1 Waves All substantive material is from Wave Motion and Sound by James Dann. http://www.ck12.org/flexr/ unless otherwise noted. Illustrations are copyright free. Objects in motion that return to the same position after a fixed period of time are said to be in harmonic motion. Objects in harmonic motion have the ability to transfer some of their energy over large distances. They do so by creating waves in a medium. Mediums A medium is the substance through which the wave travels. For example, water acts as the medium for ocean waves, while air molecules act as the medium for sound waves. When a wave passes through a medium, the medium is only temporarily disturbed. When an ocean wave travels from one side of the Mediterranean Sea to the other, no actual water molecules move this great distance. Only the disturbance propagates (moves) through the medium. 2 Types of Waves In longitudinal waves, the vibrations of the medium are in the same direction as the wave motion. A classic example is a wave traveling down a line of standing dominoes: each domino will fall in the same direction as the motion of the wave. A more commonly-seen example is a sound wave. For sound waves, high and low pressure zones move both forward and backward as the wave moves through them. In transverse waves, the vibrations of the medium are perpendicular to the direction of motion. A classic example is a wave created in a long rope: the wave travels from one end of the rope to the other, but the actual rope moves up and down, and not from left to right as the wave does. Parts of Waves The speed of a wave on a string depends on the material the string is made of, as well as the tension in the string. This fact is why tightening a string on your violin or guitar will change the sound it produces. The frequency, f, is the number of cycles an object goes through in 1 second. Frequency is measured in Hertz (Hz). I Hz = 1cycle per sec. An object oscillating with frequency f will create waves in a medium which oscillate with the same frequency f.
2 The amplitude, A, is the furthest distance moved by a particle from the medium as the wave passes over the particle. The amplitude, therefore, is half of the total distance covered by the oscillating particle. The period of a wave is the time it takes for one complete cycle to occur. For a transverse wave, a period includes one crest and one trough. For a longitudinal wave, it includes a compression and a rarefaction. Period is a measure of time so the SI units for measuring it are seconds. wave type transverse longitudinal period includes crest + trough compression + rarefaction Parts of Transverse Waves The crest is the high point of movement for a particle of medium. The trough is the low point of movement for a particle of medium. Wavelength is the distance covered by one complete crest and one complete trough. Or, another way to view it is the distance between two side-by-side crests. Parts of Longitudinal Waves A compression is the part of a wave where the particles of the medium have been shoved closer together by the energy of the wave. The rarefaction is the part of a wave where the particles of the medium have been pulled farther apart by the energy of the wave. Wavelength is the distance of one complete rarefaction and one complete compression: wavelength = compression + rarefaction. Energy Transmission Waves are energy moving through a medium. The energy is transmitted in the direction the wave moves. However, the motion of the particles in the medium may not be the same as the motion of the wave. In longitudinal waves, the medium particles oscillate in the same direction (parallel) as the wave moves. In transverse waves, the particles move at a 90 (perpendicular) to the direction the wave moves think how the molecules in a rope move compared to the motion of the wave sent down the rope. wave type movement of medium particles movement of wave movement of energy transverse left to right longitudinal left to right
3 Interference When two waves bump into each other it is called interference. Constructive interference occurs when two waves combine to create a larger wave. This occurs when the parts of the waves line up with each other: crest-to-crest and trough-to-trough or compressionto-compression and rarefaction-torarefaction. When constructive interference occurs, the result is a single wave whose amplitude is greater than the sum of the two original waves. Destructive interference occurs when two waves combine and cancel each other out. This occurs when the opposite parts of a wave line up with each other: crest-to-trough or compression-to-rarefaction. When destructive interference occurs, the result is a cancellation of the wave and no movement of the medium occurs, the wave disappears. The cancellation of the wave is because each wave tugs equally strongly on the particle of medium, but the waves pull in opposite directions. This stops the medium particle from moving. The wave can go no further. The previous examples assumed the waves were identical to each other. But if the waves are not of the same amplitude or frequency, they do not give perfect results. For example, two waves undergoing destructive interference (if they are not identical) will only cancel out part way. Some wave will be left. When two waves move in opposite directions, through each other, interference takes place. If the two waves have the same frequency and wavelength then standing waves are generated. Standing waves are so-called because they appear to be standing still. Twirling a jump rope is a way of forming a standing wave (the crest goes up and the trough goes down). If you twirl it fast enough, the wave appears to be a solid sphere which isn t moving. In comparison, snap a rope so that a wave travels down the length of the rope, this is not a standing wave. Nodes, Antinodes, and Rest Positions Before a wave travels through a medium, the particles of the medium are at rest position, sitting still. As the wave moves through, the particles of the
