lesson 1: lesson 2: properties of waves...were no disturbance moving through it. once a disturbance...

The Anatomy of a Wave

Lesson 1: The Nature of a Wave

Waves and wavelike Motion

What is a Wave?

Categories of Waves

Lesson 2: Properties of a Wave


Frequency and Period

Energy Transport and Amplitude

The Speed of a Wave

The Wave Equation

Lesson 3: Behavior of Waves

Lesson 2: Properties of Waves


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 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 to freeze the 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 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. 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 and F on the diagram represent the crests of this wave. The crest of a wave is the point on the medium which exhibits the maximum amount of positive or upwards displacement from the rest position. Points D and I on the diagram represent the troughs of this wave. The trough of a wave is the point on the medium which exhibits the maximum amount of negative or downwards displacement from the rest position.

The wave shown above can be described by a variety of properties. One such property is amplitude. The amplitude of a wave refers to the maximum amount of displacement of a 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. In the diagram above, the amplitude could be measured as the distance of a line segment which is perpendicular to the rest position and extends vertically upward from the rest position to point A.

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Boundary Behavior

Reflection, Refraction, and Diffraction

Interference of Waves

Waves Generated by Moving Sources

Lesson 4: Standing Waves

Traveling Waves vs. Standing Waves

Formation of Standing Waves

Nodes and Anti-nodes

Harmonics and Patterns

Mathematics of Standing Waves

The wavelength is another property of a wave which is portrayed in the diagram above. The wavelength of 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 has a repeating pattern. And the length of one such repetition (known as a wave 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 F, or the horizontal distance from B to G, or the horizontal distance from E to J, or the horizontal distance from D to I, or the horizontal distance from C to H. Any one of these distance measurements would suffice in determining the wavelength of this 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. If a snapshot of such a longitudinal wave could be taken so as to freeze the shape of the slinky in time, then it would look like the following diagram.

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. A compression is a point on a medium through which a longitudinal wave is traveling which has the maximum density. A region where the coils are spread apart, thus maximizing the distance between coils, is known as a rarefaction. A rarefaction is a point on a medium through which a longitudinal wave is traveling which 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.

As discussed above, the wavelength of a wave is the length of one complete cycle of a wave. For a transverse wave, the wavelength is determined by measuring from crest to crest. A longitudinal wave does not have crest; so how can its wavelength be






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determined? The wavelength can always be determined by measuring the distance between any two corresponding points on adjacent waves. 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. On the diagram above, the distance from point A to point C or from point B to point D would be representative of the wavelength.

Check Your Understanding

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 ______.

2. The amplitude of the wave in the diagram above is given by letter _____.

3. Indicate the interval which represents one full wavelength.



a. a to c2. b to d3. a to g4. c to g


● 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

Go to Lesson 3

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Frequency and Period of a Wave



What is a Wave?

Categories of Waves





The Speed of a Wave

The Wave Equation




The nature of a wave was discussed in Lesson 1 of this unit. In that lesson, it was mentioned that a wave is created in a slinky by the periodic and repeating vibration of the first coil of the slinky. This vibration creates a disturbance which moves through the slinky and transports energy from the first coil to the last coil. A single back-and-forth vibration of the first coil of a slinky introduces a pulse into the medium. But the act of continually vibrating the first coil with a back-and-forth motion in periodic fashion introduces a wave into the slinky.

Suppose 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. 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. The frequency of 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 "How frequently do you mow the lawn during the summer months?" Of course the question is an inquiry about how often the 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 quantity frequency would have units of cycles/second, waves/second, vibrations/second, or something/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 quantity frequency is often confused with the quantity period. Period refers to the time which it takes to do something. When an event occurs repeatedly, then we say that

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Boundary Behavior










the event is periodic and refer to the time for the event to repeat itself as the period. The period of 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. When a physics teacher is regular with his stools, the period of the stools is 24 hours. That doesn't mean he spends 24 hours on the stool, it merely means that it takes 24 hours before he must return to the stools to repeat the daily cycle. (Of course, this assumes that a trip to the stools is a periodic event for that teacher.)

