Lesson 1: VibrationsVibrational MotionProperties of periodic motionPendulum Motion

Waves - Chapter Outline

Lesson 2: The Nature of a Wave

PulsesWaves and Wavelike Motion What is a Wave? Categories of Waves

Lesson 3: Properties of a WaveThe Anatomy of a Wave Frequency and Period of a Wave Energy Transport and the Amplitude of a Wave The Speed of a Wave The Wave Equation

Lesson 4: Behavior of WavesBoundary Behavior Reflection, Refraction, and Diffraction Interference of WavesStanding WavesThe Doppler Effect

Lesson 1 - objectives Know: - Period of a pendulum depends only on length. - Period of a mass on a spring depends on both mass and spring

constant.

Understand: - Relationship between period of a pendulum and its mass, release

angle, and length. - Relationship between energy, position, and speed of an oscillating

system.

Be able to: - Calculate the period of a pendulum and mass on spring system. - Determine where PE and KE are maximum or minimum in an

oscillating system

Vibrational Motion• A vibrational motion is a back and forth motion.

• All vibrational motion has

1. Resting position or equilibrium position. At this position, the forces are balanced.

2. To make an object vibrate, a force must be applied to the object.

3. The object does not stop at equilibrium position because of inertia.

4. As the object reaches its maximum displacement, it stops momentarily before it moves back. This is because the object experiences a force which slows it down. This force is known as a restoring force.

5. Damping is the tendency of a vibrating object to lose its energy over time. A sustained input of energy would be required to keep the back and forth motion going.

Properties of Periodic Motion

• A vibrating object is moving over the same path over the course of time.

• The time it takes to complete one back and forth cycle is always the same amount of time.

• In Physics, a motion that is regular and repeating is referred to as a periodic motion.

The Sinusoidal Nature of a Vibration• If we collect the data on position and time of mass on a

spring and graph position vs. time, the graph would look like this:

1. The graph has the shape of a sine wave - periodic. The motion repeats itself in a regular fashion.

2. A full cycle of vibration is the movement of the mass from its resting position (A) to its maximum height (B), back down past its resting position (C) to its minimum position (D), and then back to its resting position (E). From the graph, one can determine that the time to complete one cycle of vibration is NOT changing.

3. Damping occurs with the mass-spring system, the amount of displacement of the mass at its maximum and minimum height is decreasing from one cycle to the next.

Period and Frequency

• The key measurements on the mass on a spring that have been made are measurements of:– the time for the mass to complete a cycle– the maximum displacement of the mass above (or below) the

resting position. • These two measurable quantities have names. We call these

quantities period and amplitude.• The period of the object's periodic motion is defined as the time

for the object to complete one full cycle. The standard metric unit for period is the second.

• The amplitude is the maximum displacement from resting position.

• An object in periodic motion can have a long period or a short period.

• Objects that have a relatively short period (i.e., a low value for period) are said to have a high frequency.

• The frequency describes how often an object vibrates, it is defined as the number of complete cycles occurring per period of time. Since the standard metric unit of time is the second, frequency has units of cycles/second or Hertz (Hz).

• The unit Hertz is used in honor of Heinrich Rudolf Hertz, a 19th century physicist who expanded our understanding of the electromagnetic theory of light waves.

• Frequency describe how often something occurs.

• High frequency events occur often, with little time in between each occurrence. .

• The two quantities frequency and period are inversely related to each other. They are mathematical reciprocals of each other.

Period = the time for one full cycle to complete itself; i.e., seconds/cycle Frequency = the number of cycles completed second; i.e., cycles/second

• The amplitude is defined as the maximum displacement of an object from its resting position. The resting position is that position assumed by the object when not vibrating.

• Over the course of time, the amplitude of a vibrating object tends to become less and less. The amplitude of motion is a reflection of the quantity of energy possessed by the vibrating object.

Amplitude of Vibration

Check Your Understanding1. A pendulum is observed to complete 23 full cycles in 58

seconds. Determine the period and the frequency of the pendulum.

2. A mass is tied to a spring and begins vibrating periodically. The distance between its highest and its lowest position is 38 cm. What is the amplitude of the vibrations?

Pendulum Motion•A simple pendulum consists a bob (mass) attached to a string of negligible mass. •There are two dominant forces acting upon a pendulum bob at all times during the course of its motion. There is the force of gravity and there is a tension force. We will ignore the influence of air resistance.•The gravity is always in the same direction (down) and always of the same magnitude - m∙g. However, both the direction and magnitude of the tension force change as the bob swings to and fro. •The net force is the restoring force which causes the pendulum’s periodic motion.

FTen

Fgrav

Fnet

The Sinusoidal Nature of Pendulum Motion • The resulting position vs. time plot of a pendulum look

similar to what was observed for the mass on a spring, the position of the pendulum bob (measured along the arc relative to its rest position) is a function of the sine of the time.

1. The graph is periodic. The period is NOT changing.

2. Damping occurs. The amount of displacement from equilibrium position decreases from one cycle to the next.

Energy Analysis• As the bob of a pendulum moves from one end to the other

end, there is a transformation of potential energy into kinetic energy and vise versa. However, the total amount of these two forms of energy, the total mechanical energy remains constant.

Check Your Understanding 1. A pendulum bob is pulled back to position A and released from rest.

