Chapter 22A Chapter 22A –– Sound WavesSound WavesA PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

Objectives: After completion of this Objectives: After completion of this module, you should be able to:module, you should be able to:

•• Define Define soundsound and solve problems and solve problems relating to its velocity in solids, relating to its velocity in solids, liquids, and gases.liquids, and gases.

•• Use boundary conditions to apply Use boundary conditions to apply concepts relating to concepts relating to frequenciesfrequencies in in openopen and and closedclosed pipes.pipes.

Definition of SoundDefinition of Sound

Source of sound: Source of sound: a tuning fork.a tuning fork.

Sound is a longitudinal mechanical wave that travels through an elastic medium.

SoundSound is a longitudinal is a longitudinal mechanical wave that mechanical wave that travels through an travels through an elastic medium.elastic medium.

Many things vibrate in Many things vibrate in air, producing a sound air, producing a sound wave.wave.

Is there sound in the forest Is there sound in the forest when a tree falls?when a tree falls?

Based on our definition, Based on our definition, there there ISIS sound in the sound in the forest, whether a human forest, whether a human is there to hear it or not!is there to hear it or not!

Sound is a physical disturbance in an elastic medium.

Sound is a Sound is a physical physical disturbancedisturbance in an in an elastic medium.elastic medium.

The elastic medium (air) is required!

Sound Requires a MediumSound Requires a Medium

Evacuated Bell Jar

Batteries

Vacuum pump

The sound of a ringing bell diminishes as air leaves The sound of a ringing bell diminishes as air leaves the jar. No sound exists without air molecules.the jar. No sound exists without air molecules.

Graphing a Sound Wave.Graphing a Sound Wave.

Sound as a pressure wave

The sinusoidal variation of The sinusoidal variation of pressurepressure with with distancedistance is a is a useful way to represent a sound wave graphically. useful way to represent a sound wave graphically.

Note the Note the wavelengthswavelengths

defined by the figure.defined by the figure.

Factors That Determine the Speed of Sound.Factors That Determine the Speed of Sound.

Longitudinal mechanical waves (Longitudinal mechanical waves (soundsound) have a ) have a wave speed dependent on wave speed dependent on elasticityelasticity factors and factors and densitydensity factors. Consider the following examples:factors. Consider the following examples:

A A denserdenser medium has greater medium has greater inertia resulting in inertia resulting in lowerlower wave wave speeds.speeds.

A medium that is A medium that is more elasticmore elastic springs back quicker and springs back quicker and results in results in fasterfaster speeds.speeds.

steelsteel

waterwater

Speeds for different mediaSpeeds for different media

Yv

Metal rodMetal rodYoungYoung’’s modulus,s modulus, Y

Metal density,Metal density,

43B Sv

Extended Extended SolidSolid

Bulk modulus,Bulk modulus, B Shear modulus,Shear modulus, S Density,Density,

Bv

FluidFluid

Bulk modulus,Bulk modulus, B Fluid density,Fluid density,

Example 1:Example 1: Find the speed of Find the speed of sound in a steel rod.sound in a steel rod.

vs = ?

11

3

2.07 x 10 Pa7800 kg/m

Yv

v =5150 m/s

== 7800 kg/m7800 kg/m33

Y =Y = 2.07 x 102.07 x 1011 11 PaPa

Speed of Sound in AirSpeed of Sound in AirFor the speed of sound in air, we find that:For the speed of sound in air, we find that:

P RTB P andM

= 1.4 for air

R = 8.34 J/kg mol M = 29 kg/mol

B Pv

RTvM

Note: Sound velocity increases with temperature T.Note: Sound velocity increases with temperature T.

Example 2:Example 2: What is the speed of sound What is the speed of sound in air when the temperature is in air when the temperature is 202000CC??

-3

(1.4)(8.314 J/mol K)(293 K)29 x 10 kg/mol

RTvM

Given:

= 1.4; R = 8.314 J/mol K; M = 29 g/mol

T = 200 + 2730 = 293 K M = 29 x 10-3 kg/mol

v = 343 m/sv = 343 m/s

Dependence on TemperatureDependence on TemperatureRTvM

Note: Note: vv depends on depends on

Absolute TAbsolute T::

Now v at 273 K is 331 m/s. , R, M do not change, so do not change, so a simpler formula might be:a simpler formula might be:

T331 m/s273 K

v

Alternatively, there is the approximation using Alternatively, there is the approximation using 00C:C:

0m/s331 m/s 0.6C cv t

Example 3:Example 3: What is the What is the velocity of sound in air velocity of sound in air on a day when the tempon a day when the temp-- eratureerature is equal to 27is equal to 2700C?C?