Frequency and period are distinctly 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 one 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 symbol f is used for frequency and the symbol T is used for period, these equations are also expressed as:

The quantity frequency is also confused with the quantity speed. The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. For a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given period of time. So while wave frequency refers to the number of cycles occurring per second, wave speed refers to the meters traveled per second. A wave can vibrate back and forth very


















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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. Wave speed will be discussed in more detail later in this lesson.


Throughout this unit, internalize the meaning of terms such as period, frequency, and wavelength. Utilize the meaning of these terms to answer conceptual questions; avoid a formula fixation.

1. A wave has an amplitude of 2 cm and a frequency of 12 Hz, and the distance from a crest to the nearest trough is measured to be 5 cm. Determine the period of such a wave.

2. A fly flaps its wings back and forth 150 times each second. The period of a wing flap is

a. 150 sec b. 2.5 sec c. 0.040 sec d. 0.0067 sec

3. A tennis coach paces back and forth along the sideline 10 times in 2 minutes. The frequency of her pacing is ________.

a. 5.0 Hz b. 0.20 Hz c. 0.12 Hz d. 0.083 Hz



4. The frequency of rotation of a second hand on a clock is _______.

a. 1/60 Hz b. 1/12 Hz c. 1/2 Hz d. 1 Hz e. 60 Hz

5. A kid on a playground swing makes a complete to-and-fro swing each 2 seconds. The frequency of swing is _________.

a. 0.5 Hz b. 1 Hz c. 2 Hz

6. In problem #5, the period of swing is __________.

a. 0.5 second b. 1 second c. 2 second

7. A period of 5.0 seconds corresponds to a frequency of ________ Hertz.



a. 0.2 b. 0.5 c. 0.02 d. 0.05 e. 0.002

8. A pendulum makes 40 complete back-and-forth cycles of vibration in 20 seconds. Calculate its period?

9. A child in a swing makes one complete back and forth motion in 4.0 seconds. This statement provides information about the child's

a. speed b. frequency c. period

10. The period of a 440 Hertz sound wave is ___________.

11. As the frequency of a wave increases, the period of the wave ___________.



a. decreases b. increases c. remains the same



Go to Lesson 3













Energy Transport and the Amplitude of a Wave



What is a Wave?

Categories of Waves





The Speed of a Wave

The Wave Equation




As mentioned earlier, a wave is an energy transport phenomenon which transports energy along a medium without transporting matter. A pulse or a wave is introduced into a slinky when a person holds the first coil and gives it a back-and-forth motion. This creates a disturbance within the medium; this disturbance subsequently travels from coil to coil, transporting energy as it moves. The energy is imparted to the medium by the person as he/she does work upon the first coil to give it kinetic energy. This energy is transferred from coil to coil until it arrives at the end of the slinky. If you were holding the opposite end of the slinky, then you would feel the energy as it reaches your end. In fact, a high energy pulse would likely do some rather noticeable work upon your hand upon reaching the end of the medium; the last coil of the medium would displace you hand in the same direction of motion of the coil. For the same reasons, a high energy ocean wave does considerable damage to the piers along the shoreline when it crashes upon it.

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. As discussed earlier in Lesson 2, the amplitude of a wave refers to the maximum amount of displacement of a a particle on the medium from its rest position. The logic underlying the energy-amplitude relationship is as follows: If a slinky is stretched out in a horizontal direction and a transverse pulse is introduced into the slinky, the first coil is given an initial amount of displacement. The displacement is due to the force applied by the person upon the coil to displace it a given amount from rest. The more energy that the person puts into the pulse, the more work which he/she will do upon the first coil. The more work which is done upon the first coil, the more displacement which is given to it. The more displacement which is given to the first coil, the more amplitude which it will have. So in the end, the amplitude of a transverse pulse is related to the energy which that pulse transport through the medium. Putting a lot of energy into a transverse pulse will not effect the wavelength, the frequency or the speed of the pulse. The energy imparted to a pulse will only effect the amplitude of that pulse.

