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**Introduction to Waves: Transverse and Longitudinal**

Physics Coach Stephens

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Definitions ..\Wave Terms and Snakey Spring KEY.pdf

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Parts of a Wave

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**Making Waves Using a Slinky**

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**Calculating Speed, Frequency & Wavelength**

Physics Coach Stephens

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**Measuring the Speed of Waves**

..\Measuring the Speed of Waves.pdf ..\Traveling Wave Problems.pdf Answer Keys: ..\Measuring the Speed of Waves KEY.pdf ..\Traveling Wave Problems KEY.pdf

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**Physics Coach Stephens**

Sound Waves & Sources Physics Coach Stephens

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Bell Work What causes waves? What are some properties of waves?

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**Bell Work Answer Oscillations and vibrating objects**

frequency, speed, wavelength, vibrating source transverse and longitudinal (electromagnetic, sound, and water waves)

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Simulations & Videos

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Vibrations

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**Vibrational Motion Wiggling Back and forth Vibrating Shaking**

Oscillating These phrases describe the motion of a variety of objects. They even describe the motion of matter at the atomic level. Even atoms wiggle - they do the back and forth.

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**Example of Vibrational Motion**

Bobblehead Doll -- A bobblehead doll consists of an oversized replica of a person's head attached by a spring to a body and a stand. A light tap causes it to bobble. The head wiggles; it vibrates; it oscillates. The back and forth doesn't happen forever. Over time, the vibrations tend to die off and the bobblehead stops bobbing and finally assumes its usual resting position. This usual resting position is referred to as the equilibrium position. This means that all forces acting on the object are balanced.

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Forced Vibration If a force is applied to the bobblehead, the equilibrium will be disturbed and it will begin vibrating. We could use the phrase forced vibration to describe the force which sets the resting bobblehead into motion. A short-lived, momentary force begins the motion. Each repetition of its motion is a little less vigorous than its previous repetition. The extent of its displacement from the equilibrium position becomes less and less over time. Because the forced vibration that initiated the motion is a single instance of a short-lived, momentary force, the vibrations ultimately cease. The bobblehead is said to experience damping. Damping is the tendency of a vibrating object to lose or to dissipate its energy over time. Without a sustained forced vibration, the motion of the bobblehead eventually ceases as energy is dissipated to other objects. A sustained input of energy would be required to keep the back and forth motion going. `After all, if the vibrating object naturally loses energy, then it must continuously be put back into the system through a forced vibration in order to sustain the vibration.

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The Restoring Force Why doesn't the bobblehead stop the first time it returns to the equilibrium position? According to Newton's law of inertia, “an object in motion will remain in motion and an object at rest will remain at rest unless acted upon by an unbalanced force.” An object which is moving will continue its motion if the forces are balanced. So every instant in time that the bobblehead is at the equilibrium position, the momentary balance of forces will not stop the motion. It moves past the equilibrium position towards the opposite side of its swing. As the bobblehead is displaced past its equilibrium position, then a force capable of slowing it down and stopping it exists. This force that slows the bobblehead down as it moves away from its equilibrium position is known as a restoring force. The restoring force acts upon the vibrating object to move it back to its original equilibrium position.

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**Vibrational vs. Translational Motion**

Vibrational motion is often contrasted with translational motion. In translational motion, an object is permanently displaced. The initial force that is imparted to the object displaces it from its resting position and sets it into motion. Yet because there is no restoring force, the object continues the motion in its original direction. When an object vibrates, it doesn't move permanently out of position. The restoring force acts to slow it down, change its direction and force it back to its original equilibrium position. An object in translational motion is permanently displaced from its original position. But an object in vibrational motion wiggles about a fixed position - its original equilibrium position.