Solution 1:Solution 1: T331 m/s273 K

v

300 K331 m/s273 K

v T = 270 + 2730 = 300 K;

v = 347 m/sv = 347 m/s

v = 347 m/sv = 347 m/sSolution 2:Solution 2: v = 331 m/s + (0.6)(270C);

Musical InstrumentsMusical Instruments

Sound waves in air Sound waves in air are produced by the are produced by the vibrations of a violin vibrations of a violin string. Characteristic string. Characteristic frequencies are based frequencies are based on the length, mass, on the length, mass, and tension of the and tension of the wire.wire.

Vibrating Air ColumnsVibrating Air ColumnsJust as for a vibrating string, there are characteristic wavelengths and frequencies for longitudinal sound waves. Boundary conditions apply for pipes:

Open pipeA A

The open end of a pipe must be a displacement antinode A.

Closed pipeAN

The closed end of a pipe must be a displacement node N.

Velocity and Wave Frequency.Velocity and Wave Frequency.

The period T is the time to move a distance of one wavelength. Therefore, the wave speed is:

1 but so v T v fT f

The frequency f is in s-1 or hertz (Hz).

The velocity of any wave is the product of the frequency and the wavelength:

v f vf

Possible Waves for Open PipePossible Waves for Open Pipe

L 21L Fundamental, n = 1

22L 1st Overtone, n = 2

23L 2nd Overtone, n = 3

24L 3rd Overtone, n = 4

All harmonics are possible for open pipes:

2 1, 2, 3, 4 . . .nL nn

Characteristic Frequencies Characteristic Frequencies for an Open Pipe.for an Open Pipe.

L 12vfL

Fundamental, n = 122

vfL

1st Overtone, n = 232

vfL

2nd Overtone, n = 342

vfL

3rd Overtone, n = 4

All harmonics are possible for open pipes: 1, 2, 3, 4 . . .

2nnvf nL

Possible Waves for Closed Pipe.Possible Waves for Closed Pipe.

141L Fundamental, n = 1

143L 1st Overtone, n = 3

145L 2nd Overtone, n = 5

147L 3rd Overtone, n = 7

Only the odd harmonics are allowed:

4 1, 3, 5, 7 . . .nL nn

L

Possible Waves for Closed Pipe.Possible Waves for Closed Pipe.

114vfL

Fundamental, n = 1

334

vfL

1st Overtone, n = 3

554

vfL

2nd Overtone, n = 5

774

vfL

3rd Overtone, n = 7

Only the odd harmonics are allowed:

L

1, 3, 5, 7 . . .4nnvf nL

Example 4.Example 4. What What lengthlength ofof closedclosed pipe is needed pipe is needed to resonate with a fundamental frequency of to resonate with a fundamental frequency of 256 256 HzHz? What is the ? What is the second overtonesecond overtone? Assume that ? Assume that the velocity of sound is the velocity of sound is 340 M/s340 M/s..

Closed pipeAN

L = ? 1, 3, 5, 7 . . .

4nnvf nL

11

(1) 340 m/s; 4 4 4(256 Hz)

v vf LL f

L = 33.2 cm

The second overtone occurs when n = 5:

f5 = 5f1 = 5(256 Hz) 2nd Ovt. = 1280 Hz2nd Ovt. = 1280 Hz

Summary of Formulas For Summary of Formulas For Speed of SoundSpeed of Sound

Yv

Solid rod43B Sv

Extended Solid

Bv

Liquid

RTvM

Sound for any gas:

0m/s331 m/s 0.6C cv t

Approximation Sound in Air:

Summary of Formulas (Cont.)Summary of Formulas (Cont.)

vf

v f For any wave:

Characteristic frequencies for open and closed pipes:

1, 2, 3, 4 . . .2nnvf nL

1, 3, 5, 7 . . .4nnvf nL

OPEN PIPE CLOSED PIPE

CONCLUSION: Chapter 22CONCLUSION: Chapter 22 Sound WavesSound Waves