The bob swings through its usual circular arc and is caught at position C. Determine the position (A, B, C or all the same) where the …

a. force of gravity is the greatest.b. restoring force is the greatest.c. speed is the greatest.d. potential energy is the greatest kinetic energy is the greatest.e. total mechanical energy is the greatest.

A,C

2. Use energy conservation to fill in the blanks in the following diagram.

The Period of a Pendulum• The period is the time it takes for a vibrating object to

complete one cycle. In the case of pendulum, it is the time for the pendulum to start at one extreme, travel to the opposite extreme, and then return to the original location.

• variables that might affect the period of a pendulum: – the mass of the pendulum bob, – the length of the string on which it hangs, – and the angular displacement (amplitude). The angular

displacement or arc angle is the angle that the string makes with the vertical when released from rest.

• For amplitude that less than 15o, the period of a simple pendulum is independent of mass and amplitude (angle).

• the period is directly proportional to the square root of the length of the string.

• ________________Where l is the length of the string in meters, T is the period in seconds,

g = 9.81 m/s2

T = 2π√l / g

graphs

Length (m)

perio

d

Period vs. length

perio

d√Length (m)

Period vs. √length

example

• You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12 s. how tall is the tower?

Mass on a Spring

• What are the controllable factors in creating a mass/spring system?– SPRING CONSTANT– MASS– RELEASE POSITION

• Period of a mass on a spring is the time for one complete oscillation.

• How do the controllable factors change the period?– Inversely proportional to SPRING CONSTANT– Proportional to MASS

Results

g

LT 2

k

mT 2

Period of Pendulum

Period of Mass on Spring

Lesson 2 - objectives Know: - Definition of a pulse and a wave - Wave transmit energy, but not matter. - Wave lose energy as they travel. - Definition of transverse and longitudinal waves. - Definition of mechanical and electromagnetic waves

Understand: – Relationship between energy and amplitude.

Be able to: – Determine direction of medium.

Pulse - Wave Building Blocks• A pulse is a single vibratory disturbance that

transfers energy but NOT mass.

The wave is passed from left to right from person to person, but the people did not move from left to right, people only moved up and down.

• Amplitude is the height of a pulse above equilibrium – shows ENERGY

crest

trough

• A medium is a continuum of particles of a single type– ex: Water, air, metal rod, etc.

• The medium stores then releases energy as pulses travel through it.

Waves and Wavelike Motion

A wave is started with a vibration. The vibration travels from one location to another.

WAVE – a regularly repeating pulse

Wave on a String 2.01

• Examples of waves:– microwaves, light waves, radio waves, and

sound waves.

• Wave carry energy from one location to another.

What is a wave?• A wave is a created by a vibration which caused a disturbance that

travels through a medium from one location to another location.

• A medium is a substance or material which carries the wave. A medium could be slinky, air, water, or people.

• If the first particle is given a single back-and-forth vibration, then the disturbance through the medium is called a pulse. A pulse is a single disturbance moving through a medium from one location to another location.

• If the first particle is continuously and periodically vibrated in a back-and-forth manner, a repeating disturbance is moving within the medium over some prolonged period of time. The repeating and periodic disturbance which moves through a medium from one location to another is referred to as a wave.

• Waves transport energy from one location to another. Wave lose energy as it travels, but it does not lose speed. Waves do not transport matter.

example

• A characteristic common to sound waves and light waves is that they

1. are longitudinal

2. are transverse

3. transfer energy

4. travel in a vacuum

Direction of the medium - example• As shown in the diagram, a transverse

wave is moving along a rope. In which direction will segment X move as the wave passes through it?

1. down, only

2. up, only

3. down, then up

4. up, then down

Direction of the medium - example

• The diagram shows a water wave moving in the direction shown by velocity vector v.       

• At the instant shown, a cork at point P on the water's surface is moving toward

1.A 2.B 3.C 4.D

Direction of the medium - example

In the next instant of time, indicate

1. The direction of motion of point A.

2. The direction of motion of point B.

3. The direction of motion of point C.

4. The direction of motion of point D.

Direction of the medium - example• The diagram below represents a transverse

wave traveling to the right through a medium. Point A represents a particle of the medium. In which direction will particle A move in the next instant of time?

down

• All waves are caused by a vibrating object.• For example:

– Sound waves, which is disturbance in the air, can be generated by a vibrating tuning fork, or vibrating vocal cord.

– Electromagnetic waves, such as radio waves, is produced by accelerating electrons.

example• The diagram shows an antenna emitting an

electromagnetic wave.  In what way did the electrons in the antenna produce the electromagnetic wave?

1. by remaining stationary

2. by moving at a constant speed upward, only

3. by moving at a constant speed downward, only

4. by accelerating alternately upward and downward

Check Your Understanding1. TRUE or FALSE:

In order for John to hear Jill, air molecules must move from the lips of Jill to the ears of John.

    2. Curly and Moe are conducting a wave experiment using a

slinky. Curly introduces a disturbance into the slinky by giving it a quick back and forth jerk. Moe places his cheek (facial) at the opposite end of the slinky. Using the terminology of this unit, describe what Moe experiences as the pulse reaches the other end of the slinky.

Categories of Waves - Longitudinal vs. Transverse Waves

• A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction which the wave moves. As the energy is transported from left to right, the individual coils of the medium will be displaced upwards and downwards.

