9.3. Diffraction and Interference of Water Waves

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "9.3. Diffraction and Interference of Water Waves"

Transcription

1 Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately an roll onto the shore? Water waves behave similarly to other kins of waves in many ways. Figure 1 illustrates one characteristic behaviour of waves when a part of the wave enters through a narrow opening. The section of the wave that gets through acts as a source of new waves that sprea out on the other sie of the opening, an the waves from the two sources combine in specific ways. You will learn about these characteristic behaviours of waves in this section. Figure 1 When part of a wave enters a narrow opening, such as this bay, new waves are create an the two waves combine in preictable ways. Diffraction If you observe straight wave fronts in a ripple tank, you can see that they travel in a straight line if the water epth is constant an no obstacles are in the way. If, however, the waves pass by an ege of an obstacle or through a small opening, the waves sprea out. Diffraction is the bening of a wave as the wave passes through an opening or by an obstacle. The amount of iffraction epens on the wavelength of the waves an the size of the opening. In Figure, an obstacle iffracts shorter wavelengths slightly. In Figure, the same obstacle iffracts longer wavelengths more. iffraction the bening an spreaing of a wave when it passes through an opening; epenent on the size of the opening an the wavelength of the wave short wavelengths long wavelengths Figure When waves travel by an ege, shorter wavelengths iffract less than longer wavelengths. 9.3 Diffraction an Interference of Water Waves 459

2 Although iffraction also occurs for soun an light waves, we will stuy the phenomenon first for water waves because their long wavelength allows for easier observation of the effects of iffraction. In Figure 3, you can see a wave encountering a slit with a certain with, w. In Figure 3 an Figure 3(c), the with of the opening is the same but the wavelength of the incient waves has change. Notice how the amount of iffraction changes. Therefore, as the wavelength increases but the with of the slit oes not, the iffraction also increases. In Figure 3, the wavelength is approximately one-thir of w. Notice that only the part of the wave front that passes through the slit creates the series of circular wave fronts on the other sie. In Figure 3, the wavelength is about half of w, which means that significantly more iffraction occurs, but areas to the ege exist where no waves are iffracte. In Figure 3(c), the wavelength is approximately two-thirs of w, an the sections of the wave front that pass through the slit are almost all converte to circular wave fronts. (c) Figure 3 As the wavelength increases, the amount of iffraction increases. What happens if you change the with of the slit but keep the wavelength fixe? As Figure 4 an Figure 4 show, as the size of the slit ecreases, the amount of iffraction increases. If waves are to unergo more noticeable iffraction, the wavelength must be comparable to or greater than the slit with (λ $ w). For small wavelengths (such as those of visible light), you nee to have narrow slits to observe iffraction. Figure 4(c) shows what happens to the iffraction pattern when the wavelength ecreases but the size of the slit oes not change. incient waves nearly straight-line propagation incient waves incient waves Investigation Properties of Water Waves (page 487) You have learne how observing water waves helps in unerstaning the properties of waves in general. This investigation will give you an opportunity to test conitions for iffraction to occur. large opening small opening shorter wavelength, no change in opening (c) Figure 4 an As the size of the slit (aperture) ecreases, iffraction increases. (c) With a shorter wavelength an no change in the size of the opening, there is less iffraction. The relationship between wavelength, with of the slit, an extent of iffraction is perhaps familiar for soun waves. You can hear soun through an open oor, even if you cannot see what is making the soun. The primary reason that soun waves iffract aroun the corner of the oor is that they have long wavelengths compare to the with of the oorway. Low frequencies (the longer wavelengths of the soun) iffract more than high frequencies (the shorter wavelengths of the soun). So if a soun system is in the room next oor, you are more likely to hear the lower frequencies (the bass). In the following Tutorial, you will learn how to solve problems relating to iffraction. 460 Chapter 9 Waves an Light

3 Tutorial 1 Diffraction The following Sample Problem provies an example an sample calculations for the concepts of iffraction. Sample Problem 1: Diffraction through a Slit Determine an explain the ifference between the iffractions observe in Figure 5 an Figure cm 0.5 cm w a cm w b 0.5 cm Figure 5 Given: l cm; w a 5 cm; w b cm Require: iffraction analysis Analysis: l w $ 1 Solution: For Figure 5, l 0.5 cm 5 w a c m l, 1 w a For Figure 5, l 0.5 cm 5 w b 0.5 cm l 5 1 w b Statement: Since the ratio in Figure 5 is less than 1, little iffraction occurs. Since the ratio in Figure 5 is 1, more noticeable iffraction occurs. Practice 1. Determine whether iffraction will be noticeable when water waves of wavelength 1.0 m pass through a 0.5 m opening between two rocks. T/I [ans: yes]. A laser shines re light with a wavelength of 630 nm onto an ajustable slit. Determine the maximum slit with that will cause significant iffraction of the light. T/I [ans: 6.3 * 10-7 m] 9.3 Diffraction an Interference of Water Waves 461

