What is a Wave?A wave is a series of pulses or disturbances
created in a medium from one location to another. To start a wave,
the first particle is displaced or moved out of position. For
example, a slinky is in equilibrium or rest position when you
stretch it. The coils are equally spread out. But when a pulse
occurs, the coils are unevenly spread and vibrate. After the pulses
occur, the slinky eventually goes back to its original position. A
medium is a substance or material that carries the wave. The news
media refers to various institutions (newspaper offices, television
stations, radio stations, etc.) within our society that carry the
news from one location to another. The newsmoves throughthe media.
The media doesn't make the news and the media isn't the same as the
news. The news media is thethingthat carries the news from its
source to various locations, just like how the medium carries the
wave through the slinky.The medium is made of parts that are
capable of interacting with each other. The interactions of one
particle with the next particle allow the disturbance to travel
through the medium.For example in the slinky, the first coil
becomes disturbed and begins to push or pull on the second coil;
this push or pull on the secondcoil will displace the second coil
from its equilibrium position. As the second coil becomes
displaced, it begins to push or pull on the third coil and so on.
The particles apply force to the next particle. As a disturbance
moves through a medium from one particle to the next particle,
energy is being transported from one end of the medium to the
other. In a slinky wave, a person gives energy to the first coil by
doing work. When the first coil returns to its original position,
it has the same amount of energy as it had before the pulse
occurred. A wave transports its energy without transporting matter.
Waves move through an ocean or lake and the water always returns to
its rest position. Energy is transported through the medium, yet
the water molecules are not transported. This is proof that there
is still water in the middle of the ocean. The water has not moved
from the middle of the ocean to the shore. In a stadium wave, the
fans do not get out of their seats and walk around the stadium. It
would be silly (and embarrassing) for any fan to do this. In a
stadium wave, each fan rises up and returns to the original seat.
The disturbance moves through the stadium, yet the fans are not
transported.When a wave is present in a medium (that is, when there
is a disturbance moving through a medium), the individual particles
of the medium are only temporarily displaced from their rest
position. There is always a force acting upon the particles that
restores them to their original position. In a slinky wave, each
coil of the slinky ultimately returns to its original position. In
a water wave, each molecule of the water ultimately returns to its
original position. And in astadium wave, each fan in the bleacher
ultimately returns to its original position. It is for this reason,
that a wave is said to involve the movement of a disturbance
without the movement of matter. The particles of the medium (water
molecules, slinky coils, stadium fans) simply vibrate about a fixed
position as the pattern of the disturbance moves from one location
to another location.Waves are said to be anenergy transport
phenomenon. As a disturbance moves through a medium from one
particle to its adjacent particle, energy is being transported from
one end of the medium to the other. In a slinky wave, a person
imparts energy to the first coil by doing work upon it. The first
coil receives a large amount of energy that it subsequently
transfers to the second coil. When the first coil returns to its
original position, it possesses the same amount of energy as it had
before it was displaced. The first coil transferred its energy to
the second coil. The second coil then has a large amount of energy
that it subsequently transfers to the third coil. When the second
coil returns to its original position, it possesses the same amount
of energy as it had before it was displaced. The third coil has
received the energy of the second coil. This process of energy
transfer continues as each coil interacts with its neighbor. In
this manner, energy is transported from one end of the slinky to
the other, from its source to another location.
Types of WavesAtransverse waveis a wave where particles of the
medium move in a direction that isperpendicularto the direction
that the wave moves. If a slinky is stretched out in a horizontal
direction and a pulse is introduced into the slinky on the left end
by vibrating the first coil up and down. Energy will begin to be
transported through the slinky from left to right. As the energy is
transported from left to right, the individual coils of the medium
will be displaced upwards and downwards. The particles of the
medium move perpendicular to the direction that the pulse moves.
This type of wave is a transverse wave. The particle motion of
Transverse waves is always perpendicularto wave motion.
Alongitudinal waveis a wave in which particles of the medium move
in a directionparallelto the direction that the wave moves. Suppose
that a slinky is stretched out in a horizontal direction across the
classroom and that a pulse is introduced into the slinky on the
left end by vibrating the first coil left and right. Energy will
begin to be transported through the slinky from left to right. As
the energy is transported from left to right, the individual coils
of the medium will be displaced leftwards and rightwards. In this
case, the particles of the medium move parallel to the direction
that the pulse moves. This type of wave is a longitudinal wave.
Longitudinal waves are always characterized by particle motion
beingparallelto wave motion. These waves travel in depths of the
ocean. The fans will need to sway side to side to move in stadium
to produce a longitudinal wave. Thus, as the wave travels around
the stadium they would be moving parallel to its direction of
motion. If they rise up and sit down, then they would be creating a
transverse wave.The waves that travel along the surface of the
oceans are surface waves. Asurface waveis a wave where particles of
the medium undergo a circular motion. Surface waves are neither
longitudinal nor transverse. In longitudinal and transverse waves,
all the particles in the entire bulk of the medium move in a
parallel and a perpendicular direction (respectively) relative to
the direction of energy transport. In a surface wave, it is only
the particles at the surface of the medium that undergo the
circular motion. The motion of particles tends to decrease as one
proceeds further from the surface.Anelectromagnetic waveis a wave
that can transmit its energy through a vacuum (i.e., empty space).
Electromagnetic waves are produced by the vibration of charged
particles. Electromagnetic waves are produced on the sun after
traveling to Earth through the vacuum of outer space. If
electromagnetic waves could not travel to Earth through a vacuum,
there would undoubtedly be no life on Earth. All light waves are
examples of electromagnetic waves, including light waves and sound
waves.Amechanical waveis a wave that cant transmit 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 wave. Sound waves cant travel through
a vacuum. Slinky waves, water waves, stadium waves, andjump rope
wavesare other examples that require 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.For many people, the first
thought concerning waves conjures up a picture of a wave moving
across the surface of an ocean, lake, pond or other body of water.
The waves are created by some form of a disturbance, such as a rock
thrown into the water, a duck shaking its tail in the water or a
boat moving through the water. The water wave hasa crest and a
troughand travels from one location to another. One crest is often
followed by a second crest that is often followed by a third crest.
Every crest is separated by a trough to create an alternating
pattern of crests and troughs. A duck or gull at rest on the
surface of the water is observed to bob up-and-down at rather
regular time intervals as the wave passes by. The waves may appear
to be plane waves that travel together as afrontin a straight-line
direction, perhaps towards a sandy shore. Or the waves may be
circular waves that originate from the point where the disturbances
occur; such circular waves travel across the surface of the water
in all directions. These mental pictures of water waves are useful
for understanding the nature of a wave and will be revisited later
when we begin our formal discussion of the topic. If you strike a
horizontal rod vertically from above, what can be said about the
waves created in the rod?The particles vibrate vertically,
perpendicular to the direction of the rod. Because the coils of the
slinky are vibrating longitudinally, there are regions where they
become pressed together and other regions where they are spread
apart. A region where the coils are pressed together in a small
amount of space is known as a compression. Acompressionis a point
on a medium through which a longitudinal wave is traveling that has
the maximum density. A region where the coils are spread apart,
thus maximizing the distance between coils, is known as a
rarefaction. Ararefactionis a point on a medium through which a
longitudinal wave is traveling that has the minimum density. 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.
