L 5Review of Standing Waves on a

String

Below is a picture of a standing wave on a 30 meter long string.

What is the wavelength of the running waves that the standing wave is made from?

30 mA. 30 m

B. 60 m

C. 10 m

D. 20 m

E. Impossible to tell

<ct.10.1.4>

A. Yes, n = 1

B. Yes, n = 2

C. Yes, n = 3

D. Yes, n = 4

E. No

Could you observe standing waves made from running waves with a wavelength of 2/3 m on a string of length 1 m?

(If so, what mode would that be? )

Ct 10.1.5

A string vibrates with a fundamental frequency of 220 Hz.Besides 220 Hz, which of the following are "resonant frequencies" you might also observe?

i) 110 Hzii) 330 Hziii) 440 Hz

A: i onlyB: ii onlyC: iii onlyD: i and iiE: all three

<ct.10.1.6>

If the tension is increased by a factor of 9 what happens to the speed of waves on a string?

A. Goes up by a factor of 3B. Goes up by a factor of 4.5C. Goes up by a factor of 9D. Goes up by a factor of 81E. None of these / I don’t know

<ct.10.1.8b>

T

Tv

What happens to the frequency of the fundamental?

If you want to lower the pitch of a string by two octaves, what must be done to its tension?

A. Raise it by a factor of 4

B. Lower it by a factor of 4

C. Lower it by a factor of 2

D. Lower it by a factor of 16

E. None of these

<ct.10.1.8b>

A string on an instrument plays an A (440 Hz) when plucked. If you lightly touch the string ½ way from one end, and then pluck, you are mostly likely to hear…

A: Still 440 Hz

B: 220 Hz

C: 880 Hz

D: Something entirely different

ct.10.1.10a

L

i

iiCt 10.1.4b

Which of the two points on the string oscillates with the LARGER (higher) frequency?

A) Left point (i)

B) Right point (ii)

C) They both have the same frequency

Below is a picture of a standing wave on a 30 meter long string.

What is the wavelength of running waves that the standing wave is made from?

30 m

A. 30 m

B. 60 m

C. 15 m

D. Impossible to tell

<ct.10.1.3>

Sound Waves Frequency, Harmonics,

Tone Quality (spectral content),Pitch

The “Sonic” SpectrumInfrasound:

< 20 HzSound:

20 Hz – 20,000 Hz (20kHz)Ultrasound:

> 20 kHz (~1013 Hz maximum)

Sound- a Pressure WavePhET Simulation “Wave

Interference” (Sound, Particles)

Components of Sound1) Longitudinal (along direction of

propagation) vibrations, e.g. speaker cone.

2) Material medium capable of transmission of these vibrations,

e.g. air.3) Detector of the sound wave

e.g.ear.

Longitudinal wave representation

Longitudinal wave propagation

Pressure wave amplitudeabout 10-5 atmosphere

Displacement wave amplitudeabout 10-7 m

We define 1 N/m2 to be 1 Pascal (Pa)

In air at atmospheric pressure and at 20 degrees Celsius, the speed of sound, v,

is 344 m/s

v is temperature dependent, V = 331 m/s +0.6 T,

where T is the temperature in degrees Celsius above freezing, i.e. above 0

degrees Celsius

What happens when an object exceeds the speed of sound?

Standing Sound Waves

Cylindrical tube Open at both ends

(a flute, more or less)

Easy to get overpressure in middle(Ends are just open atmosphere…)

L

http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html

overpressure

L

open tubeoverpressure

n=2, the 2nd mode of the tube

L

Displacement (not pressure) graphs. open tube

displacement

L

• Displacement is longitudinal (despite the graph going “up”)

• Pressure nodes <=> displacement antinodes (and vice versa)

“Open” TubeFrequencies and Wavelengths

f = v / λN = 1 λ = 2Lo f = v/2Lo = fo

N = 2 λ = Lo f = v/Lo = 2fo

N = 3 λ = 2Lo/3 f = 3v/2Lo = 3fo N = 4 λ = 2Lo/4 f = 4v/2Lo = 4fo N = 5 λ = 2Lo/5 f = 5v/2Lo = 5fo N = 6 λ = 2Lo/6 f = 6v/2Lo = 6fo

Pressure waves “fit” in the open tube

n (/2) = LSince f = v,

fn= n (v/2L) Same modes as a string!

Note that in the case of the string, the “v” is the speed of the wave moving down the string. Here v is the speed of the wave motion through the medium, i.e. the speed of SOUND.