4 Interference interference the phenomenon that occurs when two waves in the same meium interact constructive interference the phenomenon that occurs when two interfering waves have isplacement in the same irection where they superimpose When two waves cross paths an become superimpose, they interact in ifferent ways. This interaction between waves in the same meium is calle interference. You experience interference when you listen to music an other types of soun. For example, soun waves from two speakers may reach the listener s ears at the same time, as illustrate in Figure 6. If the crest of one wave coincies with the crest of the other, then the waves are in phase an combine to create a resultant wave with an amplitue that is greater than the amplitue of either iniviual wave resulting in a louer soun. This phenomenon is calle constructive interference. Waves are in phase. Waves are 180 out of phase. speaker 1 L 1 L speaker estructive interference the phenomenon that occurs when two interfering waves have isplacement in opposite irections where they superimpose coherent compose of waves having the same frequency an fixe phases listener constructive interference estructive interference (c) Figure 6 If L 1 an L are the same, the waves arrive at the listener in phase. Waves interfere constructively if they arrive in phase. (c) If L 1 an L iffer by, say, l, the waves arrive 1808 out of phase, an they interfere estructively. For two waves that iffer in phase by l, shown in Figure 6(c), the crest of one wave coincies with the trough of the other wave. This correspons to a phase ifference of The two waves combine an prouce a resulting wave with an amplitue that is smaller than the amplitue of either of the two iniviual waves. This phenomenon is calle estructive interference. The following conitions must be met for interference to occur: 1. Two or more waves are moving through ifferent regions of space over at least some of their way from the source to the point of interest.. The waves come together at a common point. 3. The waves must have the same frequency an must have a fixe relationship between their phases such that over a given istance or time the phase ifference between the waves is constant. Waves that meet this conition are calle coherent. Figure 7 shows water waves interfering. Figure 7 Interference of water waves in a ripple tank 46 Chapter 9 Waves an Light

5 In Figure 7, two point sources have ientical frequencies. The sources prouce waves that are in phase an have the same amplitues. Successive wave fronts travel out from the two sources an interfere with each other. Constructive interference occurs when a crest meets a crest or when a trough meets a trough. Destructive interference occurs when a crest meets a trough (resulting in zero amplitue.) Symmetrical patterns sprea out from the sources, proucing line locations where constructive an estructive interference occur. A noe is a place where estructive interference occurs, resulting in zero amplitue (a net isplacement of zero). As shown in Figure 8, the interference pattern inclues lines of maximum isplacement, cause by constructive interference, separate by lines of zero isplacement, cause by estructive interference. The lines of zero isplacement are calle noal lines. noe a point along a staning wave where the wave prouces zero isplacement noal line a line or curve along which estructive interference results in zero isplacement areas of constructive interference lines of estructive interference (noal lines) amplitue here is A crests troughs amplitue here is A noe S 1 S Figure 8 Interference pattern between two ientical sources each with positive amplitue A When the frequency of the two sources increases, the wavelength ecreases, which means that the noal lines come closer together an the number of noal lines increases. When the istance between the two sources increases, the number of noal lines also increases. The symmetry of the pattern oes not change when these two factors are change. However, if the relative phase of the two sources changes, then the pattern shifts, as shown in Figure 9, but the number of noal lines stays the same. In Figure 9, the sources are in phase, but in Figure 9, there is a phase ifference of Investigation 9.3. Interference of Waves in Two Dimensions (page 488) So far, you have rea about two-pointsource interference in theory, with iagrams an photographs. This investigation gives you an oppor tunity to see an measure interference for yourself. S 1 S S 1 S Figure 9 Effect of phase change on interference pattern for zero phase ifference an 1808 phase ifference 9.3 Diffraction an Interference of Water Waves 463

6 Mini Investigation Interference from Two Speakers Mini Investigation Skills: Performing, Observing, Analyzing, Communicating At the start of the iscussion on interference, you consiere soun waves coming from two speakers. In this investigation, your teacher will set up two speakers in the classroom, which will emit a single-frequency soun. You will examine how the soun intensity varies from place to place because of interference. Equipment an Materials: speakers; plan of classroom 1. Before you begin, ecie how to ientify areas of constructive an estructive interference. How will these manifest themselves with soun?. Mark the location of the speakers on the classroom plan. 3. Your teacher will turn on the speakers. SKILLS HANDBOOK A.1 4. Walk aroun the ifferent areas of the classroom an mark on your plan where constructive an estructive interference occur. A. How shoul your istances to the two speakers iffer to get constructive interference? K/u B. How shoul your istances to the two speakers iffer to get estructive interference? K/u C. Use your results marke on your class plan to ientify two locations for either constructive or estructive interference, estimate their istances from the speakers, an use these estimates to estimate the wavelength of the soun. A WEB LINK Mathematics of Two-Point-Source Interference You can measure wavelength using the interference pattern prouce by two point sources an evelop some mathematical relationships for stuying the interference of other waves. Figure 10 shows an interference pattern prouce by two point sources in a ripple tank. n n 1 n 1 n P 1 P S 1 S Figure 10 Ripple tank interference patterns can be use to evelop relationships to stuy interference. path length the istance from point to point along a noal line The two sources, S 1 an S, separate by a istance of three wavelengths, are vibrating in phase. The bisector of the pattern is shown as a white line perpenicular to the line that joins the two sources. You can see that each sie of the bisector has an equal number of noal lines, which are labelle n 1 an n on each sie. A point on the first noal line (n 1 ) on the right sie is labelle P 1, an a point on the secon noal line (n ) on the right sie is labelle P. The istances P 1 S 1, P 1 S, P S 1, an P S are calle path lengths. 464 Chapter 9 Waves an Light

7 If you measure wavelengths, you will fin that P 1 S 1 5 4λ an P 1 S 5 7 λ. The path length ifference, Ds 1, on the first noal line is equal to Ds 1 5 0P 1 S 1 P 1 S 0 path length ifference the ifference between path lengths, or istances 5 `4l 7 l ` Ds l This equation applies for any point on the first noal line. We use the absolute value because only the size of the ifference in the two path lengths matters, not which one is greater than the other. A noe can be foun symmetrically on either sie of the perpenicular bisector. The path length ifference, Ds, on the secon noal line is equal to Ds 5 0P S 1 P S 0 Ds 5 3 l Extening this to the nth noal line, the equation becomes Ds n 5 0P n S 1 P n S 0 Ds n 5 an 1 bl where P n is any point on the nth noal line. Thus, by ientifying a specific point on a noal line an measuring the path lengths, you can use this relationship to etermine the wavelength of the interfering waves. This technique will not work if the wavelengths are too small or the point P is too far away from the sources, because the path length ifference is too small to measure accurately. If either or both of those conitions apply, you nee to use another metho to calculate the path length ifference. For any point P n, the path length ifference is AS 1 (Figure 11): 0P n S 1 P n S 0 5 AS 1 When P n is very far away compare to the separation of the two sources,, then the lines P n S 1 an P n S are nearly parallel (Figure 11). Unit TASK BOOKMARK You can apply what you have learne about interference to the Unit Task on page 556. S 1 A S 1 S S n A Figure 11 When eveloping the equations to calculate path length ifference, it is important to consier the istance to P n to be far enough away that P n S 1 an P n S can be consiere parallel. P n 9.3 Diffraction an Interference of Water Waves 465

