Chapter 13

Mechanical Waves

Propagation of A Disturbance

What are waves? Waves are traveling disturbances or

vibrations. Waves carry energy from place to place There are 2 types of waves:

– Transverse (example: light, electromagnetic waves) Disturbance is perpendicular to direction

of wave propagation– Longitudinal (example: sound waves)

Disturbance is parallel to direction of wave propagation

Propagation of A Disturbance

What are waves? Waves are traveling disturbances or

vibrations. Waves carry energy from place to place There are 2 types of waves:

– Mechanical waves Disturb and propagate through a medium

– Electromagnetic waves Do not require a medium in order to

propagate.

All mechanical waves require1. Some source of disturbance; 2. A medium;3. Some physical mechanism through which particles of the

medium can influence one another.

sourcecompressedstretched

P

The Wave Function

PPPP

P

A

v

(a) Pulse at t=0

(b) Pulse at time t

vt

y(x,t) = y (x-vt,0)Represent y for all positions and times, measured at O:

P at x at time t has the same y positon as an element located at x-vt at time t=0

vt

O

O

y(x,t) = f (x-vt),f(x) is the shape of the pulse

The Wave Function

Wave travels to the right

y(x,t) = f (x-vt)Wave travels to the left

y(x,t) = f (x+vt)The function y called the wave function,

depends on the two variables x and t. The wave function y(x,t) represents the y

coordinate of any point P located at position x at any time t.

The Wave Modely

y

x

t

T

AA

A

• One wavelength is the minimum distance between any two identical points on a wave.

• The period T of the wave is the minimum time it takes a particle of the medium to undergo one complete oscillation.

T = 1 / f• The amplitude A is the

maximum displacement of a particle of the medium from the equilibrium position.

• The wave speed v is the distance travelled by the wave in one second.

Properties of Periodic Waves

All waves share 3 properties:– The propagation speed of the wave, v– The spacial length of a wave from crest to crest is called

its wavelength, – The rate (or frequency) of vibration describes how fast a

wave oscillates, f The propagation of a wave is related to its wavelength & its

frequency:

v = = .f The speed of wave depends on the properties of the

material (medium) where the wave travels. For a wave on a string:

Where:F = tension in the stringm = mass of the stringL = length of the string

Lm

Fv

/

The Traveling Wave

A sinusoidal waveBlue t=0Red t

20, sint y A x

tat some later time

2siny A x vt

x

y

sin 2

vT

x ty A

T

The Traveling Wave

siny A kx t

2angular wave number

2angular frequency 2

k

fT

: phase constant, determined from the initial condit

s

s

n

i n

i

o

y A kx t

Assumed y=0 at x=0 and t=0

If y≠0 at x=0 and t=0

vk

v f

The Traveling WaveA=15.0cm;A wavelength 40.0cm;A frequency 8.00Hz;The vertical displacement of the medium at t=0 and x=0 is also 15.0cm.

y(cm)

x(cm)

15.0cmcm

2 2 rad0.157rad/cm

40.0cm1 1

0.1258.00

2 2 rad*8.00 50.3rad/s

8.00 *40.0cm=320cm/s

k

T sf Hz

f Hz

v f Hz

The Traveling WaveA=15.0cm;A wavelength 40.0cm;A frequency 8.00Hz;The vertical displacement of the medium at t=0 and x=0 is also 15.0cm.

y(cm)

x(cm)

15.0cmcm

sin

15.0 15.0sin

y A kx t

sin 1

2

15.0 sin 0.157 50.3 2

15.0 cos 0.157 50.3

y cm x t

cm x t

Reflection & Transmission of Waves

If the end is fixed then the pulse is reflected and undergoes a phase change of 180 degrees or half a period, so a crest become a trough and vice versa.

Reflected pulse is inversed.

Low

er

den

sity

Reflection & Transmission of Waves

If the end is free to move then reflection but no inversion takes place.

Hig

her

den

sity

Reflection & Transmission of Waves

Inversion will always be observed when the end of the medium is fixed. When two media are connected and the pulse travelling through a medium meets a denser medium it is partially reflected and partially transmitted. The reflected part is inverted as for the fixed end.

If instead the pulse meets a lighter medium it is partially reflected and partially transmitted but undergoes no inversion.