CT 12.1. 1

How will the normal mode frequencies of an open tube compare with those of a string (with the same fundamental frequency)?

A) All different frequencies (except fundamental)

B) All the same frequencies

C) Some of the overtones will be the same and some different

CT 12.1.1c

The air in an open pipe is in the n=2 mode (shown above). A small speck of dust is located 1/2 of the way down the pipe. What does the dust do ?A) Wiggles up and down (towards /away from wall of tube)B) Wiggles back and forth (left/right, along the tube…)C) Sit still at center of the pipeD) Something else

Standing pressure wavehttp:/ www.physics.smu.edu/~olness/www/03fall1320/applet/pipe-waves.html

CT 12.1.1b

What is “v” in the formula f = v for a pipe whose both ends are open to the air?

A) The speed of sound in air, 344 m/sB) Speed of vibrations of the pipe wallC) Related to speed of sound, but depends of pipe diameter

CT 12.1.1d

The speed of sound in helium gas is considerably higher than 344 m/s. If I fill a tube with helium, what will happen to the fundamental tone produced by that tube?

A) Goes up in pitchB) Goes down in pitchC) Stays about the same

real tubes - end effectoverpressure

Outer node is a bit outside tube(about 0.3 * diameter)

L

Tube closed at one end, open at the other

Closed tubes(closed on one end)

overpressure

Closed end: pressure antinode

open end: pressure node

L

Closed tubes(closed on one end)

overpressure

Closed end: antinode

open end:node

L

CT 12.1.3

What is the wavelength of the fundamental (shown above) in a closed tube?

A) =L B) =2LC) =4L D) =L/2E) =L/4

L

Draw the next higher mode (zero at right end, antinode at left, one extra node in middle)

overpressure

Closed end: antinode

open end:node

L

Draw the next higher mode (zero at right end, antinode at left, one extra node in middle)

overpressure

Closed end: antinode

open end:node

L

CT 12.1.4

What is the wavelength of thestanding wave (shown above) in a closed tube?

A) =L B) =3L/2C) =3L/4 D) =4L/3E) Something else

L

Pressure waves “fit” in the closed tube differently:

(odd n)·(/4) = LSince f = v,

fn= (odd n)·(v/4L) Lower fundamental Missing harmonics

CT 12.1.4c

A panpipe tube is sealed at one end but open at the other.

A flute is open at both ends.

If you have a panpipe and fluteof equal lengths, and play the fundamental…

A) The flute will sound lowerB) The panpipe will sound lowerC) They will have identical pitch

A flute playing its lowest note is shown in spectrum “A”. f1 refers to the fundamental of the flute. Which spectrum best matches a panpipe (of equal length) playing its lowest note?

CT12.1.4b

A

f1,flute

ampl

itude B

f1,flute

ampl

itude

C

f1.flute

ampl

itude

6f1

6f1

6f1

D

f1,flute

ampl

itude

6f1

E) None of these looks right.

A flute playing its lowest note is shown in spectrum “A”. f1 refers to the fundamental of the flute. Which spectrum best matches a panpipe half as long as the flute?

CT12.1.4b

A

f1,flute

ampl

itude B

f1,flute

ampl

itude

C

f1.flute

ampl

itude

6f1

6f1

6f1

D

f1,flute

ampl

itude

6f1

E) None of these looks right.

CT 12.1. 1

How will the harmonics of an open tube compare with those of a stringed instrument (with the same fundamental?)A) Totally different frequenciesB) The same frequencies

Fourier SynthesisAny periodic complex wave

can be synthesized by addition of its harmonics,

each with the proper amplitude and phase

First and second harmonicswith equal amplitudes

First and Second Harmonics

First and third harmonicswith equal amplitudes

First and Third Harmonics

First and second harmonicswith unequal amplitudes

First and Second Harmonics

Synthesis with missing fundamenta

l

2nd and 3rd

harmonics

3rd and 4th

harmonics

Triangular Wave

AN = 1, 0, 1/9, 0, 1/25, 0, 1/49, …. Odd N only

Square Wave

AN = 1, 0, 1/3, 0, 1/5, 0, 1/7, …. Odd N only

Sawtooth Wave

AN = 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, …. All N

White Light and White Noise

“Noise”

Blowing gently across a microphone

http://phet.colorado.edu/simulations/sims.php?sim=Fourier_Making_Waves