• A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction which the wave moves. As the energy is transported from left to right, the individual coils of the medium will be displaced leftwards and rightwards.

•One way to categorize waves is on the basis of the direction of movement of the individual particles of the medium relative to the direction which the waves travel.

transverse wave

longitudinal wave

Examples of Longitudinal and transverse wave

• Longitudinal: – Sound wave– Wave on slinky

• Transverse: – Wave on a string– Stadium wave– Wave on a slinky– Electromagnetic waves (light, radio, microwaves,

UV rays, etc)

A sound wave is a Longitudinal wave• A sound wave traveling through air is a classic example of a

longitudinal wave. As a sound wave moves from the lips of a speaker to the ear of a listener, particles of air vibrate back and forth parallel to the direction of energy transport.

Categories of Waves - Electromagnetic versus Mechanical Waves

• Another way to categorize waves is on the basis of their ability or inability to transmit energy through a vacuum (i.e., empty space). Categorizing waves on this basis leads to two notable categories: electromagnetic waves and mechanical waves.

electromagnetic wave• An electromagnetic wave is a wave which is capable of transmitting

its energy through a vacuum (i.e., empty space).

• Electromagnetic waves are produced by the vibration of charged particles. This vibration creates a wave which has both an electric and a magnetic component. Electric field generate magnetic field and the magnetic field in turn, generate more electric field.

• Electromagnetic waves produced on the sun subsequently travel to Earth through the vacuum of outer space. All light waves are examples of electromagnetic waves.

• An electromagnetic wave transports its energy through a vacuum at a speed of 3.00 x 108 m/s (a speed value commonly represented by the symbol c). The propagation of an electromagnetic wave through a material medium occurs at a net speed which is less than 3.00 x 108 m/s.

• Electromagnetic waves are also transverse waves.

mechanical wave• A mechanical wave is a wave which is not capable of transmitting

its energy through a vacuum. Mechanical waves require a medium in order to transport their energy from one location to another.

• A sound wave is an example of a mechanical and longitudinal wave. Sound waves are incapable of traveling through a vacuum.

• Slinky waves, water waves, stadium waves, and waves on a string are other examples of mechanical waves; each requires some medium in order to exist. A slinky wave requires the coils of the slinky; a water wave requires water; a stadium wave requires fans in a stadium; and a jump rope wave requires a jump rope.

A summary of categories of waves• Category by medium

– Electromagnetic (light): travel through vacuum– Mechanical (all other waves): travel through matter

• Category by direction:– Transverse (wave on a string, ocean wave, stadium wave)– Longitudinal (sound wave, slinky wave)

• There are other categories as well.• Light waves are electromagnetic and transverse

waves.• Sound waves are mechanical and longitudinal

waves.

example• A student plucks a guitar string and the

vibrations produce a sound wave with a frequency of 650 hertz. The sound wave produced can best be described as a

1. transverse wave of constant amplitude

2. longitudinal wave of constant frequency

3. mechanical wave of varying frequency

4. electromagnetic wave of varying wavelengths

example• An electric bell connected to a battery is sealed

inside a large jar. What happens as the air is removed from the jar?

1. The electric circuit stops working because electromagnetic radiation cannot travel through a vacuum.

2. The bell's pitch decreases because the frequency of the sound waves is lower in a vacuum than in air.

3. The bell's loudness increases because of decreased air resistance.

4. The bell's loudness decreases because sound waves cannot travel through a vacuum.

example

• Which pair of terms best describes light waves traveling from the Sun to Earth?

1. electromagnetic and transverse

2. electromagnetic and longitudinal

3. mechanical and transverse

4. mechanical and longitudinal

Lesson 3 - Properties of Waves objectives

• The Anatomy of a Wave

• Frequency and Period of a Wave

• Energy Transport and the Amplitude of a Wave

• The Speed of a Wave

• The Wave Equation

Lesson 3 - objectives Know: - Definition of amplitude, wavelength, frequency, period, phase.- Wave speed equation.

Understand: - Relationship between frequency and period.- Relationship between frequency, wavelength, and speed - Relationship between energy and amplitude. - Relationship between speed and medium.

Be able to: - Use the wave speed equation to calculate speed, wavelength, or period. - Recognize or interpret diagrams of waves. - Draw diagrams of transverse waves. - Recognize or draw points on transverse waves that are in or out of phase.

The Anatomy of a transverse wave• A transverse wave is a wave in which the particles of the

medium are displaced in a direction perpendicular to the direction of energy transport.

• A transverse wave can be created in a rope.

The dashed line drawn through the center of the diagram represents the equilibrium or rest position of the string. This is the position that the string would assume if there were no disturbance moving through it.

Crest, trough and amplitude

• Crest: the highest point in a waveform.

• Trough: the lowest point in a waveform

The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. In a sense, the amplitude is the distance from rest to crest. Similarly, the amplitude can be measured from the rest position to the trough position.

Energy Transport and the Amplitude of a Wave

• A wave is an energy transport phenomenon which transports energy along a medium without transporting matter.

• The amount of energy carried by a wave is related to the amplitude of the wave. A high energy wave is characterized by a high amplitude; a low energy wave is characterized by a low amplitude.

• The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position.