4 medium will be moved by the passing wave of energy. Remember that the amplitude is the measure of how far the particles are moved from the rest position. Once the wave passes by, the medium s particles return to their rest positions. When two transverse waves meet each other they will either add or subtract their energy from each other. If a crest and a trough meet, it is called destructive interference. The smaller of the two waves will be subtracted from the larger wave. If they are the same size (amplitude) they will cancel each other out. The point where they cancel each other out is called a node. If two waves meet, and the crests meet each other or troughs meet each other, they will add their energy together and create a larger crest or trough. This is constructive interference. The result of constructive interference is an anti-node node. A standing wave can be understood as a series of nodes and anti-nodes. Reflection, Refraction, and Diffraction Reflection of a wave occurs when a wave bounces off of a barrier. An echo is an example of a sound wave being reflected. Your reflection in a mirror is formed by light waves bouncing off of the surface of the mirror. Diffraction is the bending of a wave around a barrier. Sound waves bending around a barrier explain why you are able hear noises in a different room. The wave will go around the edge of the barrier and then spread out to fill the space. Refraction occurs when waves enter a new medium. Each medium will transmit waves at a different speed. As the wave enters the new medium it will change speeds and consequently change direction. For example: a straw placed in a glass of water will appear to be in two pieces. Refraction of the light coming off of the straw accounts for this. The light traveling in only air is going at a different speed then the light traveling through the air and water. The difference in speeds makes the straw appear to be broken. Velocity of A Wave The speed (V) and wavelength (λ) of a wave depend upon the nature of the medium through which the wave travels. But regardless of the medium, the relationship between velocity, wavelength, and frequency remains the same: V = λ f or sometimes it appears as V = λ ν f = frequency (Hz) sometimes f is written as the Greek letter nu (ν) V = velocity (m/s) λ = wavelength (m)
5 Example Problem 1 Question When a particular string is vibrated at a frequency of 10Hz, a transverse wave of wavelength 0.25m is produced. Determine the speed of the wave as it travels along the string. Answer Step 1 : Determine what is given and what is required frequency of wave: f =10Hz wavelength of wave: λ =0.25m We are required to calculate the speed of the wave as it travels along the string. All quantities are in SI units. Step 2 : Determine how to approach the problem We know that the speed of a wave is: v = λ f and we are given all the necessary quantities. Step 3 : Substituting in the values v = f λ = (10 Hz)(0.25m) = 2.5m/s Step 4 : Write the final answer The wave travels at 2.5m/s in the string. [from The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Physics Grades 10 12. www.fhsst.org]
6 Example Problem 2 Question A cork on the surface of a swimming pool bobs up and down once per second on some ripples. The ripples have a wavelength of 20 cm. If the cork is 2m from the edge of the pool, how long does it take a ripple passing the cork to reach the shore? Answer Step 1 : Determine what is given and what is required We are given: frequency of wave: f = 1Hz wavelength of wave: λ = 20 cm distance of leaf from edge of pool: d = 2m We are required to determine the time it takes for a ripple to travel between the cork and the edge of the pool. The wavelength is not in SI units and should be converted. Step 2 : Determine how to approach the problem The time taken for the ripple to reach the edge of the pool is obtained from: t = d/v(from v = d/t) We know that v = λ f Therefore, t = d/ (f λ) Step 3 : Convert wavelength to SI units 20 cm = 0.2m Step 4 : Solve the problem t = d/ (f λ) =2m/[(1 Hz)(0.2m)] = 10 s Step 5 : Write the final answer A ripple passing the leaf will take 10 s to reach the edge of the pool. [from The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Physics Grades 10 12. www.fhsst.org]
7 Wave Lab # of nodes Each group needs one slinky, four pieces of tape, one stopwatch. Each group will contain the following people: 2 node people 2 tapers 1 timer The node people will each hold an end of the slinky against the floor. Stretch the slinky, but not too tight. On the end node each node person should place a piece of tape, mark the rest position with a piece of tape. The goal is to produce a standing wave. Once the standing wave is produced, each taper (they should be on the same side of the slinky) will use their piece of tape to mark adjacent (side-by-side) crests. The timer should time how long it takes for 10 oscillations (back-and-forth movement of the slinky between a crest and trough: in other words, the crest would appear 10 times in 10 oscillations) to occur. This might be easiest if one of the node people counts out each time (for 10 times) their hand moves all the way to the right. One of the node people should count the number of nodes between the end nodes (inclusive of the end nodes). Do this 3 times. Now PUT THE SLINKY AWAY, FAR FAR AWAY!!!! Fill in the following table. Time for 10 oscillations (s) # of oscillations per second (frequency) 2 x distance between 2 crests (wavelength) (m) Velocity of 1 wave (v=λf) Period (period=1/f) # of antinodes (area between 2 nodes) Calculated wavelength (divide the distance between the end nodes by the number of antinodes and multiply by 2) 4 5 6 Compare the wavelength that was measured and the one calculated. What happens to frequency as period gets larger?