Wavelength, AmplitudeA transverse wave can be created in a rope
if the rope is stretched out horizontally and the end is vibrated
back-and-forth in a vertical direction. If a snapshot of such a
transverse wave could be taken so as tofreezethe shape of the rope
in time, then it would look like the following diagram.
The dashed line drawn through the center of the diagram
represents theequilibrium or rest positionof the string. This is
the position that the string would assume if there were no
disturbance moving through it. Once a disturbance is introduced
into the string, the particles of the string begin to vibrate
upwards and downwards. At any given moment in time, a particle on
the medium could be above or below the rest position. Points A, E
and H on the diagram represent the crests of this wave. Thecrestof
a wave is the point on the medium that exhibits the maximum amount
of positive or upward displacement from the rest position. Points C
and J on the diagram represent the troughs of the wave. Thetroughof
a wave is the point on the medium that exhibits the maximum amount
of negative or downward displacement from the rest position.The
wave shown above can be described by a variety of properties. One
such property is amplitude. The amplitudeof 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 distancefrom rest
to crest. Similarly, the amplitude can be measured from the rest
position to the trough position. In the diagram above, the
amplitude could be measured as the distance of a line segment that
is perpendicular to the rest position and extends vertically upward
from the rest position to point A.
The wavelength is another property of a wave that is portrayed
in the diagram above. Thewavelengthof a wave is simply the length
of one complete wave cycle. If you were to trace your finger across
the wave in the diagram above, you would notice that your finger
repeats its path. A wave is a repeating pattern. It repeats itself
in a periodic and regular fashion over both time and space. And the
length of one such spatial repetition (known as awave cycle) is the
wavelength. 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. In the
diagram above, the wavelength is the horizontal distance from A to
E, or the horizontal distance from B to F, or the horizontal
distance from D to G, or the horizontal distance from E to H. Any
one of these distance measurements would suffice in determining the
wavelength of this wave.
Consider the diagram below in order to answer questions
#1-2.
1. The wavelength of the wave in the diagram above is given by
letter A. The wavelength is the distance from crest to crest (or
from trough to trough) (or between any two corresponding points on
adjacent waves).2. The amplitude of the wave in the diagram above
is given by letter D. The amplitude is the distance from rest to
crest or from rest to trough.
3. Indicate the interval that represents one full
wavelength.
Answer: C to G because the wavelength is the distance from crest
to crest, trough to trough, or from a point on one wave cycle to
the corresponding point on the next adjacent wave cycle.Frequency
and PeriodSuppose that a hand holding the first coil of a slinky is
moved back-and-forth two complete cycles in one second. The rate of
the hand's motion would be 2 cycles/second. The first coil, being
attached to the hand, in turn would vibrate at a rate of 2
cycles/second. The second coil, being attached to the first coil,
would vibrate at a rate of 2 cycles/second. The third coil, being
attached to the second coil, would vibrate at a rate of 2
cycles/second. In fact, every coil of the slinky would vibrate at
this rate of 2 cycles/second. This rate of 2 cycles/second is
referred to as the frequency of the wave. Thefrequencyof a wave
refers to how often the particles of the medium vibrate when a wave
passes through the medium. Frequency is a part of our common,
everyday language. For example, it is not uncommon to hear a
question like "Howfrequentlydo you mow the lawn during the summer
months?" Of course the question is an inquiry abouthow oftenthe
lawn is mowed and the answer is usually given in the form of "1
time per week." In mathematical terms, the frequency is the number
of complete vibrational cycles of a medium per a given amount of
time. Given this definition, it is reasonable that the
quantityfrequencywould have units of cycles/second, waves/second,
vibrations/second, or something/second. Another unit for frequency
is theHertz(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 quantity
frequency is often confused with the quantity period.Periodrefers
to the time that it takes to do something. When an event occurs
repeatedly, then we say that the event isperiodicand refer to the
time for the event to repeat itself as the period. Theperiodof a
wave is the time for a particle on a medium to make one complete
vibrational cycle. Period, being a time, 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 55 minutes; every 55 minutes a
class cycle begins (50 minutes for class and 5 minutes for passing
time means that a class begins every 55 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 are different, yet related, quantities.
Frequency refers to how often something happens. Period refers to
the time it takes something to happen. Frequency is a rate
quantity. Period is a time quantity. Frequency is the
cycles/second. Period is the seconds/cycle. As an example of the
distinction and the relatedness of frequency and period, consider a
woodpecker that drums upon a tree at a periodic rate. If the
woodpecker drums upon a tree 2 times in one second, then the
frequency is 2 Hz. Each drum must endure for one-half a second, so
the period is 0.5 s. If the woodpecker drums upon a tree 4 times in
a second, then the frequency is 4 Hz; each drum must endure for
one-fourth a second, so the period is 0.25 s. If the woodpecker
drums upon a tree 5 times in one second, then the frequency is 5
Hz; each drum must endure for one-fifth a second, so the period is
0.2 s. Do you observe the relationship? Mathematically, the period
is the reciprocal of the frequency and vice versa. In equation
form, this is expressed as follows.
Since the symbolfis used for frequency and the symbolTis used
for period, these equations are also expressed as:
1. A wave is introduced into a thin wire held tight at each end.
It has amplitude of 3.8 cm, a frequency of 51.2 Hz and a distance
from a crest to the neighboring trough of 12.8 cm. Determine the
period of such a wave.
Answer:0.0195 secHere is an example of a problem with a lot of
extraneous information. The period is simply the reciprocal of the
frequency. In this case, the period is 1/(51.2 Hz) which is 0.0195
seconds.2. 2. Frieda the fly flaps its wings back and forth 121
times each second. The period of the wing flapping is ____ sec.
Answer: 0.00826 seconds. The quantity 121 times/second is the
frequency. The period is the reciprocal of the frequency. T=1/(121
Hz) = 0.00826 seconds
3. A tennis coach paces back and forth along the sideline 10
times in 2 minutes. The frequency of her pacing is ________ Hz.
a. 5.0 b. 0.2 c. 0.12 d. 0.083Answer: D Frequency refers to the
number of occurrences of a periodic event per time and is measured
in cycles/second. In this case, there are 10 cycles per 2 minutes
(also known as 10 cycles per 120 seconds). So the frequency is f
=10 cycles / 120 s = 0.0833 cycles/s
4. Non-digital clocks (which are becoming more rare) have a
second hand that rotates around in a regular and repeating fashion.