• Find the amplitude of wave

1. A

2. B

3. C

example

wavelength• The wavelength of a wave the length of one complete wave

cycle. • The wavelength can be measured as the distance from crest to

crest or from trough to trough. In fact, the wavelength of a wave can be measured as the distance from a point on a wave to the corresponding point on the next cycle of the wave.

• Wavelength is the distance between …

• Find the wavelength of wave

1. A

2. B

3. C

Example

example

• The diagram represents waves A, B, C, and D traveling in the same medium.  Which two waves have the same wavelength?

1. A and B

2. A and C

3. B and D

4. C and D

example

• In the diagram, the distance between points A and B on a wave is 5.0 meters.  What is the wavelength of this wave?

example• The diagram below shows two points, A

and B, on a wave train. How many wavelengths separate point A and point B?

The Anatomy of a longitudinal wave• A longitudinal wave is a wave in which the particles of

the medium are displaced in a direction parallel to the direction of energy transport. A longitudinal wave can be created in a slinky if the slinky is stretched out horizontally and the end coil is vibrated back-and-forth in a horizontal direction.

• A region where the coils are pressed together in a small amount of space is known as a compression.

• A region where the coils are spread apart, thus maximizing the distance between coils, is known as a rarefaction.

• Points A, C and E on the diagram above represent compressions and points B, D, and F represent rarefactions. While a transverse wave has an alternating pattern of crests and troughs, a longitudinal wave has an alternating pattern of compressions and rarefactions.

Compression and rarefaction (crest and trough)

The amplitude in a longitudinal wave• The amplitude of a longitudinal wave is determined

by the pressure in the compression or by the separation of particles in rarefactions.

More compressed waves have bigger amplitude

Wavelength of a longitudinal wave• The wavelength of a wave is the length of one

complete cycle of a wave. In the case of a longitudinal wave, a wavelength measurement is made by measuring the distance from a compression to the next compression or from a rarefaction to the next rarefaction.

• Wave length is the distance …

example• A longitudinal wave moves to the right through a uniform

medium, as shown below. Points A, B, C, D, and E represent the positions of particles of the medium.

1. Describe the motion of the particle at position C.

2. Which two points represent the wavelength of this wave?

Frequency of a Wave• Wave on a String 2.01• The frequency of the wave is the same as its vibrating

source.• In mathematical terms, the frequency is the number of

complete vibrational cycles of a medium per second. the unit of frequency would be cycles/second, or waves/second, or vibrations/second.

• Another unit for frequency is the Hertz (abbreviated Hz) where 1 Hz is equivalent to 1 cycle/second.

• If a coil of slinky makes 2 vibrational cycles in one second, then the frequency is 2 Hz. If a coil of slinky makes 3 vibrational cycles in one second, then the frequency is 3 Hz. And if a coil makes 8 vibrational cycles in 4 seconds, then the frequency is 2 Hz (8 cycles/4 s = 2 cycles/s)

The period of a wave• The period of a wave is the time for a particle on a

medium to make one complete vibrational cycle. • Period is measured in units of time such as seconds,

hours, days or years. • The period of orbit for the Earth around the Sun is

approximately 365 days; it takes 365 days for the Earth to complete a cycle. The period of a typical class at a high school might be 46 minutes; every 46 minutes a class cycle begins (43 minutes for class and 3 minutes for passing time means that a class begins every 46 minutes). The period for the minute hand on a clock is 3600 seconds (60 minutes); it takes the minute hand 3600 seconds to complete one cycle around the clock.

Frequency and period relationship

• Mathematically, the period is the reciprocal of the frequency and vice versa. In equation form, this is expressed as follows.

Frequency and speed• The quantity frequency is not speed.

• The wave speed refers to how fast the wave is moving (m/s).

• The wave frequency refers to how often the medium vibrates up and down. (# of cycles/second or Hz).

• A wave can vibrate back and forth very frequently, yet have a small speed; and a wave can vibrate back and forth with a low frequency, yet have a high speed. Frequency and speed are distinctly different quantities.

• Example: what is the frequency of the wave?

example

• How are electromagnetic waves that are produced by oscillating charges and sound waves that are produced by oscillating tuning forks similar?

1. Both have the same frequency as their respective sources.

2. Both require a matter medium for propagation. 3. Both are longitudinal waves. 4. Both are transverse waves.

example

example• Determine the amplitude, frequency and period of the wave.

phase• Phase is the fraction of the wave cycle which has

elapsed relative to the origin. • Points on a periodic wave moving in the same direction

and having the same displacement from their rest position (same up or same down) are said to have the same phase, or to be “in phase.”

• Points on a periodic wave having the opposite displacement from their rest position are said to have be “out of phase”

Points C & E are out of phase.

Points B & F are not in phase b/c B is going up, F is going down

Points A & E are in phase

• If two points are in phase, they could only be multiple of wavelength apart, such as 1λ (360o), 2λ (720o), 3λ(1080o), …

• If two points are out of phase, they could only be multiple of half wavelength apart, but not whole wavelength apart, such as ½ λ (180o), 1½ λ (540o), 2½ λ (900o),…

Points in phase: points are out of phase:

example

• Points in phase: • Points out of phase:

example

• Which point on the wave diagram is in phase with point A?

1. E

2. B

3. C

4. D

example• The diagram represents a wave traveling

in a uniform medium.  Which two points on the wave are in phase?

