Chapter 21 – Mechanical Waves A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University © 2007.

Chapter 21 – Mechanical Chapter 21 – Mechanical WavesWaves

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

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

• Demonstrate your understanding of Demonstrate your understanding of transversetransverse and and longitudinallongitudinal waves. waves.

• Define, relate and apply the concepts of Define, relate and apply the concepts of frequencyfrequency, , wavelengthwavelength, and , and wave speedwave speed..

• Solve problems involving Solve problems involving massmass, , lengthlength, , tensiontension, , and and wave velocitywave velocity for transverse waves. for transverse waves.

• Write and apply an expression for determining Write and apply an expression for determining the the characteristic frequenciescharacteristic frequencies for a vibrating for a vibrating string with fixed endpoints.string with fixed endpoints.

Mechanical WavesMechanical Waves

A A mechanical wavemechanical wave is a physical is a physical disturbance in an elastic medium.disturbance in an elastic medium.

Consider a stone dropped into a lakeConsider a stone dropped into a lake.

EnergyEnergy is transferred from stone to floating log, but is transferred from stone to floating log, but only the only the disturbancedisturbance travels. travels.

Actual motion of any individual water particle is small.Actual motion of any individual water particle is small.

Energy propagation via such a disturbance is known Energy propagation via such a disturbance is known as mechanical as mechanical wave motionwave motion..

Periodic MotionPeriodic MotionSimple periodic motionSimple periodic motion is that motion in is that motion in which a body moves back and forth over a which a body moves back and forth over a fixed path, returning to each position and fixed path, returning to each position and velocity after a definite interval of time.velocity after a definite interval of time.

AmplitudeA

PeriodPeriod, T, is the time for one complete oscillation. (seconds,s)(seconds,s)

PeriodPeriod, T, is the time for one complete oscillation. (seconds,s)(seconds,s)

FrequencyFrequency, f, is the number of complete oscillations per second. Hertz (sHertz (s-1-1))

FrequencyFrequency, f, is the number of complete oscillations per second. Hertz (sHertz (s-1-1))

1f

T

Review of Simple Review of Simple Harmonic MotionHarmonic Motion

x FF

It might be helpful for you to review Chapter 14 on Simple Harmonic Motion. Many of the same terms are used in this chapter.

Example:Example: The suspended mass makes The suspended mass makes 30 complete oscillations in 15 s. What 30 complete oscillations in 15 s. What is the period and frequency of the is the period and frequency of the motion?motion?

x FF

15 s0.50 s

30 cylcesT

Period: T = 0.500 sPeriod: T = 0.500 s

1 1

0.500 sf

T Frequency: f = 2.00 HzFrequency: f = 2.00 Hz

A Transverse WaveA Transverse Wave

In a transverse wave, the vibration of the individual particles of the medium is perpendicular to the direction of wave propagation.

In a transverse wave, the vibration of the individual particles of the medium is perpendicular to the direction of wave propagation.

Motion of particles

Motion of wave

Longitudinal WavesLongitudinal Waves

In a In a longitudinal wavelongitudinal wave, the vibration of the , the vibration of the individual particles is parallel to the individual particles is parallel to the direction of wave propagation.direction of wave propagation.

In a In a longitudinal wavelongitudinal wave, the vibration of the , the vibration of the individual particles is parallel to the individual particles is parallel to the direction of wave propagation.direction of wave propagation.

Motion of particles

Motion of wave

v

Water WavesWater Waves

An ocean wave is a combi-nation of transverse and longitudinal.

An ocean wave is a combi-nation of transverse and longitudinal.

The individual particles move in ellipses as the wave disturbance moves toward the shore.

The individual particles move in ellipses as the wave disturbance moves toward the shore.

Wave speed in a string.Wave speed in a string.

v = speed of the transverse wave (m/s)

F = tension on the string (N)

or m/L = mass per unit length (kg/m)

v = speed of the transverse wave (m/s)

F = tension on the string (N)

or m/L = mass per unit length (kg/m)

The wave speed The wave speed vv in in a vibrating string is a vibrating string is determined by the determined by the tension tension FF and the and the linear density linear density , or , or mass per unit length.mass per unit length.

The wave speed The wave speed vv in in a vibrating string is a vibrating string is determined by the determined by the tension tension FF and the and the linear density linear density , or , or mass per unit length.mass per unit length.

F FLv

m

L

= m/L

Example 1:Example 1: A A 5-g5-g section of string has section of string has a length of a length of 2 M2 M from the wall to the top from the wall to the top of a pulley. A of a pulley. A 200-g200-g mass hangs at the mass hangs at the end. What is the speed of a wave in end. What is the speed of a wave in this string? this string?

200 g

F = (0.20 kg)(9.8 m/s2) = 1.96 N

(1.96 N)(2 m)

0.005 kg

FLv

m v = 28.0 m/sv = 28.0 m/s

Note:Note: Be careful to use consistent Be careful to use consistent units. The tension units. The tension FF must be in must be in newtonsnewtons, the mass m in , the mass m in kilogramskilograms, and , and the length the length LL in in metersmeters..

Note:Note: Be careful to use consistent Be careful to use consistent units. The tension units. The tension FF must be in must be in newtonsnewtons, the mass m in , the mass m in kilogramskilograms, and , and the length the length LL in in metersmeters..

Periodic Wave MotionPeriodic Wave Motion

BA

Wavelength is distance between two particles that are in phase.