The frequency of rotation of a second hand on a clock is _______
Hz.a. 1/60b. 1/12c. 1/2
d. 1e. 60
Answer:AFrequency refers to the number of occurrences of a
periodic event per time and is measured in cycles/second. In this
case, there is 1 cycle per 60 seconds. So the frequency isf = 1
cycle / (60 s) = (1 / 60) Hz5. Olive Udadi accompanies her father
to the park for an afternoon of fun. While there, she hops on the
swing and begins a motion characterized by a complete
back-and-forth cycle every 2 seconds. The frequency of swing is
_________.a. 0.5 Hzb. 1 Hzc. 2 Hz
Answer:AFrequency refers to the number of occurrences of a
periodic event per time and is measured in cycles/second. In this
case, there is 1 cycle per 2 seconds. So the frequency is 1
cycles/2 s = 0.5 Hz.
6. In problem #5, the period of swing is __________.a. 0.5
secondb. 1 secondc. 2 second
Answer:CPeriod refers to the time for something to happen. In
this case, the period is the time for one complete swing - given as
2 seconds.
7. A period of 5.0 seconds corresponds to a frequency of
________ Hertz.a. 0.2b. 0.5c. 0.02
d. 0.05e. 0.002
Answer:AFrequency is the reciprocal of the period. The period is
5 seconds, so the frequency is 1/(5 s) = 0.20 Hz.8. A common
physics lab involves the study of the oscillations of a pendulum.
If a pendulum makes 33 complete back-and-forth cycles of vibration
in 11 seconds, then its period is ______.Answer:0.33 secondPeriod
refers to the time for something to happen and is measured in
seconds/cycle. In this case, there are 11 seconds per 33
vibrational cycles. Thus the period is (11 s) / (33 cycles) = 0.33
seconds.
9. A child in a swing makes one complete back and forth motion
in 3.2 seconds. This statement provides information about the
child'sa. speedb. frequencyc. periodAnswer:B and CWe now know that
the period is 3.2 seconds and that the frequency is 0.31 Hz.
10. The period of the sound wave produced by a 440 Hertz tuning
fork is ___________.Answer:0.00227 secondsGIVEN: f = 440 HzFind TT
= 1 / f = 1 / (440 HZ) = 0.00227 s11. As the frequency of a wave
increases, the period of the wave ___________.a. decreasesb.
increasesc. remains the sameAnswer:APeriod is the reciprocal of the
frequency. So as f increases, 1 / f decreases.The Speed of a
WaveAwave is a disturbancethat moves along a medium from one end to
the other. If one watches an ocean wave moving along the medium
(the ocean water), one can observe that the crest of the wave is
moving from one location to another over a given interval of time.
The crest is observed tocoverdistance. Thespeedof an object refers
to how fast an object is moving and is usually expressed as the
distance traveled per time of travel. 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,
If the crest of an ocean wave moves a distance of 20 meters in
10 seconds, then the speed of the ocean wave is 2.0 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.he diagrams at the right show
several "snapshots" of the production of a wave within a rope. The
motion of the disturbance along the medium after every one-fourth
of a period is depicted. Observe that in the time it takes from the
first to the last snapshot, the hand has made one complete
back-and-forth motion. Aperiodhas elapsed. Observe that during this
same amount of time, the leading edge of the disturbance has moved
a distance equal to one complete wavelength. So in a time of one
period, the wave has moved a distance of one wavelength. Combining
this information with the equation for speed (speed =
distance/time), it can be said that the speed of a wave is also the
wavelength/period.
Since the period is the reciprocal of the frequency, the
expression 1/f can be substituted into the above equation for
period. Rearranging the equation yields a new equation of the
form:Speed = Wavelength FrequencyThe above equation is known as the
wave equation. It states the mathematical relationship between the
speed (v) of a wave and its wavelength () and frequency (f). Using
the symbolsv, , andf, the equation can be rewritten asv = f
1. As the wavelength of a wave in a uniform medium increases,
its speed will _____.a. decreaseb. increasec. remain the same
Answer:CIn rows 1 and 2, the wavelength was altered but the
speed remained the same. The same can be said about rows 3 and 4
and rows 5 and 6. The speed of a wave is not affected by the
wavelength of the wave.2. As the wavelength of a wave in a uniform
medium increases, its frequency will _____.a. decreaseb. increasec.
remain the same
Answer:AIn rows 1 and 2, the wavelength was increased and the
frequency was decreased. Wavelength and frequency are inversely
proportional to each other.
3. The speed of a wave depends upon (i.e., is causally affected
by) ...a. the properties of the medium through which the wave
travelsb. the wavelength of the wave.c. the frequency of the
wave.d. both the wavelength and the frequency of the
wave.Answer:AWhenever the medium is the same, the speed of the wave
is the same. However, when the medium changes, the speed changes.
The speed of these waves were dependent upon the properties of the
medium.The above example illustrates how to use the wave equation
to solve mathematical problems. It also illustrates the principle
thatwave speed is dependent upon medium propertiesand independent
of wave 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.Check Your Understanding1. Two waves on
identical strings have frequencies in a ratio of 2 to 1. If their
wave speeds are the same, then how do their wavelengths compare?a.
2:1b. 1:2c. 4:1d. 1:4
Answer:BFrequency and wavelength are inversely proportional to
each other. The wave with the greatest frequency has the shortest
wavelength. Twice the frequency means one-half the wavelength. For
this reason, the wavelength ratio is the inverse of the frequency
ratio.
2. Mac and Tosh stand 8 meters apart and demonstrate the motion
of a transverse wave on a snakey. The wave e can be described as
having a vertical distance of 32 cm from a trough to a crest, a
frequency of 2.4 Hz, and a horizontal distance of 48 cm from a
crest to the nearest trough. Determine the amplitude, period, and
wavelength and speed of such a wave.Amplitude = 16 cm(Amplitude is
the distance from the rest position to the crest position which is
half the vertical distance from a trough to a crest.)Wavelength =
96 cm(Wavelength is the distance from crest to crest, which is
twice the horizontal distance from crest to nearest trough.)Period
= 0.42 s(The period is the reciprocal of the frequency. T = 1 /
f)Speed = 230 cm/s(The speed of a wave is calculated as the product
of the frequency times the wavelength.)
3. Dawn and Aram have stretched a slinky between them and begin
experimenting with waves. As the frequency of the waves is
doubled,a. the wavelength is halved and the speed remains
constantb. the wavelength remains constant and the speed is
doubledc. both the wavelength and the speed are halved.d. both the
wavelength and the speed remain constant.Answer:ADoubling the
frequency will not alter the wave speed. Rather, it will halve the
wavelength. Wavelength and frequency are inversely related.
4. A ruby-throated hummingbird beats its wings at a rate of
about 70 wing beats per second.a. What is the frequency in Hertz of
the sound wave?b. Assuming the sound wave moves with a velocity of
350 m/s, what is the wavelength of the wave?
Answer:f = 70 Hzand = 5.0 mThe frequency is given and the
wavelength is the v/f ratio.
