Introduction to Waves: Transverse and Longitudinal

1. Introduction to Waves: Transverse and Longitudinal Physics Coach Stephens

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

3. Parts of a Wave • http://www.youtube.com/watch?feature=player_detailpage&v=Z3O2Ju3ULvo#t=0s

4. Making Waves Using a Slinky • http://www.youtube.com/watch?v=vJ8pn-FhHl8&edufilter=3vKYiNm3XpGng5TQ3-qEBw • http://www.youtube.com/watch?v=Rbuhdo0AZDU&edufilter=3vKYiNm3XpGng5TQ3-qEBw

5. Calculating Speed, Frequency & Wavelength Physics Coach Stephens

6. 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

7. Sound Waves & Sources Physics Coach Stephens

8. Bell Work • What causes waves? • What are some properties of waves?

9. Bell Work Answer • Oscillations and vibrating objects • frequency, speed, wavelength, vibrating source • transverse and longitudinal (electromagnetic, sound, and water waves)

10. Simulations & Videos • http://phet.colorado.edu/new/simulations/sims.php?sim=Wave_on_a_String • http://phet.colorado.edu/new/simulations/sims.php?sim=Sound • http://phet.colorado.edu/new/simulations/sims.php?sim=Fourier_Making_Waves • http://phet.colorado.edu/new/simulations/sims.php?sim=Radio_Waves_and_Electromagnetic_Fields.

11. Vibrations

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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)

18. Animations and Pictures

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.“

24. 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.

25. 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.

26. 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 0.00391 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 0.00195 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.

27. 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?

28. 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 = 0.0757 seconds

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

30. 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 = 0.39655 Hz = ~0.40 Hz • period = 58 seconds/23 cycles = 2.5217 sec = ~2.5 s

31. 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?

32. 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.

33. 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.

34. Answer #4 • Answer: 0.0195 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 0.0195 seconds. • Know your physics concepts to weed through the extra information.

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

36. 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) = 0.00826 s

37. 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.

38. 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).

39. Energy Transport and the Amplitude of a Wave

40. 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.

41. 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.

42. 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.

43. 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.

44. 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.

45. Answer #1 • 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)

46. 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

47. 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.

48. 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

49. 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.

50. 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: