Lecture 28 March 30. 2016. Standing waves Musical instruments, guitars, pianos, organs Doppler Effect Resonance 3/30/2016 Physics 214 Spring 2016 1
Waves on a string If we shake the end of a rope we can send a wave along the rope. The rope must be under tension in order for the wave to propagate v = (F/μ) F = TENSION (newtons) μ = MASS/UNIT LENGTH (kg/meter) 3/30/2016 Physics 214 Spring 2016 2
Standing waves If two identical waves exist on the same string but traveling in opposite directions the result can be standing waves in which some points never have a deflection, these are called nodes Some points oscillate between plus and minus the maximum amplitude, these are called antinodes. Standing waves provide the notes on musical instruments. When a string is secured at both ends and plucked or hit the generated waves will travel along the string and be reflected and set up standing waves. red blue 3/30/2016 Physics 214 Spring 2016 3
1S-41 Standing Waves in Rope Creating transverse standing waves on heavy cable How might we create different wave patterns on the cable? How is this related to tuning a guitar? BY ADJUSTING THE SPEED OF THE JIGSAW OR THE TENSION IN THE CORD, DIFFERENT STANDING WAVE PATTERNS CAN BE GENERATED BETWEEN THE FIXED ENDS. THOUGH THE WAVELENGTH OF THE STANDING WAVE IS FIXED BY THE LENGTH BETWEEN THE FIXED POINTS, THE FREQUENCY DEPENDS ON THE TENSION IN THE CABLE. YOU TUNE A GUITAR BY CHANGING THIS TENSION. 3/30/2016 Physics 214 Spring 2016 4
Musical notes on Strings f n 0 nv / 2 L, n is a positive integer, 1, 2, 3 Each end of the string must be a node so the possible standing waves must be multiples of λ/2 f n 0 nv / 2L Fundamental f = v/λ = v/2l 2 nd Harmonic f = v/λ = v/l 3 rd Harmonic f = v/λ = 3v/2L Musical sound is a mixture of harmonics modified by the body of the instrument. v = (F/μ) so a piano or a violin is tuned by changing the tension in the string 3/30/2016 Physics 214 Spring 2016 5 L 2 L 2 2 L 2 3
Musical notes on Closed organ pipe f (2n 1) (2n 1)v / 4 L, n is a positive integer, 0,1,2,3 0 Node at one end and an antinode at the other Fundamental f = v/4l 2 nd Harmonic f = 3v/4L 3 rd Harmonic f = 5v/4L The velocity of sound in air is v = 340m/s L 4 3 L 4 5 L 4 3/30/2016 Physics 214 Spring 2016 6
Doppler effect The Doppler effect is the change in frequency of a wave when the source and observer are moving. As the source is approaching the frequency increases and if it is receding the frequency decreases http://www.physics.purdue.edu/class/applets/phe/dopplereff.htm 3/30/2016 Physics 214 Spring 2016 7
4B-11 Resonance Two identical wooden have identical tuning forks attached at the center of the top of the box. When the tuning fork is struck, the sound is amplified by the resonance occurring in the box. When the one box is placed such that its opening is a few centimeters from the other s opening, striking one tuning fork sets up a sympathetic vibration in the other, which can be shown by having a ping-pong ball, attached to a thread, lightly touch the second fork. The ping-pong ball will bounce back and forth indicating that the fork is vibrating. 3/30/2016 Physics 214 Spring 2016 8
4C-01 Doppler Effect Investigating change in sound due to the Doppler effect At what point in circular movement does sound change? What is relative motion between source and listener at these points? WHEN THE SOURCE MOVES TOWARD (AWAY FROM) LISTENER, THE FREQUENCY OF SOUND, OR PITCH, INCREASES (DECREASES). 3/30/2016 Physics 214 Spring 2016 9
Summary of Chapter 15 v = λ/t = fλ Transverse or longitudinal Standing waves Fundamental f = v/λ = v/2l 2 nd Harmonic f = v/λ = v/l 3 rd Harmonic f = v/λ = 3v/2L Fundamental f = v/4l 2 nd Harmonic f = 3v/4L 3 rd Harmonic f = 5v/4L 3/30/2016 Physics 214 Spring 2016 10
Sound effects Doppler effect f increases as sound approaches Sonic boom Beats f = f 1 f 2 3/30/2016 Physics 214 Spring 2016 11
Questions Chapter 15 Q2 Waves are traveling in an eastward direction on a lake. Is the water in the lake necessarily moving in that direction? No. The main motion is just up and down Q3 A slowly moving engine bumps into a string of coupled railroad cars standing on a siding. A wave pulse is transmitted down the string of cars as each one bumps into the next one. Is this wave transverse or longitudinal? Longitudinal. The motion is back and forward in the direction of the wave 3/30/2016 Physics 214 Spring 2016 12