Applications

Ultrasound image

Sound Waves

Longitudinal wave Produced by vibrations in a medium

– The disturbance is the local change in pressure generated by the vibrating object

– It travels because of the molecular interactions. The region of increased pressure( compared to the

undisturbed pressure) is called condensation The region of lower pressure is called rarefaction. The maximum increase in pressure is the amplitude of

the pressure wave. (measurable) Frequency of the sound 20Hz to-20000Hz.

– Pressure waves below 20 Hz are called infrasonic waves

– Pressure waves over 20,000Hz are called ultrasonic waves.

The Speed of Sound

Speed of sound depends on the compressive properties of the medium.

Because of the high frequencies, the compression/expansions are fast and no heat is exchanged (adiabatic).

Sound can travel in gases, liquids and solids.

The speed of sound in gases:

The speed of sound in liquids:

The speed of sound in solids:

adiabBv

0m

kTv

Yv

The Speed of Sound

The speed of sound in gases depends on temperature. In 20 °C (68 °F) air at the sea level, the speed of sound is approximately 343 m/s (1,230 km/h; 767 mph) using the formula "v = (331 + 0.6 T) m/s".

In fresh water, also at 20 °C, the speed of sound is approximately 1,482 m/s (5,335 km/h; 3,315 mph).

In steel, the speed of sound is about 5,960 m/s (21,460 km/h; 13,330 mph).

The Doppler Effect

The frequency of the source producing the wave equals the number of cycles per second.

The frequency measured by an observer is the number of crests (condensations) encountered per second.

When both the source and the observer are at rest, the 2 frequencies are equal.

When one or both are in motion, the 2 frequencies are different.

The difference between source and observed frequency is called Doppler shift.

发射频率 s 接收频率 s ?

The Doppler Effect

Moving source, observer at rest The wavelength is changed because of the

relative motion of source and observerSource moving toward observer

Source moving away from the observer

observer ss

vf f

v v

observer ss

vf f

v v

Notes:

vs = speed of source

v = speed of sound

fs = frequency of source

The Doppler Effect

Moving observer, source at rest

The time between encountering 2 crests changes for observer

Observer moving toward source

Observer moving away from source

0observer s

v vf f

v

0observer s

v vf f

v

Notes:vo = speed of observerv = speed of soundfs = frequency of source

The Doppler Effect

The Doppler Effect

Red shift.The big bang

theory.

The Seismic Waves

P wave: longitudinal wave, higher speed S wave: transverse wave, lower speed

The Seismic Waves

Only P waves (yellow) can propagate in the liquid core.

The Seismic Waves

The Seismic Waves

P wave: longitudinal wave, higher speed

vp=9.1km/s S wave: transverse wave, lower speed

vs=3.7km/sIf the epicenter is l=20km bellow the

surface, you will have

t=l/vp-l/vs

=20/3.7 – 20/9.1 =3.21 second

to escape!

Earthquake Safety Rules

During the earthquake: Do not panic, keep calm. Douse all fires. If the earthquake catches you indoors, stay indoors. Take

cover under a sturdy piece of furniture. Stay away from glass, or loose hanging objects.

If you are outside, move away from buildings, steep slopes and utility wires.

If you are in a crowded place, do not rush for cover or to doorways.

If you are in a moving vehicle, stop as quickly as safety permits, but stay in the vehicle until the shaking stops.

If you are in a lift, get out of the lift as quickly as possible. If you are in a tunnel, move out of the tunnel to the open

as quickly as safety permits.

Earthquake Safety Rules

After the earthquake: 1. Check for casualties and seek assistance if needed. 2. If you suspect a gas leak, open windows and shut off the

main valve. Leave the building and report the gas leaks. Do not light a fire or use the telephone at the site.

3. Turn off the main valve if water supply is damaged. 4. Do not use the telephone except to report an emergency

or to obtain assistance. 5. Stay out of severely damaged buildings as aftershocks

may cause them to collapse. Report any building damage to the authorities.

6. As a precaution against tsunamis, stay away from shores, beaches and low-lying coastal areas. If you are there, move inland or to higher grounds. The upper floors of high, multi-storey, reinforced concrete building can provide safe refuge if there is no time to quickly move inland or to higher grounds.