1

Mechanical Waves

Ch 21-23

2

Waves

A wave is a disturbance in a medium

which carries energy from one point to another

without the transport of matter.The medium allows the disturbance to

propagate.

3

Transverse Wave

Particles oscillate at right angles to the direction of motion.

4

Longitudinal Waves

Particles oscillate parallel to the direction of motion.

5

Periodic Waves & Pulses

A wave pulse is a single disturbance.

A periodic wave is a series of disturbances or wave train.

6

Transverse Wave Speed

Determined by the medium and its properties.

elasticity or restoring force inertia

v =Restoring force factor

Inertia Factor

7

Wave on a mediumwith tension.

String, rope, wire, etc…T is the tension, & is the linear density,

= m/L = mass per unit length.

T

v

8

Waves

Speed:

v = f =

T

9

Wave Terminology

Frequency (f) - cycles per second. (Hz)Period (T) - Seconds per cycle.Amplitude (A) - Maximum displacement

from equilibrium.The distance that a wave travels in one

period is the wavelength ().

10

Example 1

A wave travels along a string. The time for a particular point to move from a maximum displacement to zero is 0.170 s. The wavelength is 1.40 m. What are the period, frequency, and wave speed?

11

Example 1 continued

It takes 0.680 s for one cycle, so T = 0.680 s

f = 1/T, so f = 1.47 Hz

fv m 40.1Hz 47.1s

m06.2

12

Example 2

What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N?

T

v L

mT

m

TL

13

Example 2 continued

kg 060.0

m 00.2N 500v

m/s 129v

14

Polarization

Most transverse waves are linearly polarizedThey either move just up and down

Vertically polarizedOr just side to side

Horizontally polarized

15

Circular polarization

If we combine two perpendicular waves that have equal amplitude but are out of step by a quarter-cycle, the resulting wave is circularly polarized.

16

Polarizing filters

Only let through waves that are polarized one way.

Like passing a rope through a slot in a board – only waves in the direction of the slot will get through.

17

Longitudinal Wave Speed

Depends on the pressure change and the fractional volume change

Where is the density. B is the bulk modulus of a fluid. Y is young’s modulus for a solid. See tables 12-1 and 12-2. B = 1/k

B

v Y

v

18

Longitudinal waves

Don’t have polarizationWhen the frequency is within the range

of human hearing, it is called sound.

19

Sound waves in gases

Temperature doesn’t remain constant as sound waves move through air.

So, we use the equation

Where is the ratio of heat capacities (ch 18), R is the ideal gas constant (8.314 J/mol∙K), T is temperature in K, and M is the molecular mass (ch 17).

M

RTv

20

Sound waves

Humans can hear from about 20 Hz to about 20 000 Hz.

Air is not continuous – it consists of molecules.

Like a swarm of bees.Also sort of like wave/particle duality.

21

Mathematical wave description

y(x, t) = A sin(t – kx)

(Motion to right)

or y(x, t) = A sin(t + kx)

(Motion to left)

k =2

= 2f =2

T

v = f =

k

22

Reflection

When a wave comes to a boundary, it is reflected.

Imagine a string that is tied to a wall at one end. If we send a single wave pulse down the string,when it reaches the wall, it exerts an upward

force on the wall.

23

Reflection

By Newton’s third law, the wall exerts a downward force that is equal in

magnitude.

This force generates a pulse at the wall, which travels back along the string in the opposite direction.

24

Reflection

In a ‘hard’ reflection like this,

there must be a node at the wall

because the string is tied there.

The reflected pulse is inverted from the incident wave.

25

Reflection

Now imagine that instead of being tied to a wall

the string is fastened to a ring which is free to move along a rod.

When the wave pulse arrives at the rod, the ring moves up the rod

and pulls on the string.

26

Reflection

This sort of ‘soft’ reflection

creates a reflected pulse

that is not inverted.

27

Transmission

When a wave is incident on a boundary that separates two regions of different wave speedspart of the wave is reflectedand part is transmitted.

28

Transmission

If the second medium is denser than the first the reflected wave is inverted.

If the second medium is less dense the reflected wave is not inverted.

In either case, the transmitted wave is not inverted.

29

Transmission

30

Transmission

31

Interference

32

Interference

The effect that waves have when they occupy the same part of the medium.

They can add together or cancel each other out.

After the waves pass each other, they continue on with no residual effects.

33

Constructive Interference

34

Constructive Interference

out of phase = 360° = 1 cycle = 2 rad

35

Destructive Interference

36

Destructive Interference

/2 out of phase = 180° = 1/2 cycle = rad

37

Superposition of waves

If two waves travel simultaneously along the same string

the displacement of the string when the waves overlap

is the algebraic sum of the displacements from each individual wave.

38

Standing Waves

Consider a string that is stretched between two clamps, like a guitar string.

If we send a continuous sinusoidal wave of a certain frequency along the string to the right

When the wave reaches the right end, it will reflect and travel back to the left.

39

Standing waves

The left-going wave the overlaps with the wave that is still traveling to the right.

When the left-going wave reaches the left end

it reflects again and overlaps both the original right-going wave and the reflected left-going wave.