5. Ocean waves are observed to travel along the water surface
during a developing storm. A Coast Guard weather station observes
that there is a vertical distance from high point to low point of
4.6 meters and a horizontal distance of 8.6 meters between adjacent
crests. The waves splash into the station once every 6.2 seconds.
Determine the frequency and the speed of these waves.The wavelength
is 8.6 meters and the period is 6.2 seconds.The frequency can be
determined from the period. If T = 6.2 s, thenf =1 /T = 1 / (6.2
s)f = 0.161 HzNow find speed using the v = f equation.v = f =
(0.161 Hz) (8.6 m)v = 1.4 m/s
6. Two boats are anchored 4 meters apart. They bob up and down,
returning to the same up position every 3 seconds. When one is up
the other is down. There are never any wave crests between the
boats. Calculate the speed of the waves.The diagram is helpful. The
wavelength must be 8 meters (see diagram).The period is 3 seconds
so the frequency is 1 / T or 0.333 Hz.Now use speed = f wavelength
Substituting and solving for v, you will get2.67 m/s.
Wave PhenomenonAs a wave travels through a medium, it will often
reach the end of the medium and encounter an obstacle or perhaps
another medium through which it could travel. Oneexampleof this has
already been mentioned in Lesson 2. A sound wave is known to
reflect off canyon walls and other obstacles to produce an echo. A
sound wave traveling through air within a canyon reflects off the
canyon wall and returns to its original source. What affect does
reflection have upon a wave? Does reflection of a wave affect the
speed of the wave? Does reflection of a wave affect the wavelength
and frequency of the wave? Does reflection of a wave affect the
amplitude of the wave? Or does reflection affect other properties
and characteristics of a wave's motion? The behavior of a wave (or
pulse) upon reaching the end of a medium is referred to asboundary
behavior. When one medium ends, another medium begins; the
interface of the two media is referred to as theboundaryand the
behavior of a wave at that boundary is described as its boundary
behavior. The questions that are listed above are the types of
questions we seek to answer when we investigate the boundary
behavior of waves.Fixed End ReflectionFirst consider an elastic
rope stretched from end to end. One end will be securely attached
to a pole on a lab bench while the other end will be held in the
hand in order to introduce pulses into the medium. Because the
right end of the rope is attached to a pole (which is attached to a
lab bench) (which is attached to the floor that is attached to the
building that is attached to the Earth), the lastparticleof the
rope will be unable to move when a disturbance reaches it. This end
of the rope is referred to as afixed end.If a pulse is introduced
at the left end of the rope, it will travel through the rope
towards the right end of the medium. This pulse is called
theincident pulsesince it is incident towards (i.e., approaching)
the boundary with the pole. When the incident pulse reaches the
boundary, two things occur: A portion of the energy carried by the
pulse is reflected and returns towards the left end of the rope.
The disturbance that returns to the left after bouncing off the
pole is known as thereflected pulse. A portion of the energy
carried by the pulse istransmittedto the pole, causing the pole to
vibrate.Because the vibrations of the pole are not visibly obvious,
the energy transmitted to it is not typically discussed. The focus
of the discussion will be on the reflected pulse. What
characteristics and properties could describe its motion?When one
observes the reflected pulse off the fixed end, there are several
notable observations. First the reflected pulse isinverted. That
is, if an upward displaced pulse is incident towards a fixed end
boundary, it will reflect and return as a downward displaced pulse.
Similarly, if a downward displaced pulse is incident towards a
fixed end boundary, it will reflect and return as an upward
displaced pulse.
The inversion of the reflected pulse can be explained by
returning to our conceptions of the nature of a mechanical wave.
When a crest reaches the end of a medium ("medium A"), the last
particle of the medium A receives an upward displacement. This
particle is attached to the first particle of the other medium
("medium B") on the other side of the boundary. As the last
particle of medium A pulls upwards on the first particle of medium
B, the first particle of medium B pulls downwards on the last
particle of medium A. This is merely Newtons. For every action,
there is an equal and opposite reaction. The upward pull on the
first particle of medium B has little effect upon this particle due
to the large mass of the pole and the lab bench to which it is
attached. The effect of the downward pull on the last particle of
medium A (a pull that is in turn transmitted to the other
particles) results in causing the upward displacement to become a
downward displacement. The upward displaced incident pulse thus
returns as a downward displaced reflected pulse. It is important to
note that it is theheavinessof the pole and the lab bench relative
to the rope that causes the rope to become inverted upon
interacting with the wall. When two media interact by exerting
pushes and pulls upon each other, the most massive mediumwins the
interaction. Just like in arm wrestling, the medium that loses
receives a change in its state of motion.Other notable
characteristics of the reflected pulse include: The speed of the
reflected pulse is the same as the speed of the incident pulse. The
wavelength of the reflected pulse is the same as the wavelength of
the incident pulse. The amplitude of the reflected pulse is less
than the amplitude of the incident pulse.Of course, it is not
surprising that the speed of the incident and reflected pulse are
identical since the two pulses are traveling in the same medium.
Since the speed of a wave (or pulse) is dependent upon the medium
through which it travels,two pulses in the same medium will have
the same speed. A similar line of reasoning explains why the
incident and reflected pulses have the same wavelength. Every
particle within the rope will have the same frequency. Being
connected to one another, they must vibrate at the same frequency.
Since the wavelength of a wave depends upon the frequency and the
speed, two waves having the same frequency and the same speed must
also have the same wavelength. Finally, 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. The reflected pulse is carrying less energy away
from the boundary compared to the energy that the incident pulse
carried towards the boundary. Since the amplitude of a pulse is
indicative of the energy carried by the pulse, the reflected pulse
has a smaller amplitude than the incident pulse.Free End
ReflectionNow consider what would happen if the end of the rope
were free to move. Instead of being securely attached to a lab
pole, suppose it is attached to a ring that is loosely fit around
the pole. Because the right end of the rope is no longer secured to
the pole, the lastparticleof the rope will be able to move when a
disturbance reaches it. This end of the rope is referred to as
afree end.Once more if a pulse is introduced at the left end of the
rope, it will travel through the rope towards the right end of the
medium. When the incident pulse reaches the end of the medium, the
last particle of the rope can no longer interact with the first
particle of the pole. Since the rope and pole are no longer
attached and interconnected, they will slide past each other. So
when a crest reaches the end of the rope, the last particle of the
rope receives the same upward displacement; only now there is no
adjoining particle to pull downward upon the last particle of the
rope to cause it to be inverted. The result is that the reflected
pulse is not inverted. When an upward displaced pulse is incident
upon a free end, it returns as an upward displaced pulse after
reflection. And when a downward displaced pulse is incident upon a
free end, it returns as a downward displaced pulse after
reflection. Inversion is not observed in free end reflection.
Transmission of a Pulse Across a Boundary from Less to More
DenseLet's consider a thin rope attached to a thick rope, with each
rope held at opposite ends by people. And suppose that a pulse is
introduced by the person holding the end of the thin rope. If this
is the case, there will be an incident pulse traveling in the less
dense medium (the thin rope) towards the boundary with a more dense
medium (the thick rope).