1.   A and C

2.   A and E

3.   B and D

4.   B and F  

example• The diagram shows a parked police car with a siren

on top.  The siren is producing a sound with a frequency of 680 hertz, which travels first through point A and then through point B, as shown.  The speed of the sound is 340 meters per second.

• If the sound waves are in phase at points A and B, the distance between the points could be

1. 1λ 2. ½ λ 3. 3∕2 λ 4. ¼ λ

The speed of a wave• The speed of an object refers to how fast an object is moving. In

the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time. In equation form: v = d / t

• If the crest of an ocean wave moves a distance of 20 meters in 10 seconds, then the speed of the ocean wave is 2 m/s. On the other hand, if the crest of an ocean wave moves a distance of 25 meters in 10 seconds (the same amount of time), then the speed of this ocean wave is 2.5 m/s. The faster wave travels a greater distance in the same amount of time.

• Sometimes a wave encounters the end of a medium and the presence of a different medium. In this case, some of energy will reflect off the boundary and some of the energy will travel across the boundary.

Noah stands 170 meters away from a steep canyon wall. He shouts and hears the echo of his voice one second later. What is the speed of the wave?

Since the sound wave travels 340 meters in 1 second, so the speed of the wave is 340 m/s.

Caution: in case of reflection, you need to double the distance and half the time.

Variables Affecting Wave Speed

• The speed of a wave is not dependent upon properties of the wave itself (frequency, period, amplitude, wavelength..). Rather, the speed of the wave is dependent upon the properties of the medium only. Only an alteration in the properties of the medium will cause a change in the speed.

Wave on a String 2.01

example• What is the time required for the sound

waves (v = 340 m/s) to travel from the tuning fork to point A? [show work, including equation and substitution with number and units]

Check Your Understanding1. A teacher attaches a slinky to the wall and

begins introducing pulses with different amplitudes. Which of the two pulses (A or B) below will travel from the hand to the wall in the least amount of time?

2. An automatic focus camera is able to focus on objects by use of an ultrasonic sound wave. The camera sends out sound waves that reflect off distant objects and return to the camera. A sensor detects the time it takes for the waves to return and then determines the distance an object is from the camera. The camera lens then focuses at that distance. Now that's a smart camera! If a sound wave (speed = 340 m/s) returns to the camera 0.150 seconds after leaving the camera, then how far away is the object?

The Wave Equation• A wave is produced when a vibrating source periodically

disturbs the first particle of a medium. • The frequency at which each individual particle vibrates is

equal to the frequency at which the source vibrates. Similarly, the period of vibration of each individual particle in the medium is equal to the period of vibration of the source.

• Since in one period time the wave travels one wavelength,

Speed = Frequency • Wavelength

Since f = 1 / T

v = λ / Tor

or v = f∙λ

The above equation is known as the wave equation

Practice: • Fill in the blanks in the table, analyze the data, and

answer the following questions.

Medium Wavelength Frequency Speed

Zinc, 1-in. dia. coils

1.75 m 2.0 Hz ______

Zinc, 1-in. dia. coils

0.90 m 3.9 Hz ______

Copper, 1-in. dia. coils

1.19 m 2.1 Hz ______

Copper, 1-in. dia. coils

0.60 m 4.2 Hz ______

Zinc, 3-in. dia. coils

0.95 m 2.2 Hz ______

Zinc, 3-in. dia. coils

1.82 m 1.2 Hz ______

v = f∙λ

summary

• wave speed is dependent upon medium properties

• Even though the wave speed is calculated by multiplying wavelength by frequency, an alteration in wavelength does not affect wave speed.

• Rather, an alteration in wavelength affects the frequency in an inverse manner. A doubling of the wavelength results in a halving of the frequency; yet the wave speed is not changed.

example

• As the wavelength of a wave in a uniform medium increases, its speed will _____.

a. decrease

b. increase

c. remain the same

example

• The speed of a wave depends upon (i.e., is causally affected by) ...

a. the properties of the medium through which the wave travels

b. the wavelength of the wave.

c. the frequency of the wave.

d. both the wavelength and the frequency of the wave.

example

• Light of frequency 5.0 × 1014 hertz has a wavelength of 4.0 × 10-7 meter while traveling in a certain material. What is the speed of light in the material?

Lesson 4 - Behavior of Waves

• Boundary Behavior

• Reflection, Refraction, and Diffraction

• Interference of Waves

• Standing waves

• The Doppler Effect

objectivesKnow: • Definitions of Reflection, Refraction and Diffraction.• Conditions for production of standing waves. • Definition of a node and anti-node. • Definition of Doppler Effect. Understand: • Reflection, Refraction, Diffraction and superposition of waves. • Nodes result from destructive interference. • Anti-nodes result from constructive interference. • How frequency changes due to Doppler Effect Be able to: • Determine direction of pulse reflection. • Determine amplitude of interfering pulses. • Draw and interpret diagrams of wave interference. • Recognize a node or anti-node and use them to calculate wavelength. • Use wave speed equation to solve unknown values. • Recognize and interpret diagrams showing Doppler Effect. • Determine the effect of Doppler shift on apparent frequency of a wave

source. • State or recognize the conditions under which standing waves occur.

Boundary Behavior – at fixed end reflection

• Fixed end - the last particle of the rope will be unable to move when a disturbance reaches it.