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**Other Examples of Vibrational Motion**

Bobblehead dolls are not the only objects that vibrate. It might be safe to say that all objects in one way or another can be forced to vibrate to some extent. As long as a force persists to restore the object to its original position, a displacement from its resting position will result in a vibration. A pendulum is a classic example of an object that is considered to vibrate. A simple pendulum consists of a relatively massive object hung by a string from a fixed support. It typically hangs vertically in its equilibrium position. When the mass is displaced from equilibrium, it begins its back and forth vibration about its fixed equilibrium position. An inverted pendulum (i.e. trees, skyscrapers, tennis ball on a dowel rod and support, tuning fork) is another example of vibrational motion. Another example is a mass on a spring. The mass hangs at a resting position. If the mass is pulled down, the spring is stretched. Once the mass is released, it begins to vibrate. It does the back and forth, vibrating about a fixed position. (i.e. springs inside of a mattress, car suspension systems, bathroom scales)

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**Animations and Pictures**

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**Properties of Periodic Motion**

A vibrating object is wiggling about a fixed position. Like the mass on a spring, a vibrating object is moving over the same path over the course of time. Its motion repeats itself over and over again. If it were not for damping, the vibrations would endure forever (or at least until someone catches the mass and brings it to rest). The mass on the spring not only repeats the same motion, it does so in a regular fashion. The time it takes to complete one back and forth cycle is always the same amount of time. If it takes the mass 3.2 seconds to complete the first back and forth cycle, then it will take 3.2 seconds to complete the seventh back and forth cycle. In Physics, a motion that is regular and repeating is referred to as a periodic motion. Most objects that vibrate do so in a regular and repeated fashion; their vibrations are periodic.

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**The Sinusoidal Nature of a Vibration**

Suppose that a motion detector was placed below a vibrating mass on a spring in order to detect the changes in the mass's position over the course of time. And suppose that the data from the motion detector could represent the motion of the mass by a position vs. time plot. The graphic below depicts such a graph. One obvious characteristic of the graph has to do with its shape. Many students recognize the shape of this graph from experiences in Mathematics class. The graph has the shape of a sine wave. A second obvious characteristic of the graph may be its periodic nature. The motion repeats itself in a regular fashion. A third obvious characteristic of the graph is that damping occurs with the mass-spring system. Some energy is being dissipated over the course of time.

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**Language Used to Describe the Graph**

By looking at the graph, what does the motion of the wave appear to be doing? If you said “slowing down” you are incorrect. As shown in the chart below, it took 2.3 seconds to complete the first cycle and 2.3 seconds to complete the sixth cycle. Cycle Letters Times at Beginning and End of Cycle (s) Cycle Time (s) 1st A to E 0.0 sto 2.3 s 2.3 2nd E tp I 2.3 s to 4.6 s 3rd I to M 4.6 s to 7.0 s 2.4 4th M to Q 7.0 s to 9.3 s 5th Q to U 9.3 s to 11.6 s 6th U to Y 11.6 s to 13.9 s

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**Maximum Upward Displacement Maximum Downward Displacement**

Continued… The mass will both speed up and slow down over the course of a single cycle. So to say that the mass is "slowing down" is not entirely accurate since during every cycle there are two short intervals during which it speeds up. The time to complete one cycle of vibration is NOT changing. The extent to which the mass moves above or below the resting position varies over the course of time. In the first full cycle of vibration being shown, the mass moves from its resting position (A) 0.60 m above the motion detector to a high position (B) of 0.99 m cm above the motion detector. This is a total upward displacement of 0.29 m. In the sixth full cycle of vibration that is shown, the mass moves from its resting position (U) 0.60 m above the motion detector to a high position (V) 0.94 m above the motion detector. This is a total upward displacement of 0.24 m cm. The table below summarizes displacement measurements for several other cycles displayed on the graph. Cycle Letters Maximum Upward Displacement Maximum Downward Displacement 1st A to E 0.60 m to 0.99 m 0.60 m to 0.21 m 2nd E to I 0.60 m to 0.98 m 0.60 m to 0.22 m 3rd I to M 0.60 m to 0.97 m 0.60 m to 0.23 m 6th U to Y 0.60 m to 0.94 m 0.60 m to 0.26 m

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**Continued… Over the course of time, the mass continues to vibrate.**

However, the amount of displacement of the mass is decreasing from one cycle to the next. This illustrates that energy is being lost from the mass-spring system. If given enough time, the vibration of the mass will eventually cease as its energy is dissipated. In physics (or at least in the English language), "slowing down" means to "get slower" or to "lose speed". Speed, a physics term, refers to how fast or how slow an object is moving. To say that the mass on the spring is "slowing down" over time is to say that its speed is decreasing over time. But as mentioned, the mass speeds up during two intervals of every cycle -- as the restoring force pulls the mass back towards its resting position the mass speeds up. For this reason, a physicist adopts a different language to communicate the idea that the vibrations are "dying out". We use the phrase "energy is being dissipated or lost" instead of saying the "mass is slowing down.“

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Period & Amplitude The key measurements that have been made are measurements of: Period -- the time for the mass to complete a cycle, and Amplitude -- the maximum displacement of the mass above (or below) the resting position. An object in periodic motion can have a long period or a short period. The terms fast and slow are not used since physics types reserve the words fast and slow to refer to an object's speed. Instead, we use frequent or infrequent. Here in this description we are referring to the frequency, not the speed. An object can be in periodic motion and have a low frequency and a high speed.

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Frequency Frequency is another quantity that can be used to describe the motion of an object in periodic motion. The frequency is defined as the number of complete cycles occurring per period of time. Since the standard metric unit of time is the second, frequency has units of cycles/second. The unit cycles/second is equivalent to the unit Hertz (abbreviated Hz). The unit Hertz is used in honor of Heinrich Rudolf Hertz, a 19th century physicist who expanded our understanding of the electromagnetic theory of light waves.

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**How Often Something Occurs**

Frequency is a word we often use to describe how often something occurs. You might say that you frequently check your Facebook – you check it often. In physics, frequency is used with the same meaning - it indicates how often a repeated event occurs. High frequency events that are periodic occur often, with little time in between each occurrence - like the back and forth vibrations of the tines of a tuning fork. The vibrations are so frequent that they can't be seen with the naked eye. A 256-Hz tuning fork has tines that make 256 complete back and forth vibrations each second. At this frequency, it only takes the tines about seconds to complete one cycle. A 512-Hz tuning fork has an even higher frequency. Its vibrations occur more frequently; the time for a full cycle to be completed is seconds. In comparing these two tuning forks, it is obvious that the tuning fork with the highest frequency has the lowest period. The two quantities frequency and period are inversely related to each other.

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CYU #1 According to Wikipedia, Tim Ahlstrom of Wisconsin holds the record for hand clapping. He is reported to have clapped his hands 793 times in 60.0 seconds. What is the frequency and what is the period of Mr. Ahlstrom's hand clapping during this 60.0-second period?

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Answer #1 In this problem, the event that is repeating itself is the clapping of hands; one hand clap is equivalent to a cycle. The frequency can be thought of as the number of cycles per second. Calculating frequency involves dividing the stated number of cycles by the corresponding amount of time required to complete these cycles. In contrast, the period is the time to complete a cycle. Period is calculated by dividing the given time by the number of cycles completed in this amount of time. Frequency = cycles per second = 793 cycles/60.0 seconds = 13.2 cycles/s = 13.2 Hz Period = seconds per cycle = 60.0 s/793 cycles = seconds

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CYU #2 A pendulum is observed to complete 23 full cycles in 58 seconds. Determine the period and the frequency of the pendulum.

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Answer #2 The frequency can be thought of as the number of cycles per second. Calculating frequency involves dividing the stated number of cycles by the corresponding amount of time required to complete these cycles. In contrast, the period is the time to complete a cycle. Period is calculated by dividing the given time by the number of cycles completed in this amount of time. frequency = 23 cycles/58 seconds = Hz = ~0.40 Hz period = 58 seconds/23 cycles = sec = ~2.5 s

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CYU #3 A mass is tied to a spring and begins vibrating periodically. The distance between its highest and its lowest position is 38 cm. What is the amplitude of the vibrations?