Upon reaching the boundary, the usual two behaviors will occur.
A portion of the energy carried by the incident pulse is reflected
and returns towards the left end of the thin rope. The disturbance
that returns to the left after bouncing off the boundary is known
as thereflected pulse. A portion of the energy carried by the
incident pulse is transmitted into the thick rope. The disturbance
that continues moving to the right is known as thetransmitted
pulse.The reflected pulse will be found to be inverted in
situations such as this. During the interaction between the two
media at the boundary, the first particle of the more dense 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 more dense 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. Since the more dense medium was originally at rest, an
upward pull can do nothing but cause an upward displacement. For
this reason, the transmitted pulse is not inverted. In fact,
transmitted pulses can never be inverted. Since the particles in
this medium are originally at rest, any change in their state of
motion would be in the same direction as the displacement of the
particles of the incident pulse.TheBeforeandAftersnapshots of the
two media are shown in the diagram below.
Comparisons can also be made between the characteristics of the
transmitted pulse and those of the reflected pulse. Once more there
are several noteworthy characteristics. The transmitted pulse (in
the more dense medium) is traveling slower than the reflected pulse
(in the less dense medium). The transmitted pulse (in the more
dense medium) has a smaller wavelength than the reflected pulse (in
the less dense medium). The speed and the wavelength of the
reflected pulse are the same as the speed and the wavelength of the
incident pulse.One goal of physics is to use physical models and
ideas to explain the observations made of the physical world. So
how can these three characteristics be explained? First recall
fromLesson 2that the speed of a wave is dependent upon the
properties of the medium. In this case, the transmitted and
reflected pulses are traveling in two distinctly different media.
Waves always travel fastest in the least dense medium. Thus, the
reflected pulse will be traveling faster than the transmitted
pulse. Second, particles in the more dense medium will be vibrating
with the same frequency as particles in the less dense medium.
Since the transmitted pulse was introduced into the more dense
medium by the vibrations of particles in the less dense medium,
they must be vibrating at the same frequency. So the reflected and
transmitted pulses have the different speeds but the same
frequency. Since the wavelength of a wave depends upon the
frequency and the speed, the wave with the greatest speed must also
have the greatest wavelength. Finally, the incident and the
reflected pulse share the same medium. Since the two pulses are in
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.Transmission
of a Pulse Across a Boundary from More to Less DenseFinally, let's
consider a thick rope attached to a thin rope, with the incident
pulse originating in the thick rope. If this is the case, there
will be an incident pulse traveling in the more dense medium (thick
rope) towards the boundary with a less dense medium (thin rope).
Once again there will be partial reflection and partial
transmission at the boundary. The reflected pulse in this situation
will not be inverted. Similarly, the transmitted pulse is not
inverted (as is always the case). Since the incident pulse is in a
heavier medium, when it reaches the boundary, the first particle of
the less dense medium does not have sufficient mass to overpower
the last particle of the more dense medium. The result is that an
upward displaced pulse incident towards the boundary will reflect
as an upward displaced pulse. For the same reasons, a downward
displaced pulse incident towards the boundary will reflect as a
downward displaced pulse.TheBeforeandAftersnapshots of the two
media are shown in the diagram below.
Comparisons between the characteristics of the transmitted pulse
and the reflected pulse lead to the following observations. The
transmitted pulse (in the less dense medium) is traveling faster
than the reflected pulse (in the more dense medium). The
transmitted pulse (in the less dense medium) has a larger
wavelength than the reflected pulse (in the more dense medium). The
speed and the wavelength of the reflected pulse are the same as the
speed and the wavelength of the incident pulse.These three
observations are explained using the same logic as usedabove.
Check Your UnderstandingCase 1: A pulse in a more dense medium
is traveling towards the boundary with a less dense medium.
1. The reflected pulse in medium 1 ________ (will, will not) be
inverted because _______.2. The speed of the transmitted pulse will
be ___________ (greater than, less than, the same as) the speed of
the incident pulse.3. The speed of the reflected pulse will be
______________ (greater than, less than, the same as) the speed of
the incident pulse.4. The wavelength of the transmitted pulse will
be ___________ (greater than, less than, the same as) the
wavelength of the incident pulse.5. The frequency of the
transmitted pulse will be ___________ (greater than, less than, the
same as) the frequency of the incident pulse.Answers1. will not...
because the reflection occurs for a wave in a more dense medium
heading towards a less dense medium.2. faster3. the same as4.
greater than5. the same asCase 2: A pulse in a less dense medium is
traveling towards the boundary with a more dense medium.
6. The reflected pulse in medium 1 ________ (will, will not) be
inverted because _____________.7. The speed of the transmitted
pulse will be ___________ (greater than, less than, the same as)
the speed of the incident pulse.8. The speed of the reflected pulse
will be ______________ (greater than, less than, the same as) the
speed of the incident pulse.9. The wavelength of the transmitted
pulse will be ___________ (greater than, less than, the same as)
the wavelength of the incident pulse.10. The frequency of the
transmitted pulse will be ___________ (greater than, less than, the
same as) the frequency of the incident pulse.Answers6. will...
because the reflection occurs for a wave in a less dense medium
heading towards a more dense medium.7. less than8. the same as9.
less than10. the same as
Reflection, Refraction, DefractionA linear object attached to an
oscillator bobs back and forth within the water, it becomes a
source ofstraightwaves. These straight waves have alternating
crests and troughs. As viewed on the sheet of paper below the tank,
the crests are the dark lines stretching across the paper and the
troughs are the bright lines.These waves will travel through the
water until they encounter an obstacle - such as the wall of the
tank or an object placed within the water. The diagram at the right
depicts a series of straight waves approaching a long barrier
extending at an angle across the tank of water. The direction that
these wavefronts (straight-line crests) are traveling through the
water is represented by the blue arrow. The blue arrow is called
arayand is drawn perpendicular to the wavefronts. Upon reaching the
barrier placed within the water, these waves bounce off the water
and head in a different direction. The diagram below shows the
reflected wavefronts and the reflected ray. Regardless of the angle
at which the wavefronts approach the barrier, one general law of
reflection holds true: the waves will always reflect in such a way
that the angle at which they approach the barrier equals the angle
at which they reflect off the barrier. This is known as thelaw of
reflection. The discussion above pertains to the reflection of
waves off of straight surfaces. But what if the surface is curved,
perhaps in the shape of a parabola? What generalizations can be
made for the reflection of water waves off parabolic surfaces?