• When the incident pulse reaches the boundary, two things occur:

1. A portion of the energy carried by the pulse is reflected and returns towards the left end of the rope. The disturbance which returns to the left after bouncing off the pole is known as the reflected pulse.

2. A portion of the energy carried by the pulse is transmitted to the pole, causing the pole to vibrate.

Some notable characteristics of the reflected pulse from a fixed end:

1. The reflected pulse is inverted. This inversion can be explained by Newton's third law of action-reaction. The rope is exerting a force upward on the pole, and the pole is exerting a force downward on the rope.

2. The speed of the reflected pulse is the same as the speed of the incident pulse.

3. Every particle within the rope will have the same frequency because they are connected to one another.

4. The amplitude of the reflected pulse is less than the amplitude of the incident pulse since some of the energy of the pulse was transmitted into the pole at the boundary.

Boundary Behavior – at Free End Reflection

Free End - The last particle of the rope will be able to move when a disturbance reaches it.

1. The reflected pulse have the same direction as the incident pulse.

2. The speed of the reflected pulse is the same as the speed of the incident pulse.

3. Every particle within the rope will have the same frequency because they are connected to one another.

4. The amplitude of the reflected pulse is less than the amplitude of the incident pulse since some of the energy of the pulse was transmitted into the pole at the boundary.

Some notable characteristics of the reflected pulse:

Transmission of a Pulse Across a Boundary from Less Dense to Denser

• The reflected pulse will be inverted. During the interaction between the two media at the boundary, the first particle of the denser medium overpowers the smaller mass of the last particle of the less dense medium. This causes an upward displaced pulse to become a downward displaced pulse.

• The denser medium on the other hand was at rest prior to the interaction. The first particle of this medium receives an upward pull when the incident pulse reaches the boundary. This upward pull causes an upward displacement. For this reason, the transmitted pulse is not inverted. In fact, transmitted pulses can never be inverted.

Characteristics of transmitted pulse and reflected pulse

• The transmitted pulse in the denser medium – Travels at slower speed than the incident pulse in the less

dense medium. – has a same frequency as the incident pulse– has a smaller wavelength than the incident pulse. – has a smaller amplitude than the incident pulse.

• The reflected pulse – Is inverted.– Has the same speed as the incident pulse.– Has the same wavelength as the incident pulse– Has the same frequency as the incident pulse.– has a smaller amplitude than the incident pulse.

• Waves always travel fastest in the least dense medium. Thus, the transmitted pulse will be traveling slower than the incident pulse.

• Since the transmitted pulse was introduced into the denser medium by the vibrations of particles in the less dense medium, they must be vibrating at the same frequency. Since the wavelength of a wave depends upon the frequency and the speed, the transmitted wave has slower speed and smaller wavelength.

• Finally, the incident and the reflected pulse share the same medium. They will have the same speed. Since the reflected pulse was created by the vibrations of the incident pulse, they will have the same frequency. And two waves with the same speed and the same frequency, must also have the same wavelength.

Explanation of Characteristics of transmitted pulse and reflected pulse

Transmission of a Pulse Across a Boundary from Denser to Less Dense

• The transmitted pulse in the less dense medium – is not inverted– travels faster than the incident pulse in denser medium. – has the same frequency as the incident pulse in the denser

medium). – has a larger wavelength than the incident pulse in the denser

medium. • The reflected pulse

– is not inverted– has the same speed as the incident pulse– has the same frequency as the incident pulse– has the same wavelength as the incident pulse.

• The boundary behavior of waves in ropes can be summarized by the following principles:– The wave speed is always greatest in the least

dense rope. – The wavelength is always greatest in the least

dense rope. – The frequency of a wave is not altered by crossing a

boundary. – The reflected pulse becomes inverted when a wave

in a less dense rope is heading towards a boundary with a denser rope.

– The amplitude of the incident pulse is always greater than the amplitude of the reflected pulse.

– http://paws.kettering.edu/~drussell/Demos/reflect/reflect.html

Reflection, Refraction, and Diffraction• When a wave is traveling in two and three dimensional

medium and encounter a boundary, the same boundary behavior such as reflection and transmission can be observed.

• http://www.falstad.com/ripple/

• ..\..\RealPlayer Downloads\Reflexión - Reflection.mp4

• The light rays – both incident and reflected – is drawn perpendicular to the wave fronts.

• The line of normal – the line drawn perpendicular to the boundary (surface) of the incident ray.

• The incident angle – the angle between the incident ray and the normal.

• The reflected angle – the angle between the reflected ray and the normal.

Basic terminologies

normal

The Law of reflection

The angles θi is incident angle and θr if reflected angle, they are the angle formed with the normal line.

θi = θr

• Compare to the incident wave, the reflected wave has the same speed, same frequency, and same wavelength

example

• The diagram above shows two rays of light striking a plane mirror.  Which diagram below best represents the reflected rays?

1. 2. 3. 4.

Example • A ray of light strikes a mirror at an angle

of incidence of 60°. What is the angle of reflection?

1. 0°

2. 30°

3. 60°

4. 90°

example• The diagram represents a light ray being reflected from

a plane mirror. The angle between the incident ray and the reflected ray is 70.°. What is the angle of incidence for this ray?

1. 20.°2. 35°3. 55°4. 70.°

Refraction • As a wave approaches across the boundary,

its speed changes. Since the wave approaches the boundary at an angle, the wave fronts changes its speed at different time. The part that reaches the boundary first slows down while the rest of the fronts moves ahead at the faster pace. The result is that the direction of the wave front is changed at the boundary.