Q4 A wave can be propagated on a blanket by holding adjacent corners in your hands and moving the end of the blanket up and down. Is this wave transverse or longitudinal? It is transverse the motion of the blanket is up and down Q13 Suppose that we increase the tension in a rope, keeping the frequency of oscillation of the end of the rope the same. What effect does this have on the wavelength of the wave produced? The increase in tension cause an increase in velocity so the wavelength increases 3/30/2016 Physics 214 Spring 2016 13
Q20 If we increase the tension of a guitar string, what effect does this have on the frequency and wavelength of the fundamental standing wave formed on that string? The fundamental wavelength is fixed by the length of the string. Increasing the tension increases the wave speed so the frequency increases. Q23 Is it possible for sound to travel through a steel bar? Yes. It is just a compression wave that travels through the steel. Putting your ear to the rails is a good way of hearing a train coming a long distance away 3/30/2016 Physics 214 Spring 2016 14
Q24 Suppose that we increase the temperature of the air through which a sound wave is traveling. A. What effect does this have on the speed of the sound wave? Explain. B. For a given frequency, what effect does increasing the temperature have on the wavelength of the sound wave? Explain. The speed increases and the wavelength increases Q25 If the temperature in a organ pipe increases above room temperature, thereby increasing the speed of sound waves in the pipe but not affecting the length of pipe significantly, what effect does this have on the frequency of the standing waves produced by this pipe? The length is unchanged so the wavelength is unchanged so the frequency increases 3/30/2016 Physics 214 Spring 2016 15
Q26 Is the wavelength of the fundamental standing wave in a tube open at both ends greater than, equal to, or less than the wavelength for the fundamental wave in a tube open at just one end? A tube open at both ends has half a wavelength whereas the tube open at one end has one quarter of a wavelength. Q27 A band playing on a flat-bed truck is approaching you rapidly near the end of a parade. Will you hear the same pitch for the various instruments as someone down the street who has already been passed by the truck? The pitch increases as the band approaches and decreases as it is going away 3/30/2016 Physics 214 Spring 2016 16
Ch 15 E 2 Water waves have a wavelength = 1.4 m and Period T = 0.8s. What is the velocity of the waves? v = / T = 1.4/0.8 = 1.75 m/s 3/30/2016 Physics 214 Spring 2016 17
Ch 15 E 8 String of length 0.8 m is fixed at both ends. The story is plucked so that there are two nodes along the string in addition to those at either end. What is the wavelength of the interfering waves? fixed head node node snapshot λ v ½ λ T v R At each node wave T and wave R cancel. From the picture a total of 3/2 fit on 0.8 m string. 3/2 = 0.8 m = (2/3) 0.8 = 0.53 m 3/30/2016 Physics 214 Spring 2016 18
Ch 15 E 10 What is the frequency of a sound wave with wavelength = 0.68 m travelling in room temperature air (v=340m/s)? v = f f = v/ = 340/0.68 m = 500 Hz 3/30/2016 Physics 214 Spring 2016 19
Ch 15 CP 2 A guitar string has length 1.25 m and mass 40 g. After stringing the guitar, string has 64 cm between fixed points. It is tightened to tension = 720 N. a) What is mass per unit length of string? b) What is wavespeed on tightened string? c) What is wavelength of traveling waves that interfere to form the fundamental standing wave? d) What is frequency of the fundamental standing wave? e) What are wavelength and frequency of the next harmonic? 3/30/2016 Physics 214 Spring 2016 20
Ch 15 CP2 cont. a) =M/L=0.04/1.25=0.032 kg/m b) = F/ = 720/0.032 = 22500 = 150 m/s c) v v R T = Snapshot of fundamental standing wave node antinode node 0.64 m = ½ = 1.28 m d) =1.28 m ; f = / = 150/1.28 = 117 Hz e) = 0.64 m f = / = 150/0.64 = 234 Hz node anti-node node anti-node node Snapshot of second harmonic 3/30/2016 Physics 214 Spring 2016 21