8 In Figure 11, the lines AS, S 1 S, an AS 1 form a right-angle triangle, which means that the ifference in path length can be written in terms of the angle u n, since sin u n 5 AS 1 Rearranging, sin u 1 5 AS 1 However, on the previous page we establishe that AS 1 5 0P n S 1 P n S 0 B x n P n AS 1 5 an 1 bl Combining this with the expression for AS 1 in terms of sinu 1 leas to sin u n 5 an 1 b l The angle for the nth noal line is u n, the wavelength is l, an the istance between the sources is. Using this equation, you can approximate the wavelength for an interference right bisector pattern. Since sin u n cannot be greater than 1, it follows that an 1 b l cannot be greater than 1. The number of noal lines on the right sie of the pattern is the maximum number of lines that satisfies this conition. By counting this number an measuring, you can etermine an approximate value for the wavelength. For example, if is.0 cm an n is 4 for a particular pattern, then A S 1 C S n n Figure 1 Use the triangle P n BC to measure sin u9 n. L an 1 b l L 1 a4 1 b l.0 cm L cm L 1 l l L 0.57 cm Although it is relatively easy to measure u n for water waves in a ripple tank, for light waves, the measurement is not as straightforwar. The wavelength of light is very small an so is the istance between the sources. Therefore, the noal lines are close together. You nee to be able to measure sin u n without having to measure the angle irectly. Assuming that a pair of point sources is vibrating in phase, you can use the following erivation. For a point P n that is on a noal line an is istant from the two sources, the line from P n to the mipoint of the two sources P n C can be consiere to be parallel to P n S 1. This line is also at right angles to AS. The triangle P n BC in Figure 1 is use to etermine sin u9 n. In this erivation, is the istance between the sources, x n is the perpenicular istance from the right bisector to the point P n, L is the istance from P n to the mipoint between the two sources, an n is the number of the noal line. sin ur n 5 x n L 466 Chapter 9 Waves an Light

9 Figure 13 is an enlarge version of the triangle with base CS in Figure 1. Since an 1 bl sin u n 5 an, since the right bisector, CB, is perpenicular to S 1 S, you can see in Figure 13 that u9 n 5 u n. So we can say that an 1 x n L 5 bl right bisector n n 90 n 90 n n or, n n n C S Figure 13 Enlargement of the base triangle from Figure 1 The following Tutorial will emonstrate how to solve problems involving twoimensional interference. Tutorial Interference in Two Dimensions This Tutorial provies examples an sample calculations for the concepts of two-point-source interference. Sample Problem 1: Interference in Two Dimensions The istance from the right bisector to a point P on the secon noal line in a two-point interference pattern is 4.0 cm. The istance from the mipoint between the two sources, which are 0.5 cm apart, to point P is 14 cm. Calculate the angle u for the secon noal line. Calculate the wavelength. Solution Given: n 5 ; x cm; L 5 14 cm Solution: u 5 sin x 1 L 4.0 cm 5 sin 1 a 14 cm b u Statement: The angle of the secon noal line is 178. Require: sin u Analysis: sin u n 5 x n L u 5 sin 1 x L 9.3 Diffraction an Interference of Water Waves 467

10 Given: n 5 ; x cm; L 5 14; cm Require: l Solution: l 5 x an 1 bl Analysis: x an 1 n L 5 bl x n l 5 an 1 bl 14.0 cm 10.5 cm l 5 a 3 b 114 cm l cm Statement: The wavelength is cm. Sample Problem : Determining Wave Pattern Properties Using Differences in Path Length Two ientical point sources are 5.0 cm apart, in phase, an vibrating at a frequency of 1 Hz. They prouce an interference pattern. A point on the first noal line is 5 cm from one source an 5.5 cm from the other. Determine the wavelength. Determine the spee of the waves. Solution Given: n 5 1; P 1 S cm; P 1 S cm; cm Require: l Given: f 5 1 Hz; l cm Require: v Analysis: v 5 fl Solution: v 5 fl 5 11 Hz 11.0 cm v 5 1 cm/s Statement: The spee of the waves is 1 cm/s. Analysis: 0P n S 1 P n S 0 5 an 1 bl Solution: 0P n S 1 P n S 0 5 an 1 bl 0P 1 S 1 P 1 S 0 5 a1 1 bl 05.5 cm 5.0 cm0 5 a1 1 bl 0.5 cm 5 a 1 bl l cm Statement: The wavelength is 1.0 cm. Practice 1. Two point sources, S 1 an S, are vibrating in phase an prouce waves with a wavelength of.5 m. The two waves overlap at a noal point. Calculate the smallest corresponing ifference in path length for this point. T/I A [ans: 1. m]. A point on the thir noal line from the centre of an interference pattern is 35 cm from one source an 4 cm from the other. The sources are 11. cm apart an vibrate in phase at 10.5 Hz. T/I Calculate the wavelength of the waves [ans:.8 cm] Calculate the spee of the waves [ans: 9 cm/s] 3. Two point sources vibrate in phase at the same frequency. They set up an interference pattern in which a point on the secon noal line is 9.5 cm from one source an 5.0 cm from the other. The spee of the waves is 7.5 cm/s. T/I Calculate the wavelength of the waves. [ans: 3.0 cm] Calculate the frequency at which the sources are vibrating. [ans:.5 Hz] 468 Chapter 9 Waves an Light