• The fundamental feature of the waves' motion that leads to this change in direction is the change in speed.

Refraction, or the bending of the path of the waves, is the result of a change in speed and wavelength of the waves.

Note: the refracted wave has the same frequency as the incident wave.

..\..\RealPlayer Downloads\Refraction - Refracción.mp4

example

• Refraction of a wave is caused by a change in the wave's

1. amplitude

2. frequency

3. phase

4. speed

example

• Compared to the wavelength of a wave of green light in air, the wavelength of this same wave of green light in Lucite is

1. less

2. greater

3. the same

example• What happens to the speed and frequency of a light ray when

it passes from air into water?1. The speed decreases and the frequency increases.2. The speed decreases and the frequency remains the same.3. The speed increases and the frequency increases.4. The speed increases and the frequency remains the same.

• Diffraction involves a change in direction of waves as they pass through an opening or around a barrier in their path.

Diffraction

• Parallel water wave fronts incident on a small opening are diffracted to form concentric semicircular fronts.

• The amount of diffraction is determined by the how the wavelength and the size of opening of the barrier compare.

Small opening: more diffraction

Larger opening: less diffraction

example

• The diagram shows straight wave fronts passing through an opening in a barrier.  This wave phenomenon is called

1. reflection

2. refraction

3. polarization

4. diffraction

example• The diagram shows a wave phenomenon. The

pattern of waves shown behind the barrier is the result of

1. reflection

2. refraction

3. diffraction

4. interference

example

• A wave is diffracted as it passes through an opening in a barrier. The amount of diffraction that the wave undergoes depends on both the

1. amplitude and frequency of the incident wave

2. wavelength and speed of the incident wave

3. wavelength of the incident wave and the size of the opening

4. amplitude of the incident wave and the size of the opening

• Reflection, refraction and diffraction are all boundary behaviors of waves associated with the bending of the path of a wave.

• Reflection occurs when there is a bouncing off of a barrier. Reflection of waves off straight barriers follows the law of reflection.

• Refraction is the change in direction of waves which occurs when waves travel from one medium to another. Refraction is always accompanied by a wavelength and speed change.

• Diffraction is the bending of waves around obstacles and openings. The amount of diffraction increases with increasing wavelength.

summary

example• Which diagram best illustrates wave diffraction? • Which diagram best illustrates wave reflection?  • Which diagram best illustrates wave refraction? 

A B

C D

Interference of Waves• Wave interference is the phenomenon which occurs when two waves meet

while traveling along the same medium. • The interference of waves causes the medium to take on a shape which results

from the net effect of the two individual waves upon the particles of the medium.

Nodes and antinodes

antinodes: top or bottom

Nodes: on the equilibrium line

Constructive interference• Constructive interference occurs where the two interfering

waves have a displacement in the same direction. The result is a larger amplitude.

2 units1 unit

-2 units-1 unit

Maximum constructive interference occurs when the waves are in phase (phase difference is 0o or 360o) and crest superposes on crest or trough on trough.

The point of maximum displacement of a medium when two waves are interacting is called an anti-node.

Destructive interference• Destructive interference occurs where the two interfering

waves have a displacement in the opposite direction. Destructive interferences result a smaller amplitude.

• Maximum destructive interference occurs when two waves of equal frequency and amplitude whose phase difference is 180o or ½ λ meet at a point. Maximum destructive interference results in the formation of nodes. Which are regions of zero displacement of the medium

-1 unit 0 unit

+1 unit

• The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur.

1 unit

-2 unit -1 unit

Practice : Constructive vs. destructive interference

principle of superposition• When two waves interfere, the resulting displacement of

the medium at any location is the algebraic sum of the displacements of the individual waves at that same location.

Displacement of Pulse 1

Displacement of Pulse 2

=Resulting Displacement

+1 +1 =

-1 -1 =

+1 -1 =

+1 -2 =

• In actuality, the task of determining the complete shape of the entire medium during interference demands that the principle of superposition be applied for every point (or nearly every point) along the medium.

• Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path. The two waves will meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.

example

• Two waves having the same amplitude and the same frequency pass simultaneously through a uniform medium.  Maximum destructive interference occurs when the phase difference between the two waves is

1.   0°

2.   90°

3.   180°

4.   360°

example

• Maximum constructive interference between two waves of the same frequency could occur when their phase difference is

1. 1λ

2. ¼ λ

3. ½ λ

4. 1 ½ λ

example• The diagram shows two pulses, each of length,

traveling toward each other at equal speed in a rope. Which diagram below best represents the shape of the rope when both pulses are in region AB? 

1 2 3 4

Practice - Determine the interference pattern

Two sources in phase in the same medium• ..\..\RealPlayer Downloads\Wave Motion Interference -

YouTube.flvConstructive interference: Point A, B are anti-nodes

Destructive interference: Point C, D, E, F are nodes

crests

troughs

Nodal lines

• Although both sources are repeatedly producing waves which move across the medium, a stable pattern is set up. The regions of constructive interference do not move, nor do the regions of destructive interference.

• These motionless regions have a pattern which can be measured. These measurements can be used to calculate the wavelength of the waves which are producing the pattern. In this way one can find the wavelength of a moving wave.

example• The diagram below represents shallow water

waves of wavelength λ passing through two small openings, A and B, in a barrier.