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Boundary Behavior










Consider two identical slinkies into which a pulse is introduced. If the same amount of energy is introduced into each slinky, then each pulse will have the same amplitude. But what if the slinkies are different? What if one is made of zinc and the other is made of copper? Will the amplitudes now be the same or different? If a pulse is introduced into two different slinkies by imparting the same amount of energy, then the amplitudes of the pulses will not necessarily be the same. In a situation such as this, the actual amplitude assumed by the pulse is dependent upon two types of factors: an inertial factor and an elastic factor. Two different material have different mass densities. The imparting of energy to the first coil of a slinky is done by the application of a force to this coil. More massive slinkies have a greater inertia and thus tend to resist the force; this increased resistance by the greater mass tends to cause a reduction in the amplitude of the pulse. Different materials also have differing degrees of springiness or elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.

The energy transported by a wave is directly proportional to the square of the amplitude of the wave. This energy-amplitude relationship is sometimes expressed in the following manner.

This means that a doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. The table at the right further expresses this energy-amplitude relationship. Observe that whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared. For example, changing the amplitude from 1 unit to 2 units represents a 2-fold increase in the amplitude and should be accompanied by a 4-fold (22) increase in the energy; thus 2 units becomes 4 times bigger - 8 units. As another example,


















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changing the amplitude from 1 unit to 4 units represents a 4-fold increase in the amplitude and should be accompanied by a 16-fold (42) increase in the energy; thus 2 units becomes 16 times bigger - 32 units.


1. A transverse wave is found to have a distance of 8 cm from a trough to a crest, a frequency of 12 Hz, and a distance of 6 cm from a crest to the nearest trough. Determine the amplitude, period, and wavelength of such a wave.

2. An ocean wave has an amplitude of 2.5 m. Weather conditions suddenly change such that the wave has an amplitude of 5.0 m. The amount of energy transported by the wave is __________.

a. halved2. doubled3. quadrupled4. remains the same

3. Two waves are traveling through a container of nitrogen gas. Wave A has an amplitude of .1 cm. Wave B has an amplitude of .2 cm. The energy transported by wave B must be __________ the energy transported by wave A.

a. one-fourth2. one-half3. two times larger than4. four times larger than





Go to Lesson 3













The Speed of a Wave



What is a Wave?

Categories of Waves





The Speed of a Wave

The Wave Equation



The Speed of a Wave

A wave is a disturbance which 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 to cover distance. The speed of 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 5 seconds, then the speed of the ocean wave is 4 m/s. On the other hand, if the crest of an ocean wave moves a distance of 25 meters in 5 seconds (the same amount of time), then the speed of this ocean wave is 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. For example, a wave introduced by a person into one end of a slinky will travel through the slinky and eventually reach the end of the slinky and the presence of the hand of a second person. One behavior which waves undergo at the end of a medium is reflection. The wave will reflect or bounce off the person's hand. When a wave undergoes reflection, it remains within the medium and merely reverses its direction of travel. In the case of a slinky wave, the disturbance can be seen traveling back to the original end. A slinky wave which travels to the end of a slinky and back has doubled its distance. That is, by reflecting back to the original location, the wave has traveled a distance which is equal to twice the length of the slinky.

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The Speed of a Wave

Boundary Behavior










Reflection phenomenon are commonly observed with sound waves. When you let out a holler within a canyon, you often hear the echo of the holler. The sound wave travels through the medium (air in this case), reflects off the canyon wall and returns to its origin (you); the result is that you hear the echo (the reflected sound wave) of your holler. A classic physics problem goes like this:

If an echo is heard one second after the holler and reflects off canyon walls which are a distance of 170 meters away, then what is the speed of the wave?

In this instance, the sound wave travels 340 meters in 1 second, so the speed of the wave is 340 m/s. Remember, when there is a reflection, the wave doubles its distance. In other words, the distance traveled by the sound wave in 1 second is equivalent to the 170 meters down to the canyon wall plus the 170 meters back from the canyon wall.