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Answer #3 Answer: 19 cm The distance that is described is the distance from the high position to the low position. The amplitude is from the middle position to either the high or the low position. So just divide the total distance by 2.

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CYU #4 A wave is introduced into a thin wire held tight at each end. It has an 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.

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Answer #4 Answer: sec Here 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 seconds. Know your physics concepts to weed through the extra information.

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CYU #5 Frieda the fly flaps its wings back and forth 121 times each second. The period of the wing flapping is ____ sec.

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**Answer #5 Answer: 0.00826 seconds**

The quantity 121 times/second is the frequency. The period is the reciprocal of the frequency. T=1/(121 Hz) = s

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Nature of a Wave The nature of a wave was discussed in a previous lesson 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 that 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 slinky. But the act of continually vibrating the first coil introduces a wave into the slinky.

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Frequency Review 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, in turn would vibrate at a rate of 2 cycles/second. 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. If a coil of a 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).

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**Energy Transport and the Amplitude of a Wave**

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Energy Transport As mentioned earlier, a wave is an energy transport phenomenon that 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.

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Amplitude The amount of energy carried by a wave is related to the amplitude of the wave. A high energy wave is characterized by a high amplitude; a low energy wave is characterized by a low amplitude. The logic underlying the energy-amplitude relationship is as follows: If you send a transverse pulse into the first coil of a slinky, that coil is given an initial amount of displacement. The displacement is due to the force applied by the person. The more energy that the person puts into the pulse, the more work that he/she will do upon the first coil. The more work that is done upon the first coil, the more displacement that is given to it. The more displacement that is given to the first coil, the more amplitude that it will have. So in the end, the amplitude of a transverse pulse is related to the energy which that pulse transports through the medium. Putting a lot of energy into a transverse pulse will not affect the wavelength, the frequency or the speed of the pulse. The energy imparted to a pulse will only affect the amplitude of that pulse.

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**Inertial & Elastic Factors**

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? In a situation such as this, the amplitude is dependent upon two types of factors: an inertial factor and an elastic factor. Two different materials have different mass densities. 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 and allow a greater amplitude to travel through it.

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**Energy-Amplitude Relationship**

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. 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 is accompanied by a 4-fold (22) increase in the energy; thus 2 units of energy becomes 4 times bigger - 8 units. As another example, changing the amplitude from 1 unit to 4 units represents a 4-fold increase in the amplitude and is accompanied by a 16-fold (42) increase in the energy; thus 2 units of energy becomes 16 times bigger - 32 units.

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CYU #1 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 of such a wave.

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**Answer #1 Amplitude = 16 cm Wavelength = 96 cm Period = 0.42 s**

(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)

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CYU #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. halved b. doubled c. quadrupled d. remains the same

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**Answer #2 Answer: C (quadrupled)**

The energy transported by a wave is directly proportional to the square of the amplitude. So whatever change occurs in the amplitude, the square of that affect impacts the energy. This means that a doubling of the amplitude results in a quadrupling of the energy.

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CYU #3 Two waves are traveling through a container of an inert 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-fourth b. one-half c. two times larger than d. four times larger than

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**Answer #3 Answer: D (four times larger)**

The energy transported by a wave is directly proportional to the square of the amplitude. So whatever change occurs in the amplitude, the square of that affect impacts the energy. This means that a doubling of the amplitude results in a quadrupling of the energy.

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The Speed of a Wave A wave is a disturbance that moves along a medium from one end to the other. If you watch an ocean wave moving along the medium (the ocean water), you 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:

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Changing Mediums 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 that 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 that 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 that is equal to twice the length of the slinky.

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**Echoes Reflection phenomena are commonly observed with sound waves.**

When you yell within a canyon, you often hear the echo of the yell. 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 yell. A classic physics problem goes like this: Noah stands 170 meters away from a steep canyon wall. He shouts and hears the echo of his voice one second later. What is the speed of the wave?

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Answer 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.