11 9.3 Review Summary Diffraction is the bening an spreaing of a wave whose wavelength is comparable to or greater than the slit with, or where l $ w. Interference occurs when two waves in the same meium meet. Constructive interference occurs when the crest of one wave meets the crest of another wave. The resulting wave has an amplitue greater than that of each iniviual wave. Destructive interference occurs when the crest of one wave meets the trough of another wave. The resulting wave has an amplitue less than that of each iniviual wave. Waves with shorter wavelengths iffract less than waves with longer wavelengths. A pair of ientical point sources that are in phase prouce a symmetrical pattern of constructive interference areas an noal lines. The number of noal lines in a given region will increase when the frequency of vibration of the sources increases or when the wavelength ecreases. When the separation of the sources increases, the number of noal lines also increases. The relationship that can be use to solve for an unknown variable in a two-point-source interference pattern is 0P n S 1 P n S 0 5 an 1 bl. Questions 1. Uner what conition is the iffraction of waves through a slit maximize? K/U T/I. Two louspeakers are 1.5 m apart, an they vibrate in phase at the same frequency to prouce soun with a wavelength of 1.3 m. Both soun waves reach your frien in phase where he is staning off to one sie. You realize that one of the speakers was connecte incorrectly, so you switch the wires an change its phase by How oes this affect the soun volume that your frien hears? K/U 3. Determine the maximum slit with that will prouce noticeable iffraction for waves of wavelength m. If the slit is wier than the with you calculate in, will the waves iffract? Explain your answer. K/U T/I 4. Two speakers are 1.0 m apart an vibrate in phase to prouce waves of wavelength 0.5 m. Determine the angle of the first noal line. T/I 5. What conitions are necessary for the interference pattern from a two-point source to be stable? K/U T/I 6. Two ientical point sources are 5.0 cm apart. A metre stick is parallel to the line joining the two sources. The first noal line intersects the metre stick at the 35 cm an 55 cm marks. Each crossing point is 50 cm away from the mile of the line joining the two sources. K/U T/I C A Draw a iagram illustrating this. The sources vibrate at a frequency of 6.0 Hz. Calculate the wavelength of the waves. (c) Calculate the spee of the waves if the frequency of the sources is the same as in part. 7. A stuent takes the following ata from a ripple tank experiment where two point sources are in phase: n 5 3, x cm, L 5 77 cm, cm, θ , an the istance between ientical points on 5 crests is 4. cm. From these ata, you can work out the wavelength in three ifferent ways. T/I Carry out the relevant calculations to etermine the wavelength. Which piece of ata o you think has been incorrectly recore? 9.3 Diffraction an Interference of Water Waves 469

Class 34. Interference of Light. Phase of a Wave. In Phase. 180 o Out of Phase. Constructive and Destructive Interferences.

Class 34. Interference of Light. Phase of a Wave. In Phase. 180 o Out of Phase. Constructive and Destructive Interferences. Phase of a Wave Interference of Light In interference phenomena, we often compare the phase ifference between two waves arriving at the same point from ifferent paths or sources. The phase of a wave at

More information

9.5. Interference of Light Waves: Young s double-slit Experiment

9.5. Interference of Light Waves: Young s double-slit Experiment Interference of Light Waves: Young s ouble-slit Experiment At the en of the 1600s an into the 1700s, the ebate over the nature of light was in full swing. Newton s theory of light particles face challenges

More information

Wave Interference: Young s Double-Slit Experiment

Wave Interference: Young s Double-Slit Experiment Wave Interference: Young s Double-Slit Experiment 9.5 If light has wave properties, then two light sources oscillating in phase shoul prouce a result similar to the interference pattern in a ripple tank

More information

DIFFRACTION AND INTERFERENCE

DIFFRACTION AND INTERFERENCE DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe

More information

Physics 133: tutorial week 6 Interference

Physics 133: tutorial week 6 Interference Physics 133: tutorial week 6 Interference 1. What is the angle of the sun above the horizon when light reflecte from a lake is completely plane polarize? Take the refractive inex of water as 1.33. (36

More information

Online Homework 13 Solution

Online Homework 13 Solution Online Homework 13 Solution Introuction to Two-Source Interference Consier two sinusoial waves (1 an ) of ientical wavelength, perio, an maximum amplitue. A snapshot of one of these waves taken at a certain

More information

WAVES Physics 1 (review)

WAVES Physics 1 (review) Chapter 23 ecture Wave Optics WAVES Physics 1 (review) WAVE A isturbance create by a vibrating source that travels in space & time, carrying energy! TRANSVERSE WAVES A wave whose particles vibrate perpenicularly

More information

Springs, Shocks and your Suspension

Springs, Shocks and your Suspension rings, Shocks an your Suspension y Doc Hathaway, H&S Prototype an Design, C. Unerstaning how your springs an shocks move as your race car moves through its range of motions is one of the basics you must

More information

Practical Lab 2 The Diffraction Grating

Practical Lab 2 The Diffraction Grating Practical Lab 2 The Diffraction Grating OBJECTIVES: 1) Observe the interference pattern prouce when laser light passes through multiple-slit grating (a iffraction grating). 2) Graphically verify the wavelength

More information

Lab 4: Diffraction of Light

Lab 4: Diffraction of Light Lab 4: iffraction of Light 1 Introuction Refer to Appenix for photos of the apparatus iffraction an interference are important phenomena that istinguish waves from particles. Thomas Young first emonstrate

More information

Unit 2 Particles and Waves

Unit 2 Particles and Waves North Berwick High School Department of Physics Higher Physics Unit 2 Particles and Waves Section 5 Interference and Diffraction Section 5 Interference and Diffraction Note Making Make a dictionary with

