Week 10 Waves Ch15 16 FINAL

Week 10 Chapter 15 & 16 Waves and Oscillations

1

Lecture 1:

Oscillations

- types of waves 16.2

- transverse and longitudinal waves 16.3

- wavelength, frequency and speed of wave 16.4, 16.5

Lecture 2:

Waves

- simple harmonic motion 15.2, 15.3

Lecture 3:

Waves

- wave speed on stretched string 16.6

- energy and power 16.7

Week 10

2

3

• A wave is an oscillation that moves through space or matter

transferring energy from one place to another

• Generally – material returns to its original position as the wave

passes by –there for there is no mass transport.

Waves

4

Classical waves transfer energy without transporting matter through the

medium. Waves in a medium do not move the material in the medium

from one place to another place; instead the wave's energy travels

through the medium, leaving the material in place, similar to a cork

rising and falling in one place as the wave moves past the cork

cork bobs up and

down at the same

position

Waves – Transfer of Energy

16.2 Types of Waves

• Mechanical waves. These waves have two central features: They

are governed by Newton’s laws, and they can exist only within a

material medium – NO medium NO wave: water waves, sound

waves, and seismic waves (earthquake).

• Electromagnetic waves. These waves require no material medium

to exist = PURE ENERGY. All electromagnetic waves travel

through a vacuum at the same exact speed c, speed of light = 3 x108

ms-1. Common examples include visible and ultraviolet light, radio

and television waves, microwaves, x rays, and radar (radio) waves.

• Matter waves. These waves are associated with electrons, protons,

and other fundamental particles, atoms and molecules moving at

very high speeds where they acquire wave-like properties. Basis of

quantum mechanics. Louise de Broglie (1882 – 1987) λ =ℎ

𝑚𝑣.

5

In a transverse wave, the

displacement of every such oscillating

element along the wave is

perpendicular to the direction of travel

of the wave

16.3 Transverse and Longitudinal Waves

In a longitudinal wave the motion of

the oscillating particles is parallel to

the direction of the wave’s travel

6

Note – energy of wave inversely

proportional to its energy 𝐸 ∝1

𝜆

7

l

A

Time/distance dis

pla

cem

ent

Wavelength, l, distance between two crests or troughs –

distance to complete one cycle

Amplitude – maximum displacement

Period, T – time between 2 crests/troughs – time to complete

one cycle

Frequency, f = 1/T in Hertz (Hz) – number of oscillations per second

vs

8

Velocity of a wave, vs = distance/time = l/T = l.f

Waves

q

l

A

Time/distance

dis

pla

cem

ent 1l = 1 cycle = 360 = 2p

y = Asin(q)=Acos(p/2 – q)

y y A

Angular frequency w = 2p.f angular speed, number of cycles per sec, in

radians per sec = rad.s-1

1 𝑟𝑎𝑑 = 1800

𝜋 1 =

𝜋

1800

Angular wave number, k = 2π/λ- number of cycles per unit distance

the spatial equivalent to frequency = “spatial frequency” of the wave

k = phase change of the traveling wave in terms of rad.m-1

vs

k = w/v = proportionality between w and v

9

16.4 Wavelength and Frequency Wave Motion

1s

angular frequency = number of cycles per sec = 2𝜋

𝑇 = 2pf rad.s-1

1m

wavenumber = number of cycles per metre = 2𝜋

𝜆 rad.m-1

T

l

10

Handbook of Recording

Engineering, 1986, p. 2


y = A.sin.f

f

Phase, f, relationship between

2 waves with same frequency

k = number of 2p cycles per

unit distance

Each full cycle f = 360

Note: height is the same whenever x is

an integral multiple of l

Phase angle at distance, x

f = k.x = (2p/l).x

http://www.aes.org/aeshc/docs/recording.technology.history/images4/figure1.jpg

11

Handbook of Recording

Engineering, 1986, p. 2


y = A.sin.f

f

Dx

y = Asin2π

λ. x = Asin k. x

x xt

w = k.v

http://www.aes.org/aeshc/docs/recording.technology.history/images4/figure1.jpg


12

360 = 2p

f y

y = A.sin.f

l

A

x dis

pla

cem

ent

If the wave is moving at velocity, v, it moves a distance xt =vt

A

w = k.v

y = Asin2π

λ. x = Asin k. x

16.4 Wavelength and Frequency –wave motion

13

y(x,t) = ym sin (kx ± wt –fo )

General Form

Phase Constant (radians)

- wt wave traveling to right

+ wt wave traveling to left

used to represent the wave as a function of time, t, at a specific distance, x,

or as a function of distance, x, at a specific time, t.

14

When 2 sine waves of equal amplitude and frequency propagate

through a medium in opposite directions – combine to form a

Standing Wave. Note: No Energy Transfer with Standing Waves

Wave moving to the right is: A sin(kx - ωt) & the left is: A sin(kx + ωt)

Standing wave – addition of the 2 waves

y(x,t) = A sin(kx - ωt) + A sin(kx + ωt)

Using sin A+sin B = 2sin [(A+B)/2] .cos [(A-B)/2]

y ( x, t ) = 2A cos(ωt) * sin kx

Where 2A cos(ωt) determines how the amplitude varies with time

While sin(kx) determines the standing wave’s shape.

Standing Waves

15

Standing Waves

Fundamental frequency

½ sine waveform = ½ l

2nd Harmonic

1 sine waveform = 1 l

3rd Harmonic

3/2 sine waveform = 3/2 l

4th Harmonic

2 sine waveform = 2 l

l = 2𝐿

𝑛 where n = 1, 2, 3, .......

n = number of nodes

16

Standing Waves Examples

atomic

orbitals

standing wave

solution for

electrons in an

atom

16.4 Wavelength and Frequency

The amplitude ym of a wave is the magnitude

of the maximum displacement of the elements

from their equilibrium positions as the

wave passes through them.

The phase of the wave is the argument

(kx –wt) of the sine function. As the wave

sweeps through a string element at a particular

position x, the phase changes linearly with

time t.

The wavelength l of a wave is the distance

parallel to the direction of the wave’s

travel) between repetitions of the shape of the

wave (or wave shape). It is related to the

angular wave number, k, by

:

The period of oscillation T of a wave is the

time for an element to move through one full

oscillation. It is related to the angular

frequency, w, by

The frequency f of a wave is defined as 1/T

and is related to the angular frequency w by

A phase constant f in the wave function:

y =ym sin(kx –wt+ f). The value of f can be

chosen so that the function gives some other

displacement and slope at x 0 when t 0.

17

16.4 Wavelength and Frequency: The Speed of a Traveling Wave

As the wave in Fig. 16-7 moves,

each point of the moving wave

form, such as point A marked on a

peak, retains its displacement y.

(Points on the string do not retain

their displacement, but points on the

wave form do.) If point A retains its

displacement as it moves, the phase

giving it that displacement must

remain a constant:

18

Example, Transverse Wave

19

20

Example, Transverse Wave, Transverse Velocity, and Acceleration

velocity 𝑥, 𝑡 = 𝑣𝑦 = 𝑑𝑦

𝑑𝑡 = −𝜔𝑦𝑚cos 𝑘𝑥 − 𝜔𝑡

y 𝑥, 𝑡 = 𝑦𝑚sin 𝑘𝑥 − 𝜔𝑡

acceleration 𝑥, 𝑡 = 𝑎𝑦= 𝑑𝑣

𝑑𝑡 = −𝜔2𝑦𝑚sin 𝑘𝑥 − 𝜔𝑡

y, vy & ay vs

vy,m = -wym

ay,m = -w2ym

transverse wave speed

21

y = Asinf v = -wAcosf

𝑠𝑖𝑛𝜙 =𝑦

𝐴 𝑐𝑜𝑠𝜙 = −

𝑣

𝜔𝐴

𝑠𝑖𝑛2𝜙 + 𝑐𝑜𝑠2𝜙 = 1 =𝑦2

𝐴2 +

𝑣2

𝜔2𝐴2

𝑣2=𝜔2 𝐴2 − 𝑦2

Relationship between v, w, A and y

Example, Transverse Wave, Transverse Velocity, and Acceleration

22

Chapter 15

Oscillations

23

15.1 Oscillatory motion

Motion which is periodic in time, that is, motion that

repeats itself in time.

Examples:

• Power line oscillates when the wind blows past it

• Earthquake oscillations move buildings

• Weight oscillating on a spring

• Tuning Fork

• Pendulum

• Piston moving in a car engine

• Guitar String

Sometimes the oscillations are so severe, that the

system exhibiting oscillations break apart. 24

15.2 Simple Harmonic Motion

• A particle repeatedly moves back and forth about the point x=0 where the

time taken for 1 complete oscillation is the period, T.

• In T the particle travels from x=+xm, to –xm, and then back to its original

position xm.

• The velocity vector arrows are scaled to indicate the magnitude of the

speed of the system at different times. At x=±xm, the velocity is zero. 25

mg

m

pendulum

Frequency of oscillation is the number of oscillations that

are completed in each second.

The symbol for frequency is f, and the SI unit is the Hertz

(abbreviated as Hz).

It follows that (sec)

1)(

THzf =


26

Any motion that repeats itself is periodic (= harmonic).

If the motion is a sinusoidal function of time, it is called

simple harmonic motion (SHM).

Mathematically SHM can be expressed as:

)cos()( fw = txtx m

Here,

• xm is the amplitude (maximum displacement of the system)

• t is the time

• w is the angular frequency, and

• f is the phase constant or phase angle


27

(a) The displacement of two SHM

systems that are different in

amplitudes, but have the same period.

(b) The displacement of two SHM

systems which are different in periods

but have the same amplitude.

(c) The value of the phase constant

term, f, depends on the value of the

displacement and the velocity of the

system at time t = 0. The

displacement of two SHM systems

having the same period and

amplitude, but different phase

constants.


28

(a)

(c)

(b)

For an oscillatory motion with period T,

)()( Ttxtx =

The cosine function also repeats itself when the argument

increases by 2p. Therefore,

fT

T

tTt

pp

w

pw

pww

22

2

2)(

==

=

=

Here, w is the angular frequency, and measures the

angle per unit time. Its SI unit is radians/second. To be

consistent, f must also be in radians.


29


The velocity of SHM

)sin()(

cos()(

)(

fww

fw

=

==

txtv

txdt

d

dt

tdxtv

m

m

The maximum value of velocity is: v = wxm.

The phase shift of the velocity is p/2, making

the cosine to a sine function. The acceleration

of SHM is:

)()(

)cos()(

)sin()(

)(

2

2

txta

txta

txdt

d

dt

tdvta

m

m

w

fww

fww

=

=

==

The acceleration amplitude is: a = - w2xm.

In SHM a(t) is proportional to the displacement but opposite in sign. 30

a(t) maximum where

maximum F applied

a(t) maximum

at bottom of

swing.

15.3 Force Law for SHM

From Newton’s 2nd law:

kxxmmaF === 2w

SHM is the motion executed by a system subject to a force that is

proportional to the displacement of the system but opposite in sign.

Period T of SHM oscillation =

31

F = -kx

Hooke’s law

k is the spring constant

m

k=w

22 )2( fmmk pw ==

k

mT p2=

m

kf

p2

1=

15.3 The Force Law for Simple Harmonic Motion

The block-spring system shown on

the right forms a linear SHM

oscillator.

The spring constant of the spring,

k, is related to the angular

frequency, w, of the oscillator:

k

mT

m

kpw 2==

32 mg

m

pendulum

T = 2p𝐿

𝑔

L

Simple pendulum

• Perpendicular to the string

mg sin q = -ma

a is the acceleration along the arc

sin q ≈ q and x ≈ Lq ,

mg sin q = -ma = mg q

a = -g(x/l) = -(g/l)x

as a = -w2x in SHM then w2 = g/l

• Period of oscillation T

g

L2

2T p=

w

p=

A

F=mg

mg cos q

mg sin q

q

x

T

L

33

Combination of Springs

34

F

F2 = k2x F1 = k1x

F = k1x + k2x = (k1 + k2) x

F

F1 = k1x1

F2 = k2 x2

==

21 k

1

k

1

xxkF

𝟏

𝒌 =

𝟏

𝒌𝟏+𝟏

𝒌𝟐

35

Combination of Springs

F = T1 + T2 = k1x + k2x = (k1 + k2)x

T1

Equilibrium position

T2

compression

of spring

extension

of spring

Spring

constant, k1

Spring

constant, k2

x

motion of block

Example: Force law:

36

Example, force law:

37

Example, force law:

38

Example, force law:

39

Example, force law:

40

Example, force law:

41

15.4: Energy in Simple Harmonic Motion (not in exam)

The potential energy of a linear oscillator

is associated entirely with the spring.

fw == tkxkxtU m

222 cos2

1

2

1)(

The kinetic energy of the system is

associated entirely with the speed of

the block.

fwfww === tkxtxmmvtK mm

222222 sin2

1sin

2

1

2

1)(

The total mechanical energy of

the system:

222

m kx2

1mv

2

1kx

2

1KUE ===

42 energy oscillates between KE and PE

Example, energy in SHM:

Many tall building have mass dampers,

which are anti-sway devices to prevent

them from oscillating in a wind. The

device might be a block oscillating at the

end of a spring and on a lubricated track.

If the building sways, say eastward, the

block also moves eastward but delayed

enough so that when it finally moves, the

building is then moving back westward.

Thus, the motion of the oscillator is out of

step with the motion of the building.

Suppose that the block has mass m = 2.72

x 105 kg and is designed to oscillate at

frequency f = 10.0 Hz and with amplitude

xm = 20.0 cm.

(a) What is the total mechanical energy E

of the spring-block system?

43

Example, energy, continued:

44

16.6: Wave Speed on a Stretched String

The speed of a wave, v , along a stretched ideal string

depends only on the tension, t , and linear density of the

string, m , and not on the frequency, f , of the wave.

45

m Wave

driver t = mg N

𝜇 =𝑚𝑠𝑡𝑟𝑖𝑛𝑔

𝑙𝑠𝑡𝑟𝑖𝑛𝑔 𝑣 = 𝜏

𝜇

Dl

vspeed


A small string element of length Dl

within the pulse is an arc of a circle

of radius R and subtending an angle

2q at the center of that circle.

A force with a magnitude equal to

the tension in the string, t, pulls

tangentially on this element at each

end

46

t cosq t cosq

t sinq t sinq

tangential q

𝑠𝑖𝑛𝜃 =Δ𝑙2

𝑅 = q

16.6: Wave Speed on a Stretched String The horizontal components of these forces

cancel, but the vertical components add to form

a radial restoring force . For small angles,

If m is the linear mass density of the string, and

Dm the mass of the small element,

The element has an acceleration:

Therefore, as F = ma

47

t cosq t cosq

t sinq t sinq

tangential q

𝑠𝑖𝑛𝜃 =Δ𝑙2

𝑅 = q

𝑣 =𝜔

𝑘= 𝜏

𝜇 𝜇 = 𝜏

𝑣2

𝑓 =1

𝜆𝜏𝜇 𝜏𝑘2=𝜇𝜔2


48

R R

Velocity, t2 = v2

Velocity, t1 = v1 Distance

travelled

t2 >t1

Change in

velocity

q

q Dv = v2 –v1

𝑑𝑖𝑠𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑

𝑟= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑣.Δ𝑡

𝑟 =

Δ𝑣

𝑣

Δ𝑣

Δ𝑡 = a =

𝑣2

𝑟

Since v1 and v2 are perpendicular to R, the angles of both

triangles are equal. Consequently the two triangles are

similar, the ratios of corresponding sides are equal.