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**Variables Affecting Wave Speed**

What variables affect the speed at which a wave travels through a medium? Does the frequency or wavelength of the wave affect its speed? Does the amplitude of the wave affect its speed? Or are other variables such as the mass density of the medium or the elasticity of the medium responsible for affecting the speed of the wave? Suppose a wave generator is used to produce several waves within a rope of a measurable tension. The wavelength, frequency and speed are determined. Then the frequency of vibration of the generator is changed to investigate the affect of frequency upon wave speed. Finally, the tension of the rope is altered to investigate the affect of tension upon wave speed. Sample data for the experiment are shown in the table on the next page.

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**Speed of a Wave Lab – Sample Data**

Trial Tension (N) Frequency (Hz) Wavelength (m) Speed (m/s) 1 2.0 4.05 4.00 16.2 2 8.03 2.00 16.1 3 12.30 1.33 16.4 4 1.00 5 20.2 0.800 6 5.0 12.8 25.6 7 19.3 25.7 8 25.5

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Change in Frequency 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 affect 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 affect wave speed. An increase in wave frequency caused a decrease in wavelength while the wave speed remained constant.

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Change in Rope Tension The last three trials involved the same procedure with a different rope tension. Observe that the speed of the waves in rows 6-8 is 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 higher at higher tensions. Waves travel through tighter ropes at higher speeds. So while the frequency did not affect the speed of the wave, the tension in the medium (the rope) did. In fact, the speed of a wave is not dependent upon properties of the wave itself. Rather, the speed of the wave is dependent upon the properties of the medium such as the tension of the rope.

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Wave vs. 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 lab activity described above 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.

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CYU #1 A teacher attaches a slinky to the wall and begins introducing pulses with different amplitudes. Which of the two pulses (A or B) below will travel from the hand to the wall in the least amount of time?

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Answer #1 They reach the wall at the same time. Don't be fooled! The amplitude of a wave does not affect the speed at which the wave travels. Both Wave A and Wave B travel at the same speed. The speed of a wave is only altered by alterations in the properties of the medium through which it travels.

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CYU #2 The teacher then begins introducing pulses with a different wavelength. Which of the two pulses (C or D) will travel from the hand to the wall in the least amount of time ?

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Answer #2 They reach the wall at the same time. Don't be fooled! The wavelength of a wave does not affect the speed at which the wave travels. Both Wave C and Wave D travel at the same speed. The speed of a wave is only altered by alterations in the properties of the medium through which it travels.

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CYU #3 The time required for the sound waves (v = 340 m/s) to travel from the tuning fork to point A is ____ . a second b second c second d. 2.9 second

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**Answer #3 Answer: B GIVEN: v = 340 m/s, d = 20 m and f = 1000 Hz**

Find time Use v = d / t and rearrange to t = d / v Substitute and solve.

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CYU #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-ninth b. one-third c. the same as d. three times larger than

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Answer #4 Answer: C The medium is the same for both of these waves ("the same container of nitrogen gas"). Thus, the speed of the wave will be the same. Alterations in a property of a wave (such as wavelength) will not affect the speed of the wave. Two different waves travel with the same speed when present in the same medium.

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CYU #5 An automatic focus camera is able to focus on objects by use of an ultrasonic sound wave. The camera sends out sound waves that reflect off distant objects and return to the camera. A sensor detects the time it takes for the waves to return and then determines the distance an object is from the camera. The camera lens then focuses at that distance. Now that's a smart camera! In a subsequent life, you might have to be a camera; so try this problem for practice: If a sound wave (speed = 340 m/s) returns to the camera seconds after leaving the camera, then how far away is the object?

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**Answer #5 GIVEN: v = 340 m/s, t = 0.150 s (down and back time)**

Find d (1-way) If it takes s to travel to the object and back, then it takes s to travel the one-way distance to the object. Now solve for time using the equation d = v • t. d = v • t = (340 m/s) • (0.075 s) = 25.5 m

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CYU #6 TRUE or FALSE: Doubling the frequency of a wave source doubles the speed of the waves.

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Answer #6 FALSE! The speed of a wave is unaffected by changes in the frequency.

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CYU #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 0.82 seconds after the scream. The speed of the sound wave in air is 342 m/s. Calculate the distance from Noah to the nearby canyon wall.