Suppose that a rubber tube having the shape of a parabola is placed
within the water. The diagram at the right depicts such a parabolic
barrier in the ripple tank. Several wavefronts are approaching the
barrier; the ray is drawn for these wavefronts. Upon reflection off
the parabolic barrier, the water waves will change direction and
head towards a point. This is depicted in the diagram below. It is
as though all the energy being carried by the water waves is
converged at a single point - the point is known as the focal
point. After passing through the focal point, the waves spread out
through the water. Refraction of WavesReflection involves a change
in direction of waves when they bounce off a barrier.Refractionof
waves involves a change in the direction of waves as they pass from
one medium to another. Refraction, or the bending of the path of
the waves, is accompanied by a change in speed and wavelength of
the waves. InLesson 2, it was mentioned that the speed of a wave is
dependent upon the properties of the medium through which the waves
travel. So if the medium (and its properties) is changed, the speed
of the waves is changed. The most significant property of water
that would affect the speed of waves traveling on its surface is
the depth of the water. Water waves travel fastest when the medium
is the deepest. Thus, if water waves are passing from deep water
into shallow water, they will slow down. And as mentioned inthe
previous section of Lesson 3, this decrease in speed will also be
accompanied by a decrease in wavelength. So as water waves are
transmitted from deep water into shallow water, the speed
decreases,the wavelength decreases, and the direction changes.This
boundary behavior of water waves can be observed in a ripple tank
if the tank is partitioned into a deep and a shallow section. If a
pane of glass is placed in the bottom of the tank, one part of the
tank will be deep and the other part of the tank will be shallow.
Waves traveling from the deep end to the shallow end can be seen to
refract (i.e., bend), decrease wavelength (the wavefronts get
closer together), and slow down (they take a longer time to travel
the same distance). When traveling from deep water to shallow
water, the waves are seen to bend in such a manner that they seem
to be traveling more perpendicular to the surface. If traveling
from shallow water to deep water, the waves bend in the opposite
direction. Diffraction of WavesReflection involves a change in
direction of waves when they bounce off a barrier;refractionof
wavesinvolves a change in the direction of waves as they pass from
one medium to another; anddiffractioninvolves a change in direction
of waves as they pass through an opening or around a barrier in
their path. Water waves have the ability to travel around corners,
around obstacles and through openings. This ability is most obvious
for water waves with longer wavelengths. Diffraction can be
demonstrated by placing small barriers and obstacles in a ripple
tank and observing the path of the water waves as they encounter
the obstacles. The waves are seen to pass around the barrier into
the regions behind it; subsequently the water behind the barrier is
disturbed. The amount of diffraction (the sharpness of the bending)
increases with increasing wavelength and decreases with decreasing
wavelength. In fact, when the wavelength of the waves is smaller
than the obstacle, no noticeable diffraction occurs.Diffraction of
water waves is observed in a harbor as waves bend around small
boats and are found to disturb the water behind them. The same
waves however are unable to diffract around larger boats since
their wavelength is smaller than the boat. Diffraction of sound
waves is commonly observed; we notice sound diffracting around
corners, allowing us to hear others who are speaking to us from
adjacent rooms. Many forest-dwelling birds take advantage of the
diffractive ability of long-wavelength sound waves. Owls for
instance are able to communicate across long distances due to the
fact that their long-wavelengthhootsare able to diffract around
forest trees and carry farther than the short-wavelengthtweetsof
songbirds. Diffraction is observed of light waves but only when the
waves encounter obstacles with extremely small wavelengths (such as
particles suspended in our atmosphere).Reflection, refraction and
diffraction are all boundary behaviors of waves associated with the
bending of the path of a wave. The bending of the path is an
observable behavior when the medium is a two- or three-dimensional
medium. Reflection occurs when there is a bouncing off of a
barrier. Reflection of waves off straight barriers follows the law
of reflection. Reflection of waves off parabolic barriers results
in the convergence of the waves at a focal point. Refraction is the
change in direction of waves that 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.Constructive Interference and Destructive
InterferenceWhat is Interference?Wave interferenceis the phenomenon
that occurs when two waves meet while traveling along the same
medium. The interference of waves causes the medium to take on a
shape that results from the net effect of the two individual waves
upon the particles of the medium. To begin our exploration of wave
interference, consider two pulses of the same amplitude traveling
in different directions along the same medium. Let's suppose that
each displaced upward 1 unit at its crest and has the shape of a
sine wave. As the sine pulses move towards each other, there will
eventually be a moment in time when they are completely overlapped.
At that moment, the resulting shape of the medium would be an
upward displaced sine pulse with an amplitude of 2 units. The
diagrams below depict the before and during interference snapshots
of the medium for two such pulses. The individual sine pulses are
drawn in red and blue and the resulting displacement of the medium
is drawn in green.
Constructive InterferenceThis type of interference is sometimes
called constructive interference.Constructive interferenceis a type
of interference that occurs at any location along the medium where
the two interfering waves have a displacement in the same
direction. In this case, both waves have an upward displacement;
consequently, the medium has an upward displacement that is greater
than the displacement of the two interfering pulses. Constructive
interference is observed at any location where the two interfering
waves are displaced upward. But it is also observed when both
interfering waves are displaced downward. This is shown in the
diagram below for two downward displaced pulses.
In this case, a sine pulse with a maximum displacement of -1
unit (negative means a downward displacement) interferes with a
sine pulse with a maximum displacement of -1 unit. These two pulses
are drawn in red and blue. The resulting shape of the medium is a
sine pulse with a maximum displacement of -2 units.
Destructive InterferenceDestructive interferenceis a type of
interference that occurs at any location along the medium where the
two interfering waves have a displacement in the opposite
direction. For instance, when a sine pulse with a maximum
displacement of +1 unit meets a sine pulse with a maximum
displacement of -1 unit, destructive interference occurs. This is
depicted in the diagram below.
In the diagram above, the interfering pulses have the same
maximum displacement but in opposite directions. The result is that
the two pulses completely destroy each other when they are
completely overlapped. At the instant of complete overlap, there is
no resulting displacement of the particles of the medium. This
"destruction" is not a permanent condition. In fact, to say that
the two waves destroy each other can be partially misleading. When
it is said that the two pulsesdestroy each other, what is meant is
that when overlapped, the effect of one of the pulses on the
displacement of a given particle of the medium isdestroyedor
canceled by the effect of the other pulse. Recall fromLesson 1that
waves transport energy through a medium by means of each individual
particle pulling upon its nearest neighbor. When two pulses with
opposite displacements (i.e., one pulse displaced up and the other
down) meet at a given location, the upward pull of one pulse is
balanced (canceled or destroyed) by the downward pull of the other
pulse. Once the two pulses pass through each other, there is still
an upward displaced pulse and a downward displaced pulse heading in
the same direction that they were heading before the interference.
Destructive interference leads to only a momentary condition in
which the medium's displacement is less than the displacement of
the largest-amplitude wave.The two interfering waves do not need to
have equal amplitudes in opposite directions for destructive
interference to occur. For example, a pulse with a maximum
displacement of +1 unit could meet a pulse with a maximum
displacement of -2 units. The resulting displacement of the medium
during complete overlap is -1 unit.
