Waves and Vibrations

Chapter 14

Waves are all around us in everyday life.

Sound is a Wave.

Radio, TV, and Cell Phones use waves.

Light is a wave.*

*But it can act as a particle…another story in Physics!

Main concepts

• Types of waves: – Transverse: waves on strings– Longitudinal: sound waves – water waves are more complex (combination)

• Relationship of wavelength, frequency and velocity of wave (f=v)

• Wave amplitudes can be added together.• Addition of waves leads to interference:

constructive or destructive

Motion of a Transverse Wave on a string

• Wave amplitude is y=Asin[2(x/ - ft)]

• If you sit at one location x, the wave oscillates in time.

• If you stop the action at a time t, the wave oscillates as a function of distance x.

The wave crest travels a distance in one period of time, 1/f. Thus the speed is the distance over the time, or

f=v

Motion of Longitudinal Wave

• Pressure wave• Oscillation of local pressure and gas density

Wavelength

Water waves combine motions

Complex motion: combination of transverse and longitudinal motion.

Light & Radio are Electro-magnetic Waves

Electric Field (and Magnetic Field) move TRANSVERSE to direction of propagation of energy.

++++++

------

Ant

enna

Key characteristic of these waves

• Energy (in the form of motion) can be transmitted by the wave

• The medium (the string, the air, the water) does not move at the speed of the wave—it essentially “stays put”

• The energy of the wave is transmitted through the medium from one piece of matter to another

• Note that light waves travel without the need for a medium at all!

Demonstrations

• Transverse waves (long spring)

• Transverse waves (tuning fork)

• Transverse waves (wave machine)

• Longitudinal wave/transverse wave (metal rod)

• Longitudinal wave (open tube)

• Longitudinal wave (recorder)

Superposition (addition) of waves

Wave amplitudes are added. They can get larger (constructive) or smaller (destructive) interference when they are superposed.

Wave interference

CONSTRUCTIVE DESTRUCTIVE

Demonstrations: waves on a rope.

• Reflection of wave at rigid wall

• Destructive interference

• Standing waves

Physlets Illustration 17.3 superposition of pulsesIllustration 17.4 superposition to make a standing wave.Exploration 17.4 superposition of two cosine waves to make a standing wave.

Addition of 2 waves that are close in frequency

Beat Frequency

Demonstration

• Beat frequency with tuning forks

Standing waves on strings

=2L f

=L 2f

First harmonic

Second harmonic

=2/3 L 3f Third harmonic

(one octave)

Standing waves in columns of air

4L 4/3 L 4/5L2L L 2/3L

Closed vs. open pipes.The closed pipe has a lower fundamental frequency.The closed pipe has only “odd” harmonics. The open pipe has odd and even.

An “octave” is a doubling of the frequency of a note. Our theory predicts a tube will produce a note one octave lower if it is closed off on one end. Try it!

A “harmonic” is a multiple of the fundamental frequency, f, 2f, 3f, etc.

f 3f 5f f 2f 3f

Intensity of Sound• Our perception of sound is that a sound with 10

times the intensity sounds TWICE as loud• To make it easier to compare sound levels, we

use the “decibel (dB)” scale

0

log10I

I

“Beta” is the “intensity LEVEL”. I is the “intensity”. Be careful. Intensity level (dB) is dimensionless. Intensity has units of power/area.

Various sound intensitiesLoudest sound produced in laboratory 109

Saturn V rocket at 50 m 108

Rupture of the eardrum 104

Jet engine at 50 m 10

Threshold of pain 1

Rock concert 10–1

Jackhammer at 1 m 10–3

Heavy street traffic 10–5

Conversation at 1 m 10–6

Classroom 10–7

Whisper at 1 m 10–10

Normal breathing 10–11

Threshold of hearing 10–12

120dB

0dB

50dB

20dB

110dB

dB scale of loudness

THESE ARE THE SAME:

1. Increase in sound intensity (P/A) of an order of magnitude.

2. Increase in intensity level (dB) of 10 units.

3. Double the “loudness”.

Intensity vs. distance from a point source

Sound is created at origin with power P.It gets spread over the area of an entire sphere of radius R.The sphere area is A=4r2.

Therefore, the Intensity, P/A, falls off like 1/r2.

R1

R2

P

Comparing sound levels

• The decibel (dB) is often used to compare sounds.

• The reference intensity, I0, is the weakest sound that can be heard.

Example:

A person talking has a sound level of about 50 dB. What is the sound level of 100 people talking?

dBI

I50log10

0

11

dB

I

I

I

I

70

log10100log10

*100log10

0

1

0

1100

The intensity level increases 10 dB for every 10 time increase in intensity.

Fix the noise!

• A factory has 50 machines that produce a total of 100 dB of noise. The Federal standard is that the total must be less than 90 dB.

• How many machines can you operate legally at one time?

You must reduce the total noise intensity level by 10 dB. This means a reduction in noise intensity of a factor of 10.

A: You must reduce the number of machines by 10, to 5!

Looked at another way….

Adding sound levels• Given that 50 machines produce a dB

level of 100, what is the dB level of one machine?

1

0

1

0

150

0.17

log1050log10

100*50

log10

dB

I

I

dBI

I

dB

dBdB

83

0.171001

Intensity and distance• You are standing 1

meter from a model rocket which takes off producing a sound level of about 85 dB. What is the sound level 100 meters away?

A: Using the 1/r2 law, the Intensity of the sound (P/A) 100 meters away is 104 times less. This means the sound level is reduced by 40 dB. The sound level is 45 dB at 100 meters.