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**Answer #7 GIVEN: v = 342 m/s, t = 0.82 s (2-way) Find d (1-way)**

If it takes 0.82 s to travel to the canyon wall and back (a down-and-back time), then it takes 0.41 s to travel the one-way distance to the wall. Now use d = v • t d = v • t = (342 m/s) • (0.41 s) = 140 m

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CYU #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.

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**Answer #8 GIVEN: d (1-way) =25 m, t (2-way)=25 s Find v.**

If the pool is 25 meters long, then the back-and-forth distance is 50 meters. The wave covers this distance in 25 seconds. Now use v = d / t. v = d / t = (50 m) / (25 s) = 2 m/s

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CYU #9 The water waves below are traveling along the surface of the ocean at a speed of 2.5 m/s and splashing periodically against Wilbert's perch. Each adjacent crest is 5 meters apart. The crests splash Wilbert's feet upon reaching his perch. How much time passes between each successive drenching? Answer and explain using complete sentences.

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Answer #9 If the wave travels 2.5 meters in one second then it will travel 5.0 meters in 2.0 seconds. If Wilbert gets drenched every time the wave has traveled 5.0 meters, then he will get drenched every 2.0 seconds.

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The Wave Equation As was discussed in Lesson 1, a wave is produced when a vibrating source periodically disturbs the first particle of a medium. This creates a wave pattern that 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 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.

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Production of a Wave 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 in the time it takes 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 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.

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**Speed = Wavelength • Frequency**

Speed Equation 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 • 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 •

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**Check Your Understanding**

Stan and Anna are conducting a slinky experiment. They are studying the possible affect 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 0.90 m 3.9 Hz Copper, 1-in. dia. coils 1.19 m 2.1 Hz 0.60 m 4.2 Hz Zinc, 3-in. dia. coils 0.95 m 2.2 Hz 1.82 m 1.2 Hz

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Answers Multiply the frequency by the wavelength to determine the speed. Row 1: speed = 3.5 m/s Row 2: speed = 3.5 m/s Row 3: speed = 2.5 m/s Row 4: speed = 2.5 m/s Row 5: speed = 2.1 m/s Row 6: speed = 2.2 m/s

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CYU #2 As the wavelength of a wave in a uniform medium increases, its speed will _____. a. decrease b. increase c. remain the same

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Answer #2 Answer: C In 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.

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CYU #3 As the wavelength of a wave in a uniform medium increases, its frequency will _____. a. decrease b. increase c. remain the same

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Answer #3 Answer: A In rows 1 and 2, the wavelength was increased and the frequency was decreased. Wavelength and frequency are inversely proportional to each other.

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CYU #4 The speed of a wave depends upon (i.e., is causally affected by) ... a. the properties of the medium through which the wave travels b. the wavelength of the wave. c. the frequency of the wave. d. both the wavelength and the frequency of the wave.

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Answer #4 Answer: A Whenever 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.

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CYU #5 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

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Answer #5 Answer: B Frequency 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.

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CYU #6 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 constant b. the wavelength remains constant and the speed is doubled c. both the wavelength and the speed are halved. d. both the wavelength and the speed remain constant.

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Answer #6 Answer: A Doubling the frequency will not alter the wave speed. Rather, it will halve the wavelength. Wavelength and frequency are inversely related.

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CYU #7 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?

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**Answer #7 Answer: f = 70 Hz and wavelength = 5.0 m**

The frequency is given and the wavelength is the v/f ratio.

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CYU #8 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.

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**Answer #8 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, then f =1 /T = 1 / (6.2 s) f = Hz Now find speed using the v = f • wavelength equation. v = f • wavelength = (0.161 Hz) • (8.6 m) v = 1.4 m/s

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CYU #9 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.

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Answer #9 The diagram is helpful. The wavelength must be 8 meters (see diagram). The period is 3 seconds so the frequency is 1 / T or Hz. Now use speed = f • wavelength Substituting and solving for v, you will get 2.67 m/s.

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