A vibrating metal plate produces a A vibrating metal plate produces a transverse continuous wave as transverse continuous wave as shown.shown.For one complete vibration, the wave For one complete vibration, the wave moves a distance of one moves a distance of one wavelength wavelength as illustrated.as illustrated.

Velocity and Wave Frequency.Velocity and Wave Frequency.

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

the wave speed is:the wave speed is:

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

the wave speed is:the wave speed is:

1 but so v T v f

T f

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

The The velocityvelocity of any wave is the of any wave is the product of the product of the frequencyfrequency and the and the

wavelengthwavelength::

v f

Production of a Longitudinal Production of a Longitudinal WaveWave

• An oscillating pendulum produces An oscillating pendulum produces condensations condensations and and rarefactionsrarefactions that travel that travel down the spring.down the spring.

• The The wave length lwave length l is the distance is the distance between adjacent condensations or between adjacent condensations or rarefactions.rarefactions.

Velocity, Wavelength, Velocity, Wavelength, SpeedSpeed

Frequency Frequency f f = waves = waves per second (Hz)per second (Hz)

VelocityVelocity v v (m/s) (m/s)

sv

t

Wavelength Wavelength (m) (m)

v f

Wave Wave equationequation

Example 2:Example 2: An electromagnetic An electromagnetic vibrator sends waves down a string. vibrator sends waves down a string. The vibrator makes The vibrator makes 600600 complete complete cycles in cycles in 5 s5 s. For one complete . For one complete vibration, the wave moves a distance of vibration, the wave moves a distance of 20 cm20 cm. What are the frequency, . What are the frequency, wavelength, and velocity of the wave?wavelength, and velocity of the wave?

600 cycles;

5 sf f = 120 Hzf = 120 Hz

The distance moved The distance moved during a time of one during a time of one

cycle is the wavelength; cycle is the wavelength; therefore:therefore:

= 0.020 m = 0.020 m

v = f

v = (120 Hz)(0.02 m)

v = 2.40 m/sv = 2.40 m/s

Energy of a Periodic WaveEnergy of a Periodic WaveThe The energy energy of a periodic wave in a string is of a periodic wave in a string is a function of the a function of the linear density mlinear density m , the , the frequency frequency f,f, the the velocity velocity vv, and the , and the amplitudeamplitude A A of the wave.of the wave.

f A

v

= m/L

2 2 22E

f AL

2 2 22P f A v

Example 3.Example 3. A A 2-m2-m string has a mass of string has a mass of 300 g300 g and vibrates with a frequency of and vibrates with a frequency of 20 Hz20 Hz and an and an amplitude of amplitude of 5050 mmmm. If the tension in the rope . If the tension in the rope is is 48 N48 N, how much power must be delivered to , how much power must be delivered to the string?the string?

0.30 kg0.150 kg/m

2 m

m

L

(48 N)17.9 m/s

0.15 kg/m

Fv

P = 2P = 222(20 Hz)(20 Hz)22(0.05 m)(0.05 m)22(0.15 kg/m)(17.9 m/s)(0.15 kg/m)(17.9 m/s)

2 2 22P f A v

P = 53.0 WP = 53.0 W

The Superposition The Superposition PrinciplePrinciple

• When two or more waves (When two or more waves (blueblue and and greengreen) ) exist in the same medium, each wave moves exist in the same medium, each wave moves as though the other were absent.as though the other were absent.

• The resultant displacement of these waves at The resultant displacement of these waves at any point is the algebraic sum (any point is the algebraic sum (yellowyellow) wave ) wave of the two displacements.of the two displacements.

Constructive Constructive InterferenceInterference

Destructive Destructive InterferenceInterference

Formation of Formation of a a Standing Standing Wave:Wave:Incident and reflected Incident and reflected waves traveling in waves traveling in opposite directions opposite directions produce nodes produce nodes NN and and antinodes antinodes AA..

The distance between The distance between alternate alternate nodes or anti-nodes or anti-nodes is one nodes is one wavelengthwavelength..

Possible Wavelengths for Standing Possible Wavelengths for Standing WavesWaves

Fundamental, n = 1

1st overtone, n = 2

2nd overtone, n = 3

3rd overtone, n = 4

2 1, 2, 3, . . .n

Ln

n

n = harmonics

Possible Frequencies Possible Frequencies f = v/f = v/::

Fundamental, n = 1

1st overtone, n = 2

2nd overtone, n = 3

3rd overtone, n = 4

1, 2, 3, . . .2n

nvf n

L

n = harmonics

f = 1/2L

f = 2/2L

f = 3/2L

f = 4/2L

f = n/2L

Characteristic FrequenciesCharacteristic Frequencies

Now, for a string under tension, we have:

; 1, 2, 3, . . .2n

n Ff n

L

and 2

F FL nvv f

m L

Characteristic frequencies:

Example 4.Example 4. A A 9-g9-g steel wire is steel wire is 2 m2 m long and is under a tension of long and is under a tension of 400 N400 N. . If the string vibrates in three loops, If the string vibrates in three loops, what is the frequency of the wave?what is the frequency of the wave?

400 N

For three loops: n = 3

; 32n

n Ff n

L

3

3 3 (400 N)(2 m)

2 2(2 m) 0.009 kg

FLf

L m

f3 = 224 HzThird harmonic 2nd overtone

Summary for Wave Summary for Wave Motion:Motion:

F FLv

m v f

2 2 22E

f AL

2 2 22P f A v

; 1, 2, 3, . . .2n

n Ff n

L

1f

T

CONCLUSION: Chapter 21CONCLUSION: Chapter 21Mechanical WavesMechanical Waves