More information

SOLUTIONS TO CONCEPTS CHAPTER 17

SOLUTIONS TO CONCEPTS CHAPTER 17 1. Given that, 400 m < < 700 nm. 1 1 1 700nm 400nm SOLUTIONS TO CONCETS CHATER 17 1 1 1 3 10 c 3 10 (Where, c = spee of light = 3 10 m/s) 7 7 7 7 7 10 4 10 7 10 4 10 4.3 10 14 < c/ < 7.5 10 14 4.3 10 14

More information

Chapter 38 Diffraction Patterns and

Chapter 38 Diffraction Patterns and Chapter 38 Diffraction Patterns an Polarization 38.1 Introuction to Diffraction Pattern 38. Dffraction Pattern from Narrow Slits Diffraction one kin of interference a The first zeroes in the intensity

More information

Diffraction Grating Equation with Example Problems 1

Diffraction Grating Equation with Example Problems 1 Emmett J. Ientilucci, Ph.D. Digital Imaging an Remote Sensing Laboratory Rochester Institute of Technology December 18, 2006 Diffraction Grating Equation with Example Problems 1 1 Grating Equation In Figure

More information

LECTURE 15: LINEAR ARRAY THEORY - PART I

LECTURE 15: LINEAR ARRAY THEORY - PART I LECTURE 5: LINEAR ARRAY THEORY - PART I (Linear arrays: the two-element array; the N-element array with uniform amplitue an spacing; broa - sie array; en-fire array; phase array). Introuction Usually the

More information

36-1One focus of physics in the study of light is to understand and put to

36-1One focus of physics in the study of light is to understand and put to CHAPTER 36 WHAT Fig. 36-1 This iffraction pattern appeare on a viewing screen when light that ha passe through a narrow vertical slit reache the screen. Diffraction cause the light to flare out perpenicular

More information

Reading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14.

Reading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14. 7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings

More information

Interference 1.1 INTRODUCTION 1.2 SUPERPOSITION OF WAVES

Interference 1.1 INTRODUCTION 1.2 SUPERPOSITION OF WAVES 1 Interference 1.1 INTROUCTION In this chapter we will iscuss the phenomena associate with the interference of light waves. At any point where two or more wave trains cross one another they are sai to

More information

ELECTRON DIFFRACTION

ELECTRON DIFFRACTION 17 Jul 12 ElectDiff.1 ELECTRON DIFFRACTION In this experiment the existence of wave properties of particles is emonstrate through observation of electron iffraction. The wavelengths of electrons iffracting

More information

** View All Solutions Here **

** View All Solutions Here ** QUESTIONS 981 Huygens Principle The three-imensional transmission of waves, incluing light, may often be preicte by Huygens principle, which states that all points on a wavefront serve as point sources

More information

λ = d If d is decreased then W is increased (inverse proportionality with λ and L constant).

λ = d If d is decreased then W is increased (inverse proportionality with λ and L constant). Saple Probles: WAVES: Interference. Consier the arrangeent for Young s ouble slit experients using onochroatic light. What effect woul ecreasing the istance between the two slits have on the istance between

More information

Diffraction. AP Physics B

Diffraction. AP Physics B Diffraction AP Physics B Superposition..AKA.Interference One of the characteristics of a WAVE is the ability to undergo INTERFERENCE. There are TWO types. We call these waves IN PHASE. We call these waves

More information

Topic 2: The Trigonometric Ratios Finding Sides

Topic 2: The Trigonometric Ratios Finding Sides Topic 2: The Trigonometric Ratios Fining Sies Labelling sies To use the Trigonometric Ratios, commonly calle the Trig Ratios, it is important to learn how to label the right angle triangle. The hypotenuse

More information

M147 Practice Problems for Exam 2

M147 Practice Problems for Exam 2 M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work

More information

Young s Double Slit Experiment

Young s Double Slit Experiment Young s Double Slit Experiment Apparatus optics bench laser slit film screen white paper and tape pencil metric ruler Ocean Optics spectrometer and fiber optics cable Goal In this experiment, you will

More information

Advanced Optics with Laser Pointer and Metersticks

Advanced Optics with Laser Pointer and Metersticks Avance Optics with aser Pointer an s Roan Ya. Kezerashvili New York City College of Technology, The City University of New York 300 Jay Street, Brooklyn NY, 1101 Eail: Rkezerashvili@citytech.cuny.eu Abstract

More information

Purpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of

Purpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose

More information

Interference. Physics 102 Workshop #3. General Instructions

Interference. Physics 102 Workshop #3. General Instructions Interference Physics 102 Workshop #3 Name: Lab Partner(s): Instructor: Time of Workshop: General Instructions Workshop exercises are to be carried out in groups of three. One report per group is due by

More information

Experiments with Diffraction

Experiments with Diffraction Experiments with iffraction Abbie Tippie (tippie@optics.rochester.edu) and Tammy Lee (talee@optics.rochester.edu) What is diffraction? When parallel waves of light are obstructed by a very small object

More information

Notes on tangents to parabolas

Notes on tangents to parabolas Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,

More information

EE 119 Homework 11 Solution

EE 119 Homework 11 Solution EE 9 Homework Solution Professor: Jeff Bokor TA: Xi Luo General iffraction questions (a) How many wavelengths wie must a single slit be if the first Fraunhofer iffraction minimum occurs at an angular istance

More information

Lecture L25-3D Rigid Body Kinematics

Lecture L25-3D Rigid Body Kinematics J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

More information

Mathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400

Mathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400 hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents

More information

Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

More information

i( t) L i( t) 56mH 1.1A t = τ ln 1 = ln 1 ln 1 6.67ms

i( t) L i( t) 56mH 1.1A t = τ ln 1 = ln 1 ln 1 6.67ms Exam III PHY 49 Summer C July 16, 8 1. In the circuit shown, L = 56 mh, R = 4.6 Ω an V = 1. V. The switch S has been open for a long time then is suenly close at t =. At what value of t (in msec) will

More information

CHAPTER 5 : CALCULUS

CHAPTER 5 : CALCULUS Dr Roger Ni (Queen Mary, University of Lonon) - 5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective

More information

And although two objects like these bumper boats cannot be in the same place on the water at the same time, water waves can. Why is that?