What variables effect the speed at which a wave travels through a medium? Does the frequency or wavelength of the wave effect its speed? Does the amplitude of the wave effect its speed? Or are other variables such as the mass density of the medium or the elasticity of the medium responsible for effecting the speed of the wave? These questions were investigated in the Speed of a Standing Wave Lab performed in class. A wave generator was used to produce several waves within a rope of a measurable tension. The wavelength, frequency and speed were determined. Then the frequency of vibration of the generator was systematically changed to investigate the effect of frequency upon wave speed. Finally, the tension of the rope was altered to investigate the effect of tension upon wave speed. Sample data for the experiment are shown below.

Speed of a Standing Wave Lab - Sample Data

Trial

Tension

(N)

Frequency

(Hz)

Wavelength

(m)

Speed

(m/s)

1 2.0 4.05 4.00 16.2

2 2.0 8.03 2.00 16.1

3 2.0 12.30 1.33 16.4

4 2.0 16.25 1.00 16.3


















The Speed of a Wave

5 2.0 20.25 0.800 16.2

6 5.0 12.8 2.00 25.6

7 5.0 19.3 1.33 25.7

8 5.0 25.45 1.33 25.5

In the first five trials, the tension of the rope was held constant and the frequency was systematically changed. The data in rows 1-5 of the table above demonstrate that a change in the frequency of a wave does not effect the speed of the wave. The speed remained a near constant value of approximately 16.2 m/s. The small variations in the values for the speed were the result of experimental error, rather than a demonstration of some physical law. The data convincingly show that wave frequency does not effect wave speed.

The last three trials involved the same procedure with a different rope tension. Observe that the speed of the waves in rows 6-8 are distinctly different than the speed of the wave in rows 1-5. The obvious cause of this difference is the alteration of the tension of the rope. The speed of the waves was significantly faster at higher tensions. So while the frequency did not effect the speed of the wave, the tension in the medium (the rope) did.

A similar study was conducted in the Exploring Waves Simulation conducted in class. In this simulation, various properties of a wave (frequency, wavelength, and amplitude) were systematically altered to see if they effected the wave speed. Then various properties of the medium (mass density, spring constant, and damping coefficient) through which the wave traveled were altered to see if they effected the wave speed. The outcome of the study revealed that the speed of a wave was not dependent (causally effected by) the properties of the wave; rather the speed of the wave was dependent upon the properties of the medium.

One theme of this unit has been that "a wave is a disturbance moving through a medium." There are two distinct objects in this phrase - the "wave" and the "medium." The medium could be water, air, or a slinky. These media are distinguished by their properties - the material they are made of and the physical properties of that material such as the density, the temperature, the elasticity, etc. Such physical properties describe the material itself, not the wave. On the other hand, waves are distinguished from each other by their properties - amplitude, wavelength, frequency, etc. These properties describe the wave, not the material through which the wave is moving. The lesson of the Exploring Waves Simulation and the Speed of a Standing Wave Lab is that wave speed depends upon the medium through which the wave is moving. Only an alteration in the properties of the medium will cause a change in the speed.


The Speed of a Wave


1. A teacher attaches a slinky to the wall and begins introducing pulses with different amplitude. Which of the two pulses (A or B) below will reach the wall first? Justify your answer.

2. The teacher then begins introducing pulses with a different wavelength. Which of the two pulses (C or D) will reach the wall first? Justify your answer.

3. The time required for the waves (v = 340 m/s) to travel from the tuning fork to point A is


The Speed of a Wave

a. 0.020 sec.2. 0.059 sec.3. 0.59 sec.4. 2.9 sec.

4. Two waves are traveling through the same container of nitrogen gas. Wave A has a wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The speed of wave B must be ________ the speed of wave A.

a. one-ninth2. one-third3. the same as4. three times larger than

5. An automatic focus camera is able to focus on objects by use of an ultrasonic sound wave. The camera sends out sound waves which 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. If a sound wave (speed = 340 m/s) returns to the camera 0.150 seconds after leaving the camera, how far away is the object?


The Speed of a Wave

6. Doubling the frequency of a wave source doubles the speed of the waves.

a. True2. False

7. While hiking through a canyon, Noah Formula lets out a scream. An echo (reflection of the scream off a nearby canyon wall) is heard 1.6 seconds after the scream. The speed of the sound wave in air is 345 m/s. Calculate the distance from Noah to the nearby canyon wall.