49


Vibration frequency of a stretched string depends on:

• The length of the string that is free to oscillate - longer

string lower frequency

• The tension in the string - higher tension higher frequency

• The mass of the string - higher mass lower frequency

𝑓 =1

𝜆 𝜏

𝜇 =

𝑛

2𝑙

𝜏 . 𝐿

𝑚 𝑓 =

𝑛

2

𝜏

𝑚𝐿 l =

2𝐿

𝑛

16.6: Energy and Power of a Wave Traveling along a String

50

Kinetic energy of string element

at each position depends on the

transverse velocity of the element

Potential energy depends on the

amount by which the string element

is stretched as the wave passes

through it

Snapshot of a travelling wave on a

string at t = 0

l

T

Energy, El

51

• In one period T, energy has moved one wavelength, l

• Transported Power, P = 𝐸𝜆

𝑇

𝑷 =𝟏

𝟐 𝝁𝝕𝟐𝑨𝟐

𝝀

𝑻=𝟏

𝟐 𝝁𝝕𝟐𝑨𝟐𝒗


𝑬𝝀 = 𝟏

𝟐 𝝁𝝕𝟐𝑨𝟐 𝝀

52


𝑷𝝀 =𝟏

𝟐 𝝁𝒘𝟐𝑨𝟐𝒗𝒔𝒑𝒆𝒆𝒅

Mass per unit length

of the string (kg/m)

Angular frequency of

wave (radians/sec)

wave velocity

speed (m/s)

Amplitude of wave (m)

Power (Watts)

53


l

vs

Wave motion

A y

vy

𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 = ∆𝐾𝐸 = 1

2𝑚𝑣𝑦

2 = 1

2𝜇 Δ𝑥 𝑣𝑦

2

𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = ∆𝑈 =1

2𝐹. 𝑦 =

1

2(𝑚. 𝑎). 𝑦 =

1

2𝑚𝜔2𝑦2

𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = ∆𝑈 = 1

2𝑚𝜔2𝑦2 =

1

2 m Dx 𝜔2𝑦2

𝜇 =𝑚

Δ𝑥

54


y = Asin(kx – wt)

∆𝑈 = 1

2𝑚𝜔2𝑦2 =

1

2𝜔2𝑦2mDx =

1

2m𝜔2A2sin2(kx-wt)Dx

Wave equation – travelling wave

Potential energy for one wavelength along the string is found by

integrating over dx at t=0

𝑈𝜆 =1

2 𝜇𝜔2𝐴2 𝑠𝑖𝑛2 𝑘𝑥 𝑑𝑥 =

1

2 𝜇𝜔2𝐴2

1

2𝑥 −

1

4𝑘𝑠𝑖𝑛 2𝑘𝑥

0

𝜆𝜆

0

= 1

4𝜇𝜔2𝐴2𝜆

Total Energy with one wavelength = El = Ul + KEl = 1

2𝜇𝜔2𝐴2𝜆

Δ𝐾𝐸 =1

2𝑚𝑣2 =

1

2𝜇𝑣2Δ𝑥 v = wAcos(kx–wt)

∆𝐾𝐸 = 1

2m𝜔2A2cos2(kx-wt)Dx 𝐾𝐸𝜆 =

1

2 𝜇𝜔2𝐴2 𝑐𝑜𝑠𝑑𝑥

𝜆

0

= 1

4𝜇𝜔2𝐴2𝜆

Similarly

𝑠𝑖𝑛2 𝑘𝑥 = 1

21 − 𝑐𝑜𝑠 𝑘𝑥


The average power, which is the average rate at which energy of both kinds

(kinetic energy and elastic potential energy) is transmitted by the wave, is:

55

Velocity of string Kinetic energy of string

ym = A

Example, Transverse Wave:

56

57

A string of mass of 20 mg and length of 1000 mm carries a wave

with a period 65 ms and a wavelength 60 cm. What is the average

power transmitted along the string if the waves amplitude is 12

mm?

𝑃𝜆 =1

2 𝜇𝑤2𝐴2𝑣

v = l .T -1 w = 2p .T -1

𝑃𝜆(𝑊𝑎𝑡𝑡) =1

2 𝜇4𝜋2

𝑇2𝐴2𝜆

𝑇

Here A = 12 x10-3 m, l = 0.6 m, T = 6.5 x10-2 s, μ = 2 x10-3 kg/m

P = 12.42 Watts

Documents

Week 10 Waves Ch15 16 FINAL