And although two objects like these bumper boats cannot be in the same place on the water at the same time, water waves can. Why is that? Physics 1103 Wave Interactions (picture of beach on screen) Today, we go back to the beach to investigate more wave interactions. For example, what makes these waves change direction as they approach the

More information

SPH3UW Superposition and Interference Page 1 of 6

SPH3UW Superposition and Interference Page 1 of 6 SPH3UW Superposition and Interference Page 1 of 6 Notes Physics Tool box Constructive Interference: refers to when waves build up each other, resulting in the medium having larger amplitude. Destructive

More information

Experiment 11: Interference and Diffraction

Experiment 11: Interference and Diffraction MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Experiment 11: Interference and Diffraction OBJECTIVES 1. To explore the diffraction of light through a variety of apertures 2. To learn

More information

WAVES: WAVE BEHAVIOUR QUESTIONS

WAVES: WAVE BEHAVIOUR QUESTIONS WAVES: WAVE BEHAVIOUR QUESTIONS ROPES AND A MIRAGE (2015;3) Tom and his friend Ellen hold each end of a rope. Each of them sends a pulse along the rope in opposite directions. The grid below shows the

More information

Interference and the wave nature of light

Interference and the wave nature of light Interference and the wave nature of light Fig 27.2 Sound waves combining result in constructive and destructive interference, and Light waves (electromagnetic waves) do, too! The waves emitted by source

More information

Jitter effects on Analog to Digital and Digital to Analog Converters

Jitter effects on Analog to Digital and Digital to Analog Converters Jitter effects on Analog to Digital an Digital to Analog Converters Jitter effects copyright 1999, 2000 Troisi Design Limite Jitter One of the significant problems in igital auio is clock jitter an its

More information

INTERFERENCE and DIFFRACTION

INTERFERENCE and DIFFRACTION Course and Section Date Names INTERFERENCE and DIFFRACTION Short description: In this experiment you will use interference effects to investigate the wave nature of light. In particular, you will measure

More information

Kater Pendulum. Introduction. It is well-known result that the period T of a simple pendulum is given by. T = 2π

Kater Pendulum. Introduction. It is well-known result that the period T of a simple pendulum is given by. T = 2π Kater Penulum ntrouction t is well-known result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity.

More information

Applications of Global Positioning System in Traffic Studies. Yi Jiang 1

Applications of Global Positioning System in Traffic Studies. Yi Jiang 1 Applications of Global Positioning System in Traffic Stuies Yi Jiang 1 Introuction A Global Positioning System (GPS) evice was use in this stuy to measure traffic characteristics at highway intersections

More information

Lesson 30: Diffraction & Interference

Lesson 30: Diffraction & Interference Lesson 30: Diffraction & Interference The wave model of light eventually replaced Newton's particle theory. This did not happen overnight, and it certainly wasn't the work of one person. Three specific

More information

ConcepTest Superposition. If waves A and B are superposed (that is, their amplitudes are added) the resultant wave is

ConcepTest Superposition. If waves A and B are superposed (that is, their amplitudes are added) the resultant wave is ConcepTest 24.1 Superposition If waves A and B are superposed (that is, their amplitudes are added) the resultant wave is 1) 2) 3) 4) ConcepTest 24.1 Superposition If waves A and B are superposed (that

More information

Traffic Delay Studies at Signalized Intersections with Global Positioning System Devices

Traffic Delay Studies at Signalized Intersections with Global Positioning System Devices Traffic Delay Stuies at Signalize Intersections with Global Positioning System Devices THIS FEATURE PRESENTS METHODS FOR APPLYING GPS DEVICES TO STUDY TRAFFIC DELAYS AT SIGNALIZED INTERSECTIONS. MOST OF

More information

Chapter 2 Review of Classical Action Principles

Chapter 2 Review of Classical Action Principles Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger

More information

State of Louisiana Office of Information Technology. Change Management Plan

State of Louisiana Office of Information Technology. Change Management Plan State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change

More information

IB PHYSICS HL REVIEW PACKET: WAVES & VIBRATIONS

IB PHYSICS HL REVIEW PACKET: WAVES & VIBRATIONS NAME IB PHYSICS HL REVIEW PACKET: WAVES & VIBRATIONS 1. This question is about waves and wave properties. (a) By making reference to waves, distinguish between a ray and a wavefront.......... (3) The diagram

More information

Pre-Lab Assignment: Interference, Measuring Wavelengths, and Atomic Spectra

Pre-Lab Assignment: Interference, Measuring Wavelengths, and Atomic Spectra Name: Lab Partners: Date: Pre-Lab Assignment: Interference, Measuring Wavelengths, and Atomic Spectra (Due at the beginning of lab) Directions: Read over the lab handout and then answer the following questions

More information

Introduction to Integration Part 1: Anti-Differentiation

Introduction to Integration Part 1: Anti-Differentiation Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction

More information

Double Slit Experiment. and diffrac5on gra5ngs

Double Slit Experiment. and diffrac5on gra5ngs Double Slit Experiment and diffrac5on gra5ngs Diffraction Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending

More information

CURVES: VELOCITY, ACCELERATION, AND LENGTH

CURVES: VELOCITY, ACCELERATION, AND LENGTH CURVES: VELOCITY, ACCELERATION, AND LENGTH As examples of curves, consier the situation where the amounts of n-commoities varies with time t, qt = q 1 t,..., q n t. Thus, the amount of the commoities are

More information

Ch. 27. Interference and the Wave Nature of Light

Ch. 27. Interference and the Wave Nature of Light Ch. 27. Interference and the Wave Nature of Light Up to now, we have been studying geometrical ti optics, where the wavelength of the light is much smaller than the size of our mirrors and lenses and the

More information

DIFFRACTION OF LIGHT

DIFFRACTION OF LIGHT Laboratory Exercise 4. DIFFRACTION OF LIGHT Diffraction Gratings. Determining the Wavelength of Laser Light Using a Diffraction Grating. Refraction. Observation of Atomic Spectra. Theoretical background:

More information

Measures of distance between samples: Euclidean

Measures of distance between samples: Euclidean 4- Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate

More information

Example Optimization Problems selected from Section 4.7

Example Optimization Problems selected from Section 4.7 Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point

More information

Wave Optics. b. the crest from one wave overlaps with the d. darkness cannot occur as the two waves are coherent.