• How much longer is the length of path AP than the length of path BP?

1. 1λ 2. 2λ 3. 3λ 4. 4λ

• The diagram below represents shallow water waves of constant wavelength passing through two small openings, A and B, in a barrier. Which statement best describes the interference at point P?

1. It is constructive, and causes a longer wavelength. 2. It is constructive, and causes an increase in amplitude. 3. It is destructive, and causes a shorter wavelength. 4. It is destructive, and causes a decrease in amplitude.

example

example• The diagram shows two sources, A and B,

vibrating in phase in the same uniform medium and producing circular wave fronts.  Which phenomenon occurs at point P?

1. destructive interference

2. constructive interference

3. reflection

4. refraction

example• The diagram represents shallow water waves of

wavelength λ passing through two small openings, A and B, in a barrier.  Compared to the length of path BP, the length of path AP is how many wavelength longer?

Standing Waves• If a wave is introduced into a confined small medium, as the wave

reaches the end of the boundary it will reflect and travel back in the opposite direction. The reflected wave will then interfere with the incident wave towards the fixed end.

• http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html• When the proper frequency is used, the interference of the incident

wave and the reflected wave occur in such a manner that there are specific points along the medium that appear to be standing still. There are other points along the medium whose displacement changes over time, but in a regular manner. The vibrations occur at regular time intervals such that the motion of the medium is regular and repeating.

• Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern.

• A standing wave pattern is formed as the result of the perfectly timed interference of two waves passing through the same medium. A standing wave is not actually a wave; rather it is the pattern.

• Standing wave patterns are only created within the medium at specific frequencies of vibration. These frequencies are known as harmonics.

• Standing waves can be created for both transverse and longitudinal waves.

1st harmonic

2nd harmonic

3rd harmonic

Nodes and anti-nodes in a standing wave

Nodes: the points of zero displacement of the resultant wave

Antinotes: the points of maximum displacement of a medium

The distance between two successive nodes is ½ λ

standingWaveDiagrams1/StandingWaveDiagrams1.html

Harmonic

# of Nodes

# of Antinodes

Pattern

1st 2 1                                                       

2nd 3 2                                                       

3rd 4 3                                                       

4th 5 4                                                       

5th 6 5                                                       

6th 7 6                                                       

nth n + 1 n --

Standing waves in water• Standing waves in water are produced

most often by periodic water waves reflecting from a barrier.

example• What is the number of nodes and

antinodes in the standing wave shown in the diagram?  

exampleThe diagram represents a wave moving toward the right.                    

Which wave shown below could produce a standing wave with the original wave?

1 2 3 4

example• Two waves traveling in the same medium and

having the same wavelength (λ) interfere to create a standing wave. What is the distance between two consecutive nodes on this standing wave?

1. λ

2. ½ λ

3. ¼ λ

4. ¾ λ

Doppler Effect

Stationary source

Wave fronts: A wave front is the curve of all adjacent points on a wave that are in phase.

To observer A and observer B, the frequency is the same everywhere.

Moving source

Higher frequency, short λ

Lower frequency, longer λ

The Doppler effect can be described as the effect produced by a moving source of waves, the observer, or both –

an apparent upward shift in frequency if the observers and the source is approaching each otheran apparent downward shift in frequency if the observers and the source is moving away from each other.

• It is important to note that both the speed and the frequency of the source does not change. Using the example above, the bug is still producing disturbances at the same rate; it just appears to the observer whom the bug is approaching that the disturbances are being produced at a frequency greater.

• The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc.

• car horn - coming and going• As the car approached with its siren blasting, the pitch of

the siren sound (a measure of the siren's frequency) was high; and then suddenly after the car passed by, the pitch of the siren sound was low. That was the Doppler effect - an apparent shift in frequency for a sound wave produced by a moving source.

Blue shift and red shiftThe human eye perceives light waves of different frequencies as differences in color. The lowest frequency we can see is red and the highest frequency we can see is blue-violet.Due to Doppler effect, the apparent color of an approaching light source is shifted toward the blue end of the spectrum, while that of a receding source is shifted toward the red end.

Applications of the Doppler Effect• Police work – the speed of a car is determined by a radar system.

– when a car is at rest, the sent out frequency is the same as received frequency.

– If the car is moving toward the source of radar, the reflected waves have higher frequency, the greater the car’s speed, the greater the Doppler shift in frequency.

– If the car is moving away from the source of radar, the reflected waves have lower frequency.

• Weather stations – Doppler radars are used to determine the location and intensity of precipitation as well as directions and speed of the winds blowing around rain drops.

example

• A police officer's stationary radar device indicates that the frequency of the radar wave reflected from an automobile is less than the frequency emitted by the radar device. This indicates that the automobile is

1. moving toward the police officer 2. moving away from the police officer 3. not moving

example• The diagram shows radar waves being emitted from a

stationary police car and reflected by a moving car back to the police car. The difference in apparent frequency between the incident and reflected rays is an example of

1. constructive interference 2. refraction 3. the Doppler effect 4. total internal reflection

example

• As observed from the Earth, the light from a star is shifted toward lower frequencies. This is an indication that the distance between the Earth and the star is

1. decreasing

2. increasing

3. constant