8 Review Questions: Review questions are from the SD Achievement Series. 1. The bending of waves around the edges of barriers is called A. diffraction B. refraction C. reflection D. dispersion 2. Destructive interference occurs when what happens? A. the crests of two waves overlap B. two waves of the same color overlap C. two waves of the same wavelength meet D. the crest of one wave meets the trough of another wave 3. Which of the following may be produced during destructive interference of waves? A. a node B. a reflection C. a higher crest D. a lower trough 4. The change in wave direction at the boundary of two different media is A. incidence B. refraction C. reflection D. diffraction
9 5. The spreading of waves around the edge of a barrier is A. incidence B. refraction C. reflection D. diffraction 6. When does constructive interference occur? A. the crests of two waves overlap B. two waves of the same color overlap C. two waves of the same wavelength meet D. the crest of one wave meets the trough of another wave 7. At which point will destructive interference occur on the concentric waves below? A. 1 B. 2 C. 3 D. 4 8. A sound wave is an example of which type of wave? A. surface wave B. inverted wave C. transverse wave D. longitudinal wave
10 9. Both transverse and longitudinal waves A. transfer energy through a medium B. have compressions and rarefactions C. are capable of moving the medium a long distance D. move at right angles to the vibrations of the medium 10. The frequency of longitudinal waves may be measured by counting the number of successive compression zones that pass a point in a given time interval. What is the similar part of a transverse wave that may also be used to measure frequency A. crests B. wavelengths C. amplitudes D. rarefactions 11. If the frequency of a given wave doubles, what happens to its wavelength? A. it doubles B. it is halved C. it quadruples D. it stays the same 12. If 300 waves pass a point in 60 seconds, what is their frequency? A. 5 Hz B. 30 Hz C. 300 Hz D. 18000 Hz
11 13. A wave has a frequency of 4.0 hertz and a wavelength of 15 mm. What is its speed? A..27 mm/s B. 3.8 mm/s C. 60 mm/s D. 60 Hertz/s 14. A light wave is an example of which type? A. surface wave B. inverted wave C. transverse wave D. compressional wave 15. The types of waves produced by a piano are classified as A. radio waves B. micro waves C. transverse waves D. longitudinal waves 16. You are creating a wave on a spring. If you start shaking the spring more quickly, what happens to its wavelength? A. it increases B. it decreases C. it remains the same D. it depends on its amplitude 17. What is the frequency of a wave if 250 waves pass a point in 0.5 seconds? A. 125 Hz B. 250 Hz C. 500 Hz D. 5000 Hz
12 18. Which of the following wave interactions will result in constructive interference? A. B. C. D. 19. Which of the following occurs as a result of constructive interference? A. a wave that has a smaller amplitude B. a wave that has a larger amplitude C. a wave that has a larger wavelength D. a wave that has a smaller wavelength
13 20. Which of the following diagrams correctly shows constructive interference? A. B. C. D. 21. Why does the straw in the glass of water shown below look different above water than it does underwater? A. The light from the straw is being refracted because the light scatters as it hits the surface of the water, causing some of the light to reflect and leaving only part of the light to be seen underwater. B. The light from the straw is being refracted because as the light travels from air into water, it slows down and changes its direction. C. The light is scattered as it enters the water, so it spreads out underwater, causing the straw to look larger and bent. D. The light from the straw above the surface of the water reflects off the curved bottom of the glass, bending the light slightly and making the straw seem bent.