This is still destructive interference since the two interfering
pulses have opposite displacements. In this case, the destructive
nature of the interference does not lead to complete
cancellation.Interestingly, the meeting of two waves along a medium
does not alter the individual waves or even deviate them from their
path. This only becomes an astounding behavior when it is compared
to what happens when two billiard balls meet or two football
players meet. Billiard balls might crash and bounce off each other
and football players might crash and come to a stop. Yet two waves
will meet, produce a net resulting shape of the medium, and then
continue on doing what they were doing before the interference.
The Doppler EffectSuppose that there is a happy bug in the
center of a circular water puddle. The bug is periodically shaking
itslegs in order to produce disturbances that travel through the
water. If these disturbances originate at a point, then they would
travel outward from that point in all directions. Since each
disturbance is traveling in the same medium, they would all travel
in every direction at the same speed. The pattern produced by the
bug'sshaking would be a series of concentric circles as shown in
the diagram at the right. These circles would reach the edges of
the water puddle at the same frequency. An observer at point A (the
left edge of the puddle) would observe the disturbances to strike
the puddle's edge at the same frequency that would be observed by
an observer at point B (at the right edge of the puddle). In fact,
the frequency at which disturbances reach the edge of the puddle
would be the same as the frequency at which the bug produces the
disturbances. If the bug produces disturbances at a frequency of 2
per second, then each observer would observe them approaching at a
frequency of 2 per second.Now suppose that our bug is moving to the
right across the puddle of water and producing disturbances atthe
same frequency of 2 disturbances per second. Since the bug is
moving towards the right, each consecutive disturbance originates
from a position that is closer to observer B and farther from
observer A. Subsequently, each consecutive disturbance has a
shorter distance to travel before reaching observer B and thus
takes less time to reach observer B. Thus, observer B observes that
the frequency of arrival of the disturbances is higher than the
frequency at which disturbances are produced. On the other hand,
each consecutive disturbance has a further distance to travel
before reaching observer A. For this reason, observer A observes a
frequency of arrival that is less than the frequency at which the
disturbances are produced. The net effect of the motion of the bug
(the source of waves) is that the observer towards whom the bug is
moving observes a frequency that is higher than 2
disturbances/second; and the observer away from whom the bug is
moving observes a frequency that is less than 2
disturbances/second. This effect is known as theDoppler effect.
What is the Doppler Effect?The Doppler Effect is observed
whenever the source of waves is moving with respect to an observer.
The Doppler effectcan be described as the effect produced by a
moving source of waves in which there is an apparent upward shift
in frequency for observers towards whom the source is approaching
and an apparent downward shift in frequency for observers from whom
the source is receding. It is important to note that the effect
does not result because of anactualchange in the frequency of the
source. Using the example above, the bug is still producing
disturbances at a rate of 2 disturbances per second; it just
appears to the observer whom the bug is approaching that the
disturbances are being produced at a frequency greater than 2
disturbances/second. The effect is only observed because the
distance between observer B and the bug is decreasing and the
distance between observer A and the bug is increasing.The Doppler
effect can be observed for any type of wave - water wave, sound
wave, light wave, etc. We are most familiar with the Doppler Effect
because of our experiences with sound waves. Perhaps you recall an
instance in which a police car or emergency vehicle was traveling
towards you on the highway. 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.The Doppler Effect in AstronomyThe Doppler effect is of
intense interest to astronomers who use the information about the
shift in frequency of electromagnetic waves produced by moving
stars in our galaxy and beyond in order to derive information about
those stars and galaxies. The belief that the universe is expanding
is based in part upon observations of electromagnetic waves emitted
by stars in distant galaxies. Furthermore, specific information
about stars within galaxies can be determined by application of the
Doppler effect. Galaxies are clusters of stars that typically
rotate about some center of mass point. Electromagnetic radiation
emitted by such stars in a distant galaxy would appear to be
shifted downward in frequency (ared shift) if the star is rotating
in its cluster in a direction that is away from the Earth. On the
other hand, there is an upward shift in frequency (ablue shift) of
such observed radiation if the star is rotating in a direction that
is towards the Earth.
Standing WaveIt is however possible to have a wave confined to a
given space in a medium and still produce a regular wave pattern
that is readily discernible amidst the motion of the medium. For
instance, if an elastic rope is held end-to-end and vibratedat just
the right frequency, a wave pattern would be produced that assumes
the shape of a sine wave and is seen to change over time. The wave
pattern is only produced when one end of the rope is vibrated at
just the right frequency. 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. Because the observed wave pattern is
characterized by points that appear to be standing still, the
pattern is often called astanding wave pattern. There are other
points along the medium whose displacement changes over time, but
in a regular manner. These points vibrate back and forth from a
positive displacement to a negative displacement; the vibrations
occur at regular time intervals such that the motion of the medium
is regular and repeating. A pattern is readily observable.The
diagram at the right depicts a standing wave pattern in a medium. A
snapshot of the medium over time is depicted using various colors.
Note that point A on the medium moves from a maximum positive to a
maximum negative displacement over time. The diagram only shows
one-half cycle of the motion of the standing wave pattern. The
motion would continue and persist, with point A returning to the
same maximum positive displacement and then continuing its
back-and-forth vibration between the up to the down position. Note
that point B on the medium is a point that never moves. Point B is
a point of no displacement. Such points are known asnodes. The
standing wave pattern that is shown at the right is just one of
many different patterns that could be produced within the rope.
What are Nodes and Antinodes?One characteristic of every standing
wave pattern is that there are points along the medium that appear
to be standing still. These points, sometimes described as points
of no displacement, are referred to asnodes. There are other points
along the medium that undergo vibrations between a large positive
and large negative displacement. These are the points that undergo
the maximum displacement during each vibrational cycle of the
standing wave. In a sense, these points are the opposite of nodes,
and so they are calledantinodes. A standing wave pattern always
consists of an alternating pattern of nodes and antinodes. The
animation shown below depicts a rope vibrating with a standing wave
pattern. The nodes and antinodes are labeled on the diagram. When a
standing wave pattern is established in a medium, the nodes and the
antinodes are always located at the same position along the medium;
they arestanding still. It is this characteristic that has earned
the pattern the namestanding wave.Standing Wave DiagramsThe
positioning of the nodes and antinodes in a standing wave pattern
can be explained by focusing on the interference of the two waves.
The nodes are produced at locations where destructive interference
occurs. For instance, nodes form at locations where a crest of one
wave meets a trough of a second wave; or ahalf-crestof one wave
meets ahalf-troughof a second wave; or aquarter-crestof one wave
meets aquarter-troughof a second wave; etc. Antinodes, on the other
hand, are produced at locations where constructive interference
occurs. For instance, if a crest of one wave meets a crest of a
second wave, a point of large positive displacement results.
Similarly, if a trough of one wave meets a trough of a second wave,
a point of large negative displacement results. Antinodes are
always vibrating back and forth between these points of large
positive and large negative displacement; this is because during a
complete cycle of vibration, a crest will meet a crest; and then
one-half cycle later, a trough will meet a trough. Because
antinodes are vibrating back and forth between a large positive and
large negative displacement, a diagram of a standing wave is
sometimes depicted by drawing the shape of the medium at an instant
in time and at an instant one-half vibrational cycle later. This is
done in the diagram below.
Nodes and antinodes should not be confused with crests and
troughs. When the motion of atraveling waveis discussed, it is
customary to refer to a point of large maximum displacement as
acrestand a point of large negative displacement as atrough. These
represent pointsof the disturbancethat travel from one location to
another through the medium. An antinode on the other hand is a
pointon the mediumthat is staying in the same location.
Furthermore, an antinode vibrates back and forth between a large
upward and a large downward displacement. And finally, nodes and
antinodes are not actually part of a wave. Recall that a standing
wave is not actually a wave but rather a pattern that results from
the interference of two or more waves. Since a standing wave is not
technically a wave, an antinode is not technically a point on a
wave. The nodes and antinodes are merely unique points on the
medium that make up the wave pattern.Check Your Understanding1.
Suppose that there was arideat an amusement park that was titledThe
Standing Wave. Which location - node or antinode - on the ride
would give the greatest thrill?Answer:The antinodeThe antinode is
continually vibrating from a high to a low displacement - now that
would be a ride.
2. A standing wave is formed when ____.a. a wave refracts due to
changes in the properties of the medium.b. a wave reflects off a
canyon wall and is heard shortly after it is formed.c. red, orange,
and yellow wavelengths bend around suspended atmospheric
particles.d. two identical waves moving different directions along
the same medium interfere.Answer:D3. The number of nodes in the
standing wave shown in the diagram at the right is ____.a. 6b.
7
c. 8d. 14
Answer:C (8 nodes)There are eight positions along the medium
which have no displacement. Be sure to avoid the common mistake of
not counting the end positions.
4. The number of antinodes in the standing wave shown in the
diagram above right is ____.a. 6b. 7c. 8d. 14
Answer:B (7 antinodes)There are seven positions along the medium
which have vibrate between a large positive and a large negative
displacement.Be sure to avoid the common mistake of counting the
antinodal positions twice. An antinode is simply a point along a
medium which undergoes maximum displacement above and below the
rest position. Do not count these positions twice.
Consider the standing wave pattern at the right in answering
these next two questions.5. The number of nodes in the entire
pattern is ___.a. 7b. 8
c. 9d. 16
Answer:C (9 nodes)There are nine positions along the medium
which have no displacement. (Be sure to avoid the common mistake of
not counting the end positions.)
6. Of all the labeled points, destructive interference occurs at
point(s) ____.a. B, C, and Db. A, E, and Fc. A only
d. C onlye. all points
Answer:ADestructive interference has occurred at points B, C and
D to produce the nodes which are seen at these pointsFirst Harmonic
Standing Wave Pattern Second Harmonic Standing Wave Pattern
Third Harmonic Standing Wave PatternOne full wave equals two
loops. So then each loop is equal one half a wave length.
n represents the number of antinodes in a standing waveThe
Electromagnetic and Visible SpectraElectromagnetic wavesare waves
that are capable of traveling through a vacuum. Unlikemechanical
wavesthat require a medium in order to transport their energy,
electromagnetic waves are capable of transporting energy through
the vacuum of outer space. Electromagnetic waves are produced by a
vibrating electric charge and as such, they consist of both an
electric and a magnetic component. The precise nature of such
electromagnetic waves is not discussed in The Physics Classroom
Tutorial. Nonetheless, there are a variety of statements that can
be made about such waves.Electromagnetic waves exist with an
enormous range of frequencies. This continuous range of frequencies
is known as theelectromagnetic spectrum. The entire range of the
spectrum is often broken into specific regions. The subdividing of
the entire spectrum into smaller spectra is done mostly on the
basis of how each region of electromagnetic waves interacts with
matter. The diagram below depicts the electromagnetic spectrum and
its various regions. The longer wavelength, lower frequency regions
are located on the far left of the spectrum and the shorter
wavelength, higher frequency regions are on the far right. Two very
narrow regions within the spectrum are the visible light region and
the X-ray region. You are undoubtedly familiar with some of the
other regions of the electromagnetic spectrum.
Visible Light SpectrumThe focus of Lesson 2 will be upon the
visible light region - the very narrow band of wavelengths located
to the right of the infrared region and to the left of the
ultraviolet region. Though electromagnetic waves exist in a vast
range of wavelengths, our eyes are sensitive to only a very narrow
band. Since this narrow band of wavelengths is the means by which
humans see, we refer to it as thevisible light spectrum. Normally
when we use the term "light," we are referring to a type of
electromagnetic wave that stimulates the retina of our eyes. In
this sense, we are referring to visible light, a small spectrum
from the enormous range of frequencies of electromagnetic
radiation. This visible light region consists of a spectrum of
wavelengths that range from approximately 700 nanometers
(abbreviated nm) to approximately 400 nm. Expressed in more
familiar units, the range of wavelengths extends from 7 x 10-7meter
to 4 x 10-7meter. This narrow band of visible light is
affectionately known asROYGBIV.Each individual wavelength within
the spectrum of visible light wavelengths is representative of a
particular color. That is, when light of that particular wavelength
strikes the retina of our eye, we perceivethat specific color
sensation. Isaac Newton showed thatlight shining through a prism
will be separated into its different wavelengthsand will thus show
the various colors that visible light is comprised of. The
separation of visible light into its different colors is known
asdispersion. Each color is characteristic of a distinct
wavelength; and different wavelengths of light waves will bend
varying amounts upon passage through a prism. For these reasons,
visible light is dispersed upon passage through a prism. Dispersion
of visible light produces the colors red (R), orange (O), yellow
(Y), green (G), blue (B), and violet (V). It is because of this
that visible light is sometimes referred to as ROY G.
BIV.(Incidentally, the indigo is not actually observed in the
spectrum but is traditionally added to the list so that there is a
vowel in Roy's last name.) The red wavelengths of light are the
longer wavelengths and the violet wavelengths of light are the
shorter wavelengths. Between red and violet, there is a continuous
range or spectrum of wavelengths.The visible light spectrum is
shown in the diagram below.
When all the wavelengths of the visible light spectrum strike
your eye at the same time, white is perceived. The sensation of
white is not the result of a single color of light. Rather, the
sensation of white is the result of a mixture of two or more colors
of light. Thus, visible light - the mix of ROYGBIV - is sometimes
referred to as white light. Technically speaking, white is not a
color at all - at least not in the sense that there is a light wave
with a wavelength that is characteristic of white. Rather, white is
the combination of all the colors of the visible light spectrum. If
all the wavelengths of the visible light spectrum give the
appearance of white, then none of the wavelengths would lead to the
appearance of black. Once more, black is not actually a color.
Technically speaking, black is merely the absence of the
wavelengths of the visible light spectrum. So when you are in a
room with no lights and everything around you appears black, it
means that there are no wavelengths of visible light striking your
eye as you sight at the surroundings.