Pipes and Standing Sound Waves

HW #5HW check tomorrow!

I. Waves on a String

The subject of standing waves has already been introduced with the introduction to waves packet. The results were generalized to standing waves on a taut string. The results were that only half integer amounts of wavelengths can fit the length of the string. This integer, n, is called the harmonic number. The wavelength, n, is 2L/n. Since the speed of the wave ties wavelength and frequency, v = f, the frequency of the nth harmonic is:

2n

nvf

L

The speed of the wave depends on the tension force, F, in the string and the linear mass density, . The linear mass density can be computed by dividing the mass of the string by the length of the string. The equation for the speed is:

Fv

2n

L

n n nv f

II. Sound Waves in a Pipe

Standing waves can also be set up in pipes, with generated sound waves interfering constructively and destructively with reflected waves from the other end. The details for the structure of the standing waves, namely the location of nodes and antinodes, depends on whether the pipe is open at both ends or closed at one end.

An open pipe is open to the atmosphere on both ends.

A closed pipe has one open end and one end closed to (or sealed from) the atmosphere.

Nodes and antinodes refer to the motion of air molecules carrying the sound waves in the tubes.

The structure of the waves can be characterized by either the motion of _____________________ or by fluctuations in _____________ . The properties of the two methods are one quarter of a wavelength different. Consider the vibrations of air molecules in the pipe below:

air moleculespressure

Where the air molecules have the greatest vibrations or amplitude, the pressure fluctuation is minimal. As a result, the location of an _______________ for the motion of air molecules is also the location for a __________ in the pressure fluctuations. Where the air molecules move the least, they are either squeezed by surrounding incoming air waves or pulled by surrounding outgoing molecules. Thus, __________ for the motion of air molecules corresponds to _______________ , or greatest fluctuation, in the pressure. The structure of the wave is usually given in terms of the motion of air molecules.

The conditions for a standing wave depend on whether the ends of the pipe are both open or one end closed. For the open end, the pressure must be the same as the surrounding atmosphere. For the closed end, the motion of the air molecules must be ________ . In terms of the motion of air molecules, an open end of a pipe must be an ________________ and a closed end must have a ___________ .

antinodenode

nodesantinodes

zero

antinodenode

For the open pipe, the results are essentially the same as for the taut string. You will have half integer number of wavelengths to the length of the pipe. If n is the harmonic number, then the wavelength and frequency of the nth harmonic is:

2n

L

n

2n

nvf

L

Here, the speed of the wave, v, is the speed of ____________ . If no speed or temperature is specified in the homework problems, assume v = 345 m/s.

sound

1,2,3n

For a pipe closed at one end, there must be an ____________ at the open end and a ___________ at the closed end. The minimum distance between a node and an antinode is _______________________ of a wavelength. The amount of a wavelength in a pipe is odd integer amounts of quarter waves, or:

antinodenode

one quarter

4n

L

n

4n

nvf

L 1,3,5n Thus, closed pipes only

have odd harmonics! Again, v is the speed of sound.

Ex. #1: The windpipe of a typical Apatosaurus is about 3.5 m long. What is the lowest resonant frequency of this pipe, assuming that it is closed at one end? Assume a temperature of 37°C.

4n

nvf

L 331 1

273Cm

air s

Tv

C

37331 1 352.7

273m m

air s s

Cv

C

1

1 352.7

4 3.5

msf

m 25.2Hz

Ex. #2: A pipe open at both ends has a fundamental frequency of 300 Hz when the temperature is 0°C.

(a) What is the length of the pipe?

2n

nvf

L 331 1

273Cm

air s

Tv

C

0331 1 331

273m m

air s s

Cv

C

1

1 3311

2 2 300

msv

Lf Hz

0.552m

(b) What is the fundamental frequency at a temperature of 30°C?

331 1273

Cmair s

Tv

C

30331 1 348.7

273m m

air s s

Cv

C

30 30

0 02C C

nC C

f vnvf

L f v

30

348.7300

331

ms

C ms

f Hz 316Hz

Ex. #3: The longest pipe used in cathedral organs is typically a closed pipe 10.55 m in length. (a) What is the fundamental frequency of this length of pipe? Assume 345 m/s for the speed of sound.

4n

nvf

L

1 345

4 10.55

ms

m 8.18Hz

(b) What would be the fundamental frequency of an open pipe of the same length?

2n

nvf

L

1 345

2 10.55

ms

m 16.4Hz

Which notes are these?

o 440Hz

o 261.63Hz

o 130.81Hz

o 65.41Hz

o 32.70Hz

o 16.35Hz

o 8.18Hz

III. Beats and Beat Frequency

When two waves of different frequency combine, the amplitudes interfere with one another. The result can be a rather complex wave. The effect can be heard readily with sound waves. If two sound waves interfere, a warble in the loudness can be heard.The closer the two frequencies, the longer and more pronounced the warble, or beat. The sound heard has a frequency equal to the average of the frequencies of the two waves, and the amplitude of the sound wave warbles or beats with a frequency equal to the difference between the frequencies of the two waves. This is the definition of the beat frequency.

1 2beatf f f

The pattern on the bottom has a longer duration (period), corresponding to a lower beat frequency.

The two frequencies for the bottom pattern are closer to each other than in the upper pattern.

Ex. #4: One clarinet is tuned to 440 Hz. When this clarinet and another sound together, a beat frequency of 4 Hz is produced. What are the possible frequencies of the other clarinet?

1 2beatf f f

24 440Hz Hz f

2 436 444f Hz or Hz

Ex. #5: Two open pipes have the same length of 378 cm and internal temperature of 22.0 °C. Both pipes are sounded at their 3rd harmonic at the same time. One of the pipes is accidentally exposed to direct sunlight, raising its internal temperature to 30.0 °C. What is the resulting beat frequency produced by the sounding of the two pipes?

331 1273

Cmair s

Tv

C

22.0331 1 344.1

273m m

air s s

Cv

C

30.0331 1 348.7

273m m

air s s

Cv

C

2n

nvf

L

3 344.1137

2 3.78

ms

coldf Hzm

3 348.7138.

2 3.78

ms

warmf Hzm

1 2 137 138beatf f f Hz Hz

1beatf Hz