Very soon, we have many overlapping waves which interfere with each other.

40

Standing waves

For certain frequenciesthe interference produces a standing wave

patternwith nodes and large antinodes.This is called resonanceand those certain frequencies are called

resonant frequencies.

41

Standing waves

A standing wave looks like a stationary vibration pattern,

but is the result of waves moving back and forth on a medium.

42

Standing waves

Superposition of reflected waves which have a maximum amplitude and appear to be a stationary vibration pattern.

y1 + y2 = -2Acos(t)sin(kx)

43

Standing Waves

If the string is fixed at both endsthere must be nodes there.The simplest pattern of resonance that can

occur is one antinode at the center of the string.

44

Standing Waves on Strings

Nodes form at a fixed or closed end.Antinodes form at a free or open end.

45

Standing waves

For this pattern, half a wavelength spans the distance L.

This is called the 1st harmonic. It is also called the fundamental mode of vibration.

46

Standing waves

For the next possible pattern, a whole wavelength spans the distance L.

This is called the 2nd harmonic, or the 1st overtone.

47

Standing Waves

For the next possible pattern, one and a half wavelengths span the distance L.

This is called the 3rd harmonic, or the 2nd overtone.

48

Standing waves

,.......3,2,1for ,2

nn

L

,.......3,2,1for ,2

nL

vn

vf

In general, we can write

frequency.resonant 1st theis where 11 fnff

or

number harmonic

theis where n

49

Standing Waves

50

Standing Waves on a String

51

Overtones

52

String fixed at ONE end

Note: Only the odd harmonics exist!

=4L

n

53

Example

The A-string of a violin has a linear density of 0.6 g/m and an effective length of 330 mm.

(a) Find the tension required for its fundamental frequency to be 440 Hz.

(b) If the string is under this tension, how far from one end should it be pressed against the fingerboard in order to have it vibrate at a fundamental frequency of 495 Hz, which corresponds to the note B?

54

Example

= 0.6 g/m = 6 x 10–4 kg/mL = 330 mm = 0.33 ma) Ft = ?

b) 0.33 m – L2 = ?

55

Example - A)

v Ft

v2

Ft

Ft v2 v f

Ft 2f 2 2L

Ft 4L2f 2 51N

56

Example - B)

v1 = v2

f11 = f22

2 f11

f2

0.59m

L2 0.29m

0.330m – 0.293m 0.037m

57

Wave Example 1

The stainless steel forestay of a racing sailboat is 20 m long, and its mass is 12 kg. To find its tension, it is struck by a hammer at the lower end and the return of the pulse is timed. If the time interval is 0.20 s, what is the tension in the stay?

58

Example 1

L = 20 m, t = 0.20 s, m = 12 kgFind: F

59

Example 1

v xt

2Lt

mL

v F

F mv2

L

mL

(2Lt

)2 4mL

t2

F (4)(12kg)(20m)

(0.20s)2 = 2.4 x 104 N

60

Note:

Wave speed is determined by the medium.

Wave frequency is determined by the source.

61

Sound Waves

p = BkAcos(t - kx)If y is written as a sine function, P is

written as a cosine function because the displacement and the pressure are/2 rad out of phase.

pmax = BkA

62

Waves in 3 Dimensions

x

A 0 1 4 9 16

For Spherical Wavefronts: A = 4r2

63

Intensity

Power per unit areaW/m2

B

p

v

pkABI

222

1 2max

2max2

A

PI

64

Loudness of Sound

Also called intensity levelDetermined by the intensitywhich is a function of the sound's

amplitude.The human ear does not have a linear

response to the intensity of sound.The response is nearly logarithmic.

65

Decibel Scale (dB)

= 10 logI

Io

Where: Io = 1 x 10-12 W/m2

66

Common decibel levels

Threshold of hearing0 dB = 1 x 10-12 W/m2

Whisper20 dB = 1 x 10-10 W/m2

Conversation65 dB = 3.2 x 10-6 W/m2

Threshold of pain120 dB = 1 W/m2

67

EXAMPLE

How many times more intense is an 80-dB sound than a 40-dB sound?

= 10logI

IologI

Io=

10

logI – logIo =

10

68

EXAMPLE

logI =

10+ logIo

logI1 = 8 +– 12 = –4 I1 = 10

–4

logI2 = 4 +– 12 = –8 I2 = 10

-8

69

EXAMPLE

Number of times greater = I1/I2

I1I2=10–4

10 -8= 104 = 10, 000

70

Beats

When two sound waves that are at nearly the same frequency interfere with each other, they form a beat pattern.

It is an amplitude variation.The beat frequency

21 fffbeat

71

The Doppler effect

When a source of sound is moving towards you, it sounds higher pitched (higher frequency).

When it moves away, it sounds lower pitched.

72

The Doppler Effect

S

S

L

L

vv

f

vv

f

The S’s stand for the source of the sound. The L’s stand for the listener. v by itself stands for the speed of sound. Be careful with the signs on your velocities!!

The direction from listener toward source is positive The direction from source toward listener is

negative