Sound Waves

Sound Waves

A type of longitudinal wave that must travel through some medium (like air).

The speed of sound depends on the medium it is traveling through. It is faster through more dense materials.

ex. v

 345 m/s

v

 1500 m/s (in solids  even faster)

If a string is vibrated at exactly the right frequency, a crest moving toward one end and a reflected trough meet at some point.

2

3

Two opposing waves

(red & blue) combine

to form a standing

wave (black).

Antinodes exist midway between nodes at the point with the largest amplitudes.

(Constructive interference )

Nodes are points created where two waves

cancel out. The wave appears to stand still at this point. (Destructive interference)

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A standing wave. The red dots are the wave nodes.

L 2 λ



Distance from one node to the next  half of a wavelength

Standing Waves on a Vibrating String

L = length of string (m)

 = wavelength (m)

L

Fundamental Frequency (f

) or 1

Harmonic

Ends of String  can’t vibrate  nodes Center of String  most displacement  antinode

simplest vibration 

Since v = f, so f = v/, and f = v/2L  this

frequency is AKA OR

 = 2L

Examples: Guitars, Violins

 λ L

2

Harmonic (f

) 

3 nodes instead of 2

f

= 2f

L

- one full wavelength

Standing Waves on a Vibrating String

When a guitar player presses down on a guitar

string at any point  that point becomes a node  only a portion of the string vibrates

A single string can be used to create a variety of fundamental frequencies

2 L 3 λ



3

Harmonic (f

)

2 λ L 

4

Harmonic (f

)

f

= 4f

-1.5 wavelengths

L

L - 2 full wavelengths

f

= 3f

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Standing Waves on a Vibrating String

……and so on 8

2

Harmonic 1

Harmonic

3

Harmonic

4

Harmonic

The first harmonic frequency of a violin string is 440 Hz. Find the next two harmonic

frequencies (overtones) of this string.

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2

Harmonic: f

= 2f

= 2(440 Hz) = 880 Hz 3

Harmonic: f

= 3f

= 3(440 Hz) = 1320 Hz There are some points on a standing wave that never move. What are these points called?

1. Harmonics

2. Normal Modes 3. Nodes

4. Anti-nodes

5. Interference

The frequency of the third harmonic of a string is 1. one-third the frequency of the fundamental.

2. equal to the frequency of the fundamental.

3. three times the frequency of the fundamental.

4. nine times the frequency of the fundamental.

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An open-open tube of air supports standing waves at frequencies of 300 Hz and 400 Hz, and at no

frequencies between these two. The second harmonic of this tube has frequency.1. 800 Hz.

2. 600 Hz.

3. 400 Hz.

4. 200 Hz.

5. 100 Hz.

A 52.0 cm long guitar string has a

fundamental frequency of 444 Hz.

b) What should be the string’s length in order to produce a fundamental note of 650 Hz?

(assume the same speed of sound as in part a)

L = 52.0 cm = 0.52 m f₁ = 444 Hz v = ?

v = 2L

f

a) What is the speed of sound in the string according to this data?

v = 2(0.52m)

444 Hz v = 461.76 m/s

f₁ = 650 Hz

v = 461.76 m/s L = ?

L = v =

2f

461.76 m/s =

2(650 Hz) 0.355 m

A standing wave is set up on a 2 m long

string as shown below and has a frequency of 84 Hz.

a) What is the harmonic of this wave?

b) What is the wavelength of the wave?

c) What is the fundamental frequency of this wave?

7

harmonic L = 7  = 3.5 

2

m

 2

 = 0.57 m

f

= 7f

7

f₁  f⁷

= 12 Hz

7

Hz

 84

Standing Waves in an Air Column

Standing waves can also be set up in a tube of air.

Examples: inside a trumpet, column of a sax, pipes of an organ, many other brass instruments/woodwinds.

 standing waves are produced when waves traveling down the tube combine with reflected waves coming back up.

If both ends of a pipe are open all harmonics are present

Examples: Flutes, Pipe Organs

Exact opposite of a string fixed at both ends

Each end is an antinode

L

antinode

antinode node

But like the vibrating string  the entire harmonic series is present

Can use the same equations we used for vibrating string

If one end of a pipe is closed  only odd harmonics are present.

Standing Waves in an Air Column

Examples: Trumpets, Saxophones, Clarinets

Closed end restricts air molecule movement  node

Open end  antinode

L

node antinode

In this case, the wavelength of the standing wave equals four times the length of the pipe ( = 4L)

The fundamental frequency equals velocity divided by four times the pipe length (f₁ = v/ = v/4L)

So for, f₃ = v/₃ = v/^4/3L = 3v/4L =3f₁

Only odd-numbered harmonics vibrate in a pipe closed at one end.

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An Alp horn behaves like a pipe with one end closed. If the frequency of the fifteenth harmonic is 26.7 Hz, how long is the alphorn? The speed of sound in air is 334 m/s.

If a standing wave with a 19th harmonic is produced in an 8.6 m open pipe, what is its fundamental frequency?

The speed of sound in air is 334 m/s.

f₁₅ = 26.7 Hz v = 334 m/s L = ?

f₁₅ = 15f₁

f₁ = 26.7 Hz/15 f₁ = 1.78 Hz

 = v/f₁

 = 334^m/s/1.78^Hz

 = 187.64 m

For 1 end closed

L = /4

L = 187.64 ^m/4 L = 46.91 m

L = 8.6 m f₁ = ?

v = 334 m/s

For open pipe

L = 19/2

 = 2L/19

 = 0.905 m

 = v/f₁

f₁ = v/ = 334^m/s/0.905 ^m f₁ = 369.06 Hz

Find the fundamental frequency of an

open pipe of length 85 cm. The speed of sound in air is approximately 340 m/s.

1) Draw a picture.

2) List your known & unknown info.

L = 85 cm = 0.85 m v = 340 m/s

f₁ = ?

3) Identify the equation for the first harmonic (fundamental harmonic) 4) Solve for the fundamental frequency.

One object can cause another object to vibrate if the original object vibrates at the natural

vibrating frequency of the second object.

 This will increase the amplitude of the vibration of the first object

Examples: Pushing someone on a swing Singer breaking a glass

Tuning Forks microwave

Tacoma Narrows Bridge (WA)

Loma Prieta Earthquake (CA)

If a wave source and observer are moving relatively towards each other, a higher

frequency is heard, if they are moving relatively away from each other, a lower frequency is heard.

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Stationary source Moving source

Source moving at

speed of sound Source moving faster than speed of sound

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Is the source of the sound moving toward or away from you when you are standing at point:

A:

B:

C:

away away toward

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A police radar gun sends out a radar wave.

The returning wave comes back at a higher frequency. Is the car coming

closer or moving farther away? Explain how you know.

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Moving Closer  waves are getting squeezed because the distance between the 2 is

getting smaller; squeezed waves  shorter

wavelengths  higher frequencies

Monkey Joe watches a frog swimming

through the water on a pond. He notices that the frequency of the waves created by the frog is less than that of the

swimming stroke of the frog. If Monkey Joe had paid attention in Physics class, what conclusion about the frog’s motion could he draw? Explain.

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Moving Away  waves are getting stretched because the distance between the 2 is

getting longer; stretched waves  longer

wavelengths  lower frequencies

Amy and Zack are both listening to the source of sound waves that is moving to the right.

Compare the frequencies each hears.

1. f

< f

2. f

= f

3. f

> f

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Interference effect that results from the superposition of two waves with slightly

different frequencies.

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Beat Frequency

The number of beats (loud/soft) you hear per second.

High beat

frequency Low beat

frequency

As two sound waves get closer in frequency, the beat frequency decreases.

The 2 sound waves alternate between being in

phase (loud) and out-of-phase (soft) .

Two sound waves of nearly equal frequencies

are played simultaneously. What is the name of the acoustic phenomena you hear if you listen to these two waves?

1. Beats

2. Diffraction 3. Harmonics 4. Chords

5. Interference

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Hearing Nodes

1

2

If the sound from speaker 2 travels ½ of a wavelength farther (or 3/2 or 5/2) then we get destructive interference (node).

Notice that the sound waves from speaker 2 must travel farther.

27 produces a repetitive signal

These two loudspeakers are in phase. They emit equal-amplitude sound waves with a

wavelength of 1.0 m. At the point indicated, is the interference maximum constructive,

perfect destructive or something in between?

1. maximum constructive 2. perfect destructive

3. something in between

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