Wave Optics. b. the crest from one wave overlaps with the d. darkness cannot occur as the two waves are coherent. Wave Optics 1. Two beams of coherent light are shining on the same piece of white paper. With respect to the crests and troughs of such waves, darkness will occur on the paper where: a. the crest from

More information

physics 112N interference and diffraction

physics 112N interference and diffraction physics 112N interference and diffraction the limits of ray optics shadow of the point of a pin physics 112N 2 the limits of ray optics physics 112N 3 the limits of ray optics physics 112N 4 this is how

More information

Solutions to the Exercises of Chapter 9

Solutions to the Exercises of Chapter 9 9A. Vectors an Forces Solutions to the Exercises of Chapter 9. F = 5 sin 5.9 an F = 5 cos 5 4.8.. a. By the Pythagorean theorem, the length of the vector from to (, )is + = 5. So the magnitue of the force

More information

Experiment #3: Interference and Diffraction

Experiment #3: Interference and Diffraction Experiment #3: Interference and Diffraction EYE HAZARD: never look directly into a laser! Starting with experiment #4, Please always bring a formatted high-density PC diskette with you to the lab. Purpose:

More information

Physics 9 Fall 2009 DIFFRACTION

Physics 9 Fall 2009 DIFFRACTION Physics 9 Fall 2009 NAME: TA: SECTION NUMBER: LAB PARTNERS: DIFFRACTION 1 Introduction In these experiments we will review and apply the main ideas of the interference and diffraction of light. After reviewing

More information

11 CHAPTER 11: FOOTINGS

11 CHAPTER 11: FOOTINGS CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit

More information

12. PRELAB FOR INTERFERENCE LAB

12. PRELAB FOR INTERFERENCE LAB 12. PRELAB FOR INTERFERENCE LAB 1. INTRODUCTION As you have seen in your studies of standing waves, a wave and its reflection can add together constructively (peak meets peak, giving large amplitude) or

More information

v = fλ PROGRESSIVE WAVES 1 Candidates should be able to :

v = fλ PROGRESSIVE WAVES 1 Candidates should be able to : PROGRESSIVE WAVES 1 Candidates should be able to : Describe and distinguish between progressive longitudinal and transverse waves. With the exception of electromagnetic waves, which do not need a material

More information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

More information

Lesson 18: Diffraction and Interference!

Lesson 18: Diffraction and Interference! Lesson 18: Diffraction and Interference Part 1: The Double Slit Experiment What is light? - A particle? - A wave? In 1801, Thomas Young s Double Slit Experiment confirmed the wave nature of light: If light

More information

1/26/2016. Chapter 21 Superposition. Chapter 21 Preview. Chapter 21 Preview

1/26/2016. Chapter 21 Superposition. Chapter 21 Preview. Chapter 21 Preview Chapter 21 Superposition Chapter Goal: To understand and use the idea of superposition. Slide 21-2 Chapter 21 Preview Slide 21-3 Chapter 21 Preview Slide 21-4 1 Chapter 21 Preview Slide 21-5 Chapter 21

More information

Solutions of Laplace s equation in 3d

Solutions of Laplace s equation in 3d Solutions of Laplace s equation in 3 Motivation The general form of Laplace s equation is: Ψ = 0 ; it contains the laplacian, an nothing else. This section will examine the form of the solutions of Laplaces

More information

Homework 8. problems: 10.40, 10.73, 11.55, 12.43

Homework 8. problems: 10.40, 10.73, 11.55, 12.43 Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks

More information

19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes

19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is so-calle because we rearrange the equation

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 230.01, Fall 2012: HW 1 Solutions Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

More information

Investigation of interference effects of Young s double slits and diffraction gratings

Investigation of interference effects of Young s double slits and diffraction gratings Name: Teacher: Class: Investigation of interference effects of Young s double slits and diffraction gratings Practical skills covered A use of analogue equipment to measure lengths J use of laser or light

More information

Cutnell/Johnson Physics

Cutnell/Johnson Physics Cutnell/Johnson Physics Classroom Response System Questions Chapter 17 The Principle of Linear Superposition and Interference Phenomena Interactive Lecture Questions 17.1.1. The graph shows two waves at

More information

SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH

SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH Basic Definitions In homework set # one of the questions involves basic unerstaning of the hyperbolic functions sinh an cosh. We will use this

More information

Waves Review Checklist Pulses 5.1.1A Explain the relationship between the period of a pendulum and the factors involved in building one

Waves Review Checklist Pulses 5.1.1A Explain the relationship between the period of a pendulum and the factors involved in building one 5.1.1 Oscillating Systems Waves Review hecklist 5.1.2 Pulses 5.1.1A Explain the relationship between the period of a pendulum and the factors involved in building one Four pendulums are built as shown

More information

PRACTICE Q6--Quiz 6, Ch15.1 &15.2 Interference & Diffraction

PRACTICE Q6--Quiz 6, Ch15.1 &15.2 Interference & Diffraction Name: Class: Date: ID: A PRACTICE Q6--Quiz 6, Ch5. &5. Interference & Diffraction Multiple Choice Identify the choice that best completes the statement or answers the question.. The trough of the sine

