

energy by doing mechanical work on the system. 

Waves transport energy, but not matter, from one region to 

another.

11

MATHEMATICAL

12

Harmonic oscillator and waves

13

One position: SHM One time shot:position in space

Have a description NEED ONE)cos()( tAtx14

Wavelength  (lambda), 16

Harmonic wave

wavelength

=Wave speed

=v

Wave speed = v =distancetime

wavelengthperiod= =

T = f

but 1/T=f

V=for f=V/

Shake end ofstring up & down

with SHM period = T

this is the “Golden Rule” for waves 17

Example: wave on a string

A wave moves on a string at a speed of 4 cm/sA snapshot of the motion shows that the wavelength, is 2 

cm, what is the frequency,  ?v =  , so  =  v / =  (4 cm/s ) / (2 cm) = 2 Hz

T = 1 / f = 1 / (2 Hz)  = 0.5 s

2 cm 2 cm 2 cm

18

Look at:

x=5

x=10

)cos()( tAtx

Waves

)0(2cos)( xAxy

)5(2cos)( xAxy

)10(2cos)( xAxy

position = 5 t= vt

t

fAtx 2cos)(

shm

)(2cos),( vtxAtxy

move right

)(2cos),( vtxAtxy

move left

Movement of wave

19

)(2cos),( vtxAtxy

vtxAtxy

22cos),( fv

tfxAtxy 22cos),(

tkxAtxy cos),(2k

Wave number

Alternate notation

“+” if wave travels toward –x

“-” if wave travels toward +x 20

Definitions

Amplitude ‐ (A, ym) Maximum value of the displacement of a particle in a medium 

(radius of circular motion).

Wavelength ‐ () The spatial distance between any two points that behave identically, 

i.e. have the same amplitude, move in the same direction (spatial period)

Wave Number ‐ (k) Amount the phase changes per unit length of wave travel.  (spatial 

frequency, angular wavenumber)  

Period ‐ (T) Time for a particle/system to complete one cycle.

Frequency ‐ (f) The number of cycles or oscillations completed in a period of time

Angular Frequency ‐ Time rate of change of the phase.

Phase ‐ kx ‐ t Time varying argument of the trigonometric function.

Phase Velocity ‐ (v) The velocity at which the disturbance is moving through the 

medium 21

The Wave Equation

If we see an equation

that looks like:

... we can write down the

amplitude, frequency,

velocity, and

wavelength of the

wave it describes.

xty m 5796.4 s 8.1570cosm15.0 -1-1

direction.x in the m/s, 343

m 1.372 m 5796.42Hz 250 s 1570.8 2 m 15.0

22cos

1-

1-

fv

ffA

xftAy

)cos()cos(

22

General description SHM

• The force described by Hooke’s Law is the net force in Newton’s Second Law.

)()(2

2

tkxdt

txdm

1. The acceleration is proportional to the displacement of the block

2. The force is conservative. In the absence of friction, the motion will continue forever.

mk

tAtx

= where

)cos()(

23

General description waves

2y(x, t)x2 1

v2

2y(x, t)t2

Wave Equation:

position time

Partial Differential Equation (PDE)

tkxAtxy cos),(A solution is:

We will not try to solve this equation

24

General description waves

2y(x, t)x2 1

v2

2y(x, t)t2

Wave Equation:

tkxAtxy cos),(

tkxAty sin

tkxAty

cos2

2

2

tkxkAxy sin

tkxAkxy

cos2

2

2

tkxAv

tkxAk coscos 2

22

And 2k

fv kv25

Sinusoidal Wave on a String, final

The maximum magnitudes of the transverse speed and

transverse acceleration are

vy, max = A

ay, max = 2 A

The speed v is constant for a uniform medium,

whereas vy varies sinusoidally.

Tgolf‐getal k

v

vy

26

27

28

Rope: mass + springs

mktAtx = where)cos()(

1) Mass rope: m larger > slower

2) More tension > k larger > faster

How fast will it be transmitted

29

Speed of a Wave on a String

The speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected.

•This assumes that the tension is not affected by the pulse.•This does not assume any particular shape for the pulse.

tension force

(string mass / string length)

F

lengthmasstensionv

/

30

31

• The average power:

Energy in a Wave

22

21 AFPave

• When a duck swims, it necessarily produces waves on the surface of the water. • The faster the duck swims, the larger the wave amplitude and the more power the duck must supply to produce these waves.• The maximum power available from their leg muscles limits the maximum swimming speed of ducks to only about 0.7 m/s (2.5 km/h)

Duck swimming

32

Waves Intensity

Wave intensity for a three dimensional wave from a point source:

22 W/mofunitsin

4 rPI

22

212

1 44 IrIr

21

22

2

1

rr

II

Example: earthquake, sound waves33

Wave boundary/

interference

34

35

The principle of superpositionWhen two waves overlap, the actual displacement of anypoint at any time is obtained by adding the displacementthe point would have if only the first wave were present andthe displacement it would have if only the second wave werepresent:

),(),(),( 21 txytxytxy

36

Superposition of equal amplitude waves

Constructive Destructive

37

Reflection (from a fixed end)

Animations courtesy of Dr. Dan Russell, Kettering University 38

Reflection (from a loose end)

Animations courtesy of Dr. Dan Russell, Kettering University 39

Superposition of equal amplitude waves

Constructive Destructive

40

Standing Wave

42

Adding waves

harmonic waves in opposite directions

incident wave

reflected wave

resultant wave

(standing wave)

Animations courtesy of Dr. Dan Russell, Kettering University 43

44

Standing Waves ‐ Resonance

tkxAtxy cos),(1

tkxAtxy cos),(2

tkxAtkxAtxy coscos),(

cos x - cos y = -2 sin ((x - y)/2) sin( (x + y)/2 )

)sin()sin(2),( tkxAtxy 45

Nodes and Antinodes

Node – position of no displacement

Antinode – position of maximum displacement

y 2Asin kx cos t

2kx x 0, , 2 ,3 ...

3x 0, , , ,.....2 2

Nodes

46

Confined waves

Only waves with wavelengths that just fit in survive(all others cancel themselves out)

nL

n2

2nL 12

nfLvnfn

2kx x 0, , 2 ,3 ...

3x 0, , , ,.....2 2

47

Allowed frequencies

=(2/3)L

f0=V/ = V/2L

f1=V/ = V/L=2f0

= 2L

=L

=(2/5)L

=L/2

f2=V/=V/(2/3)L=3f0

f3=V/=V/(1/2)L=4f0

f4=V/=V/(2/5)L=5f0

Fundamental tone

1st overtone

3rd overtone

4th overtone

2nd overtone

48

VIDEO + UITWERKING

HznLvnfn 7.4

2

L = 1.8 mString mass= 188 grTension = 3N

µ =0.00810 kg/m

smTv /92.16

4.7 9.4 14.1 18.8 23.5 28.2 32.9 36.6 42.3 47.0 51.7 56.4 61.1 65.81 2 3 4 5 6 7 8 9 10 11 12 13 14

4.7 9.4 14.1 18.8 23.5 28.2 32.9 36.6 42.3 47.0 51.7 56.4 61.1 65.81 2 3 4 5 6 7 8 9 10 11 12 13 14

50

Fundamental, 2nd, 3rd... Harmonics

Fig 14.18, p. 443

Slide 25

Fundamental (n=1)

2nd harmonic

3rd harmonic

n 2 L

51

Loose Ends

(Organ pipes open at one end)

L 2n 1 4 L

4

L 34

L 5 4

)12(4

n

Ln

1)12(4

)12(

fnLvnfn

52

Harmonics with a bridge

53

Stringed instruments

Three types

Plucked: guitar, bass, harp, harpsichord

Bowed: violin, viola, cello, bass

(Bowing excites many vibration modes  

simultaneously mixture of tones (richness) 

Struck: piano

All use strings that are fixed at both ends

• Use different diameter strings (mass per unit 

length is different)

• The string tension is adjustable – tuning  T

Lf

21

1

54

A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz.  What is the speed of the waves on this string? 

1. 220 m/s     

2. 440 m/s     

3. 660 m/s     

4. 880 m/s     

5. 1.10 km/s

55

A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz.  What is the speed of the waves on this string? 

1. 220 m/s     

2. 440 m/s     

3. 660 m/s     

4. 880 m/s     

5. 1.10 km/s

12nf

Lvnfn

56

Lfv 21

A stretched string is fixed at points 1 and 5.  When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points 

1. 1 and 5. 

2. 1, 3, and 5. 

3. 1 and 3. 

4. 2 and 4. 

5. 1, 2, 3, 4, and 5. 57

A stretched string is fixed at points 1 and 5.  When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points 

1. 1 and 5. 

2. 1, 3, and 5. 

3. 1 and 3. 

4. 2 and 4. 

5. 1, 2, 3, 4, and 5. 

58

A stretched string of length L, fixed at both ends, is vibrating in its third harmonic.  How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration? 

1. L/6     

2. L/4     

3. L/5     

4. L/2     

5. L/3 59

A stretched string of length L, fixed at both ends, is vibrating in its third harmonic.  How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration? 

1. L/6     

2. L/4     

3. L/5     

4. L/2     

5. L/3 60

Drum: 2D standing waves

Mode (0,1) Mode (1,1)

Mode (2,1) Mode (0,2)Animations courtesy of Dr. Dan Russell, Kettering University 61

62

Applied Natural Sciences

Thank you for your attention

Merry Christmas

Have a happy holiday and see you next year

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Summary

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Summary

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