Chapter 21 Chapter 21 –– Mechanical WavesMechanical Waves

A 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:

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

Mechanical WavesMechanical WavesA 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 which a is that motion in which a body moves back and forth over a fixed path, body moves back and forth over a fixed path, returning to each position and velocity after a returning to each position and velocity after a definite interval of time.definite interval of time.

Amplitude A

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

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

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

FrequencyFrequency, f, is the number of complete oscillations per second. Hertz (sHertz (s--11))

1fT

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

x FF

15 s 0.50 s30 cylces

T

Period: T = 0.500 sPeriod: T = 0.500 s

1 10.500 s

fT

Frequency: f = 2.00 HzFrequency: f = 2.00 Hz

A Transverse WaveA Transverse WaveIn 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 longitudinal wave, the vibration of the individual particles is parallel to the 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 v in a vibrating string is determined by the tension F and the linear density , or 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 FLvm

L

= m/L

Example 1:Example 1: A A 55--gg section of string has a section of string has a length of length of 2 M2 M from the wall to the top of a from the wall to the top of a pulley. A pulley. A 200200--gg mass hangs at the end. mass hangs at the end. What is the speed of a wave in this string? What is the speed of a wave in this string?

200 g

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

(1.96 N)(2 m)0.005 kg

FLvm

v = 28.0 m/sv = 28.0 m/s

Note: Be careful to use consistent units. The tension F must be in newtons, the mass m in kilograms, and the length L in meters.

Note:Note: Be careful to use consistent units. Be careful to use consistent units. The tension The tension FF must be in must be in newtonsnewtons, the mass , the mass m in m in kilogramskilograms, and the length , and 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 shown.transverse continuous wave as shown.

For one complete vibration, the wave moves For one complete vibration, the wave moves a distance of one a distance of one wavelength wavelength

as illustrated.as illustrated.

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

1 but so v T v fT f

The The frequency frequency ff is in sis in s--11 or or hertz (Hz)hertz (Hz)..

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

v f

Production of a Longitudinal WaveProduction of a Longitudinal Wave

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

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

Velocity, Wavelength, SpeedVelocity, Wavelength, Speed

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

VelocityVelocity vv (m/s)(m/s)

svt

Wavelength Wavelength

(m)(m)

v f

Wave equationWave equation

Example 2:Example 2: An electromagnetic vibrator An electromagnetic vibrator sends waves down a string. The vibrator sends waves down a string. The vibrator makes makes 600600 complete cycles in complete cycles in 5 s5 s. For one . For one complete vibration, the wave moves a complete vibration, the wave moves a distance of 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 s

f f = 120 Hzf = 120 Hz

The distance moved during The distance moved during a time of one cycle is the a time of one cycle is the wavelength; therefore:wavelength; 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 Wave

The The energy energy of a periodic wave in a string is a of a periodic wave in a string is a function of the 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 22--mm string has a mass of string has a mass of 300 g300 g and and vibrates with a frequency of vibrates with a frequency of 20 Hz20 Hz and an amplitude and an amplitude of of 5050 mmmm. If the tension in the rope is . If the tension in the rope is 48 N48 N, how , how much power must be delivered to the string?much power must be delivered to the string?

0.30 kg 0.150 kg/m2 m

mL

(48 N) 17.9 m/s0.15 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 PrincipleThe Superposition Principle•• When two or more waves (When two or more waves (blueblue and and greengreen) exist in ) exist in

the same medium, each wave moves as though the the same medium, each wave moves as though the other were absent.other were absent.

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

Constructive InterferenceConstructive Interference Destructive InterferenceDestructive Interference

Formation of a Formation of a Standing Wave:Standing 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 antinodes or anti-- nodes is one nodes is one wavelengthwavelength..

Possible Wavelengths for Standing WavesPossible Wavelengths for Standing Waves

Fundamental, n = 1

1st overtone, n = 2

2nd overtone, n = 3

3rd overtone, n = 4

2 1, 2, 3, . . .nL n

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, . . .2nnvf 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, . . .2nn Ff nL

and 2

F FL nvv fm L

Characteristic frequencies:

Example 4.Example 4. A A 99--gg steel wire is steel wire is 2 m2 m long long and is under a tension of and is under a tension of 400 N400 N. If the . If the string vibrates in three loops, what is the string vibrates in three loops, what is the frequency of the wave?frequency of the wave?

400 N

For three loops: n = 3

; 32nn Ff nL

33 3 (400 N )(2 m )

2 2(2 m ) 0.009 kgFLf

L m

f3 = 224 HzThird harmonic 2nd overtone

Summary for Wave Motion:Summary for Wave Motion:

F FLvm

v f

2 2 22E f AL

2 2 22P f A v

; 1, 2, 3, . . .2nn Ff nL

1fT

CONCLUSION: Chapter 21CONCLUSION: Chapter 21 Mechanical WavesMechanical Waves