More information

4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet

4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet 4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet Required: READ Hamper pp 115-134 SL/HL Supplemental: Cutnell and Johnson, pp 473-477, 507-513 Tsokos, pp 216-242 REMEMBER TO. Work through all

More information

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar

More information

10.3 The Diffraction Grating

10.3 The Diffraction Grating diffraction grating a device with a large number of equally spaced parallel slits that produces interference patterns 10.3 The Diffraction Grating It is difficult to measure the wavelength of light accurately

More information

Solutions to Examples from Related Rates Notes. ds 2 mm/s. da when s 100 mm

Solutions to Examples from Related Rates Notes. ds 2 mm/s. da when s 100 mm Solutions to Examples from Relate Rates Notes 1. A square metal plate is place in a furnace. The quick temperature change causes the metal plate to expan so that its surface area increases an its thickness

More information

Objectives 450 CHAPTER 10 LIGHT AND OPTICAL SYSTEMS

Objectives 450 CHAPTER 10 LIGHT AND OPTICAL SYSTEMS Objectives Use wave properties to explain interference and diffraction of light. Explain how double slits, a diffraction grating, a single slit, and an aperture produce interference patterns. Use measurements

More information

UNIT 31: INTERFERENCE AND DIFFRACTION

UNIT 31: INTERFERENCE AND DIFFRACTION Name St.No. - Date(YY/MM/DD) / / Section Group # UNIT 31: INTERFERENCE AND DIFFRACTION Interference of two circular waves, snapshots of absolute value of (real,scalar) wave field for different wave lengths

More information

INTERFERENCE. Physics 122 June 2, 2006

INTERFERENCE. Physics 122 June 2, 2006 Physics 122 June 2, 2006 http://www.physics.sfsu.edu/~manuals/ph122/ I. Theory INTERFERENCE This laboratory will investigate the phenomenon of interference. The interference and diffraction of light waves

More information

Waves Physics Leaving Cert Quick Notes

Waves Physics Leaving Cert Quick Notes Waves Physics Leaving Cert Quick Notes Waves A wave is a means of transferring energy from one point to another Waves can be classified as mechanical where the wave must have a medium to travel through,

More information

A NATIONAL MEASUREMENT GOOD PRACTICE GUIDE. No.107. Guide to the calibration and testing of torque transducers

A NATIONAL MEASUREMENT GOOD PRACTICE GUIDE. No.107. Guide to the calibration and testing of torque transducers A NATIONAL MEASUREMENT GOOD PRACTICE GUIDE No.107 Guie to the calibration an testing of torque transucers Goo Practice Guie 107 Measurement Goo Practice Guie No.107 Guie to the calibration an testing of

More information

Lesson 19: Mechanical Waves!!

Lesson 19: Mechanical Waves!! Lesson 19: Mechanical Waves Mechanical Waves There are two basic ways to transmit or move energy from one place to another. First, one can move an object from one location to another via kinetic energy.

More information

10.2 Systems of Linear Equations: Matrices

10.2 Systems of Linear Equations: Matrices SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix

More information

The Range of Alpha Particles in Air. Lennon Ó Náraigh, Date of Submission: 5 th April Abstract:

The Range of Alpha Particles in Air. Lennon Ó Náraigh, Date of Submission: 5 th April Abstract: The ange of lpha Particles in ir Lennon Ó Náraigh, Date of Submission: 5 th pril 4 bstract: This eperiment assumes that the notion of ange is applicable to alpha particles an sets out to calculate sai

More information

Solutions to modified 2 nd Midterm

Solutions to modified 2 nd Midterm Math 125 Solutions to moifie 2 n Miterm 1. For each of the functions f(x) given below, fin f (x)). (a) 4 points f(x) = x 5 + 5x 4 + 4x 2 + 9 Solution: f (x) = 5x 4 + 20x 3 + 8x (b) 4 points f(x) = x 8

More information

Achieving quality audio testing for mobile phones

Achieving quality audio testing for mobile phones Test & Measurement Achieving quality auio testing for mobile phones The auio capabilities of a cellular hanset provie the funamental interface between the user an the raio transceiver. Just as RF testing

More information

Chapter 30 Reflection and Refraction

Chapter 30 Reflection and Refraction Chapter 30 Reflection and Refraction Slide 30-1 Geometrical optics When light or other electromagnetic waves interact with systems much larger than the wavelength, it s a good approximation to Neglect

More information

Stress Concentration Factors of Various Adjacent Holes Configurations in a Spherical Pressure Vessel

Stress Concentration Factors of Various Adjacent Holes Configurations in a Spherical Pressure Vessel 5 th Australasian Congress on Applie Mechanics, ACAM 2007 10-12 December 2007, Brisbane, Australia Stress Concentration Factors of Various Ajacent Holes Configurations in a Spherical Pressure Vessel Kh.

More information

Superposition and Interference

Superposition and Interference Superposition and Interference Bởi: OpenStaxCollege These waves result from the superposition of several waves from different sources, producing a complex pattern. (credit: waterborough, Wikimedia Commons)

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Two radio antennas are 120 m apart on a north-south line. The two antennas radiate in

More information

Diffraction and Color Spectra

Diffraction and Color Spectra Diffraction and Color Spectra Bob Somers 5/18/05 Per. 4 Mark Saenger, Cecily Cappello Purpose In this lab we applied all the underlying wave behavior theory we ve learned about diffraction and interference

More information

Chapter 35, example problems:

Chapter 35, example problems: Chapter 35, example problems: (35.02) Two radio antennas A and B radiate in phase. B is 120 m to the right of A. Point Q along extension of line AB. 40 m to the right of B. Frequency and wavelength can

More information

Chapter 24. Wave Optics

Chapter 24. Wave Optics Chapter 24 Wave Optics Wave Optics The wave nature of light is needed to explain various phenomena. Interference Diffraction Polarization The particle nature of light was the basis for ray (geometric)

More information