14 22. Use the diagram below to answer the question. Which diagram illustrates the direction of a particle's motion in the transverse wave above? A. B. C. D. 23. Which term describes the area of the longitudinal wave indicated by the arrow? A. compression B. rarefaction C. node D. antinode 24. What part of a longitudinal wave is under the least amount of compression? A. the area that is experiencing compression B. the crest of the wave C. the trough of the wave D. the area that is experiencing rarefaction
15 25. The frequency of a wave traveling at 15 m/s with an amplitude of 0.030 m is 6.0 Hz. What is its wavelength? A. 2.5 m B. 90 m C. 0.40 m D. 83 m 26. What is the velocity of a wave if its frequency is 250 Hz, its wavelength is 20.0 m, and its amplitude is 0.150 m? A. 37.5 m/s B. 0.080 m/s C. 1667 m/s D. 5000 m/s 27. How would a wave be affected if its wavelength increased and its frequency remained constant as it traveled through a new medium? A. Its velocity would increase. B. Its velocity would decrease. C. Its amplitude would increase. D. Its amplitude would decrease. 28. How would a wave be affected if its frequency increased and its wavelength remained constant as the wave entered a new medium? A. Its amplitude would decrease. B. Its velocity would increase. C. Its velocity would decrease. D. Its amplitude would increase. 29. How would a wave's frequency be affected if its velocity and its wavelength were doubled? A. the frequency would decrease B. the frequency would double C. the frequency would remain the same D. the frequency would increase slightly
16 Advanced Questions: 1) Summarize the motion of a particle in a medium during the transmission of a wave. 2) Compare/contrast longitudinal and transverse waves. 3) If v=λf is true and if the speed of light is 3 x 10 8 m/s, predict what will happen to the frequency of a wave of light as its wavelength increases. 4) If two pebbles are tossed near each other in a pond, describe the interaction of the resulting waves.
Waves Teaux 17 C A D B Choose the letter which best identifies the specified part of a wave. 1) Wavelength 2) Amplitude 3) Crest 4) Trough Choose the best answer. 5) If a wave has a wavelength of 650 nm and a frequency of 77 cycles per second, what is its velocity? a) 5005 m/s b) 50000 m/s c) 0.00005 m/s 6) Two beams of light have different wavelengths. Wave A has a wavelength of 33000 nm while wave B has a wavelength of 499 m. Which has the greater frequency? a) A b) B 7) What is the velocity of an electromagnetic wave whose frequency is 88.5 hertz. a) 3x10 8 m/s b) 88.5 m/s c) unsolvable d) 9.8 m/s/s 8) What is the wavelength of a wave whose frequency is 8000000 hz and whose velocity is 216 m/s? a) 1728000000 m b) 2.7x10-5 m c) 3704 m 9) What is the frequency of a ray of light whose wavelength is 997 nm? a) 3x10 8 hz b) 3x10 14 hz c) unsolvable 10) What is the wavelength of light whose frequency is 7.45 x 10 13 hz? a) 4x10-6 m b) 2.2x10 22 m/s c) 1.3x10-14 11) For any given wave traveling at a constant velocity, as its wavelength lengthens, its frequency a) increases b) decreases 12) As a wave's frequency decreases, the total energy of the wave a) increases b) decreases 13) As a wave's wavelength increases, the total energy of the wave a) increases b) decreases E
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19 Bye, Bye, Waves Choose the best answer. 1) If a radio wave has a frequency of 4.8 khz, what is its wavelength in m? a) 7.9x10-5 b) 6.25x10 4 c) 1.8x10 4 d) 6.25x10 3 2) If an EM wave has a wavelength of 2.3 mm, how much energy does it carry in Joules? a) 1.3x10 8 b) 8.6x10-26 c) 8.6x10-29 d) all of the above 3) Planck's equation allows what to be calculated, assuming you can find values for the proper variables? a) energy of an EM wave b) frequency of an EM wave c) the value of Planck's constant d) all of the above 4) If a photon has a wavelength of 330km, what is its energy in J? a) 6x10-31 b) 6x10-28 c) 6x10-19 d) 9.1x10 5 5) What is the frequency of a compression wave traveling at 600 m/s and having a wavelength of 220m a) 2.7 b) 132000 c) 0.37 d) none of the above
20 6) For which of the following do the particles of its medium oscillate parallel to the transmission of energy? a) transverse wave b) sound wave c) compression wave d) both a and b e) both b and c 7) For which of the following do the particles of its medium oscillate vertically to the transmission of energy? a) transverse wave b) sound wave c) compression wave d) both a and b e) both b and c 8) Which of the following is the same type of wave as a light wave? a) transverse wave b) sound wave c) compression wave d) both a and b e) both b and c 9) As wavelength for a photon increases, frequency a) stays the same b) increases c) decreases 10) For a sound wave, as frequency increases, wavelength a) stays the same b) increases c) decreases 11) For a photon, as frequency increases, velocity a) stays the same b) increases c) decreases