8. Mac and Tosh are resting on top of the water near the end of the pool when Mac creates a surface wave. The wave travels the length of the pool and back in 25 seconds. The pool is 25 meters long. Determine the speed of the wave.

9. The water waves below are traveling along the surface at a speed of 2 m/s and splashing periodically against Wilbert's perch. Each adjacent crest is 4 meters apart and splashes Wilbert's feet upon reaching his perch. How much time passes between each successive drenching? Answer and explain using complete sentences.


The Speed of a Wave



Go to Lesson 3












The Speed of a Wave



The Wave Equation



What is a Wave?

Categories of Waves





The Speed of a Wave

The Wave Equation



The Wave Equation

As was discussed in Lesson 1, a wave is produced when a vibrating source displaces the first particle of a medium. This creates a disturbance which begins to travel along the medium from particle to particle. The frequency at which each individual particle vibrates is equal to the frequency at which the 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. In one period, the source is able to displace the first particle upwards from rest, back to rest, downwards from rest, and finally back to rest. This complete back-and-forth movement constitutes one complete wave cycle.

The 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 it takes that from the first to the last snapshot, the hand has made one complete back-and-forth motion. A period has elapsed. Observe that during this same amount of time, 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

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The Wave Equation

Boundary Behavior










the form:

Speed = Wavelength * Frequency

The 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 symbols v, , and f, the equation can be rewritten as

v = f *

As a test of your understanding of the wave equation and its mathematical use in analyzing wave motion, consider the following three-part question:

Stan and Anna are conducting a slinky experiment. They are studying the possible effect of several variables upon the speed of a wave in a slinky. Their data table is shown below. 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 _________


Copper, 1-in. dia. coils 1.19 m 2.1 Hz _________

Copper, 1-in. dia. coils 0.60 m 4.2 Hz _________



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

a. decrease b. increase c. remain the same


















The Wave Equation

b. As the wavelength of a wave in a uniform medium increases, its frequency will _____.

a. decrease b. increase c. remain the same

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

i. the properties of the medium through which the wave travels2. the wavelength of the wave.3. the frequency of the wave.4. both the wavelength and the frequency of the wave.

The above example illustrates how to use the wave equation to solve mathematical problems. It also illustrates the principle that wave speed is dependent upon medium properties and independent of wave properties. Even though the wave speed is calculated by multiplying wavelength by frequency, an alteration in wavelength does not effect wave speed. Rather, an alteration in wavelength effects 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.


The Wave Equation


1. 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. 2:1 b. 1:2 c. 4:1 d. 1:4

2. A transverse wave is found to have a distance of 4 cm from a trough to a crest, a frequency of 12 Hz, and a distance of 5 cm from a crest to the nearest trough. Determine the amplitude, period, wavelength and speed of such a wave.

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 constant2. the wavelength remains constant and the speed is doubled3. both the wavelength and the speed are halved.4. both the wavelength and the speed remain constant.

4. The annoying sound from a mosquito is produced when it beats its wings at the average rate of 600 wing beats per second.

a. What is the frequency in Hertz of the sound wave?


The Wave Equation

b. Assuming the sound wave moves with a velocity of 340 m/s, what is the wavelength of the wave?

5. A marine weather station reports waves along the shore that are 2 meters high, 8 meters long, and reach the station 8 seconds apart. Determine the frequency and the speed of these waves.

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 Wave Equation



Go to Lesson 3
















What is a Wave?

Categories of Waves





The Speed of a Wave

The Wave Equation




What happens when two waves meet while they travel through the same medium? What effect will the meeting of the waves have upon the appearance of the medium? Will the two waves bounce off each other upon meeting (much like two billiard balls would) or will the two waves pass through each other? These questions involving the meeting of two or more waves along the same medium pertain to the topic of wave interference.

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. 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 crest has an amplitude of +1 unit (the positive indicates an upward displacement as would be expected for a crest) and has the shape of a sine wave. As the sine crests 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 a sine crest with an amplitude of +2 units. The diagrams below depict the before- and during interference snapshots of the medium for two such crests. The individual sine crests are drawn in red and blue and the resulting displacement of the medium is drawn in green.

This type of interference is sometimes called constructive interference. Constructive interference is a type of interference which 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 which is greater than the displacement of the two interfering pulses. Constructive interference is observed when a crest meets a crest; but it is also observed when a trough meets a trough as shown in the diagram below.














Boundary Behavior










In this case, a sine trough with an amplitude of -1 unit (negative means a downward displacement) interferes with a sine trough with a displacement of -1 unit. These two troughs are drawn in red and blue. The resulting shape of the medium is a sine trough with a maximum displacement of -2 units.

Destructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine crest with an amplitude of +1 unit meets a sine trough with an amplitude of -1 unit, destructive interference occurs. This is depicted in the diagram below.

In the situation 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 disturbance in 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 pulses "destroy 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 is "destroyed" or canceled by the effect of the other pulse. Recall from Lesson 1 that 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., a crest and trough) meet at a given location, the upward pull of the crest is balanced (canceled or "destroyed") by the downward pull of the trough. Once the two pulses pass through each other, there is still a crest and a trough heading in the same direction which they were heading before 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 crest with an amplitude of +1 unit could meet a trough with an amplitude of -2 units; the resulting displacement of the medium during complete overlap is -1 unit.


















http://www.glenbrook.k12.il.us/gbssci/phys/CLass/waves/u10l1b.html#particle


This is still destructive interference since the two interfering waves have opposite displacement. 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 waves meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.

The task of determining the shape of the resultant demands that the principle of superposition is applied. The principle of superposition is sometimes stated as follows:

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.

In the cases above, the summing the individual displacements for locations of complete overlap was mad out to be an easy task - as easy as simple arithmetic:

Displacement of Pulse 1 Displacement of Pulse 2 = Resulting Displacement

+1 +1 = +2

-1 -1 = -2

+1 -1 = 0



+1 -2 = -1

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. As an example of the complexity of this task, consider the two interfering waves at the right. A snapshot of the shape each of the individual waves at a particular instant in time is shown. To determine the precise shape of the medium at this given instant in time, the principle of superposition must be applied to several locations along the medium. A short-cut involves measuring the displacement from equilibrium at a few strategic locations. Thus, approximately 20 locations have been picked and labeled as A, B, C, D, etc. The actual displacement of each individual wave can be counted by measuring from the equilibrium position up to the particular wave. At position A, there is no displacement for either individual wave; thus, the resulting displacement of the medium will be 0 units. At position B, the smaller wave has a displacement of approximately 1.4 units; the larger wave has a displacement of approximately 2 units; thus, the resulting displacement of the medium will be approximately 3.4 units. At position C, the smaller wave has a displacement of approximately 2 units; the larger wave has a displacement of approximately 4 units; thus, the resulting displacement of the medium will be approximately 6 units. At position D, the smaller wave has a displacement of approximately 1.4 units; the larger wave has a displacement of approximately 2 units; thus, the resulting displacement of the medium will be approximately 3.4 units. This process can be repeated for every position. When finished, a dot (done in green below) can be marked on the "graph" to note the displacement of the medium at each given location. The actual shape of the medium can then be sketched by estimating the position between the various marked points and sketching the wave. This is shown as the green line in the diagram below.




1. Several positions along the medium are labeled with a letter. Categorize each labeled position along the medium as being a position where either constructive or destructive interference occurs.

2. Twin water bugs Jimminy and Johnny are both creating a series of circular waves by jiggling their legs in the water. The waves undergo interference and create the pattern represented in the diagram at the right. The thick lines in the diagram represent wave crests and the thin lines represent wave troughs. Several of positions in the water are labeled with a letter. Categorize each labeled position as being a position where either constructive or destructive interference occurs.




● Boundary Behavior● Reflection, Refraction, and Diffraction● Interference of Waves● The Doppler Effect

Go to Lesson 4