Electromagnetic waves EM waves consist of oscillating magnetic and electric fields transmitted through vacuum at a constant speed of c = 3 x 10 8 m/s. They are produced whenever there is a changing electric or magnetic field. The acceleration of electric charge produces EM waves such as a broadcast antenna, AC wiring and lightning. A simple EM wave in vacuum has f = c/λ where c = 3 x 10 8 m/s In a medium like glass the velocity decreases, f stays the same so λ also decreases and f = v/λ The factor for the decrease is the refractive index which for glass is typically 1.4 to 2.0 http://www.physics.purdue.edu/class/applets/phe/emwave.htm 3/30/2016 Physics 214 Spring 2016 22
Electromagnetic spectrum Almost all the information we receive from outside the earth is in the form of EM radiation. Different parts of the spectrum correspond to different physical processes We can understand what is going on in the Universe and back in time to near the beginning of the Universe using a variety of earth and space based telescopes. 3/30/2016 Physics 214 Spring 2016 23
The telescopes 3/30/2016 Physics 214 Spring 2016 24
How do we see color An image is formed at the back of the eye like a camera and there are receptors called cones that respond to different wavelengths. The brain interprets the mixture of the three signals as color. If we look at an object the color we see is the reflected light so the other colors were absorbed. If we are looking through a colored object then the object lets that color be transmitted and the other colors are absorbed 3/30/2016 Physics 214 Spring 2016 25
Color Mixing The process of mixing two different wavelengths of light, such as red and green, to produce a response interpreted as another color, such as yellow, is additive color mixing. Combining the three primary colors blue, green, and red in different amounts can produce responses in our brains corresponding to all the colors we are used to identifying. Red and green make yellow, blue and green make cyan, and blue and red make magenta. Combining all three colors produces white. Pigments used in paints or dyes work by selective color mixing. They absorb some wavelengths of light more than others. 3/30/2016 Physics 214 Spring 2016 26
Blue Sky Red Sunset The sun produces white light but the shorter wavelengths of blue light are scattered by gas molecules in the atmosphere more than longer wavelengths such as red light. The blue light enters our eyes after being scattered multiple times, so appears to come from all parts of the sky. Looking directly at the sun near sunset one sees red because the blue light has been scattered in all directions 3/30/2016 Physics 214 Spring 2016 27
Refractive Index The speed of electromagnetic waves in vacuum is 3 x 10 8 m/sec and no energy can be transferred faster than this speed. When an electromagnetic wave passes through a transparent medium such as glass the speed is reduced by a factor n that we call the refractive index. If the speed in glass is v then n = c/v. The frequency of the wave does not change so that the speed is reduced by the factor n and therefore since c = fλ vacuum and v = fλ glass then c/v = n and λ vacuum /λ glass = n and λ glass is smaller than λ vacuum 3/30/2016 Physics 214 Spring 2016 28
Interference When two coherent beams of light are brought together they will add just like two waves on a string. So when two peaks coincide the light will be a maximum and when a peak coincides with a valley there will be no light. If we use two slits then as we move on the screen the path difference between the two beams will vary and we will get bright and dark fringes. Bright fringes are located at positions given by dy/x = nλ where n is an integer Dark fringes occur when dy/x = nλ/2 where n is an odd integer 3/30/2016 Physics 214 Spring 2016 29
Thin film interference Thin film interference occurs when light is reflected from the top surface and the underneath surface. This provides the two beams of coherent light that interfere. Since we normally observe this with white light we see colors because the path difference varies depending on the angle of observation So different wavelengths (colors) have constructive and destructive interference at different places on the film. 3/30/2016 Physics 214 Spring 2016 30
Diffraction Interference occurs even for a single aperture and this is called diffraction. The pattern shown is from a square aperture and light from different parts of the aperture are interfering. As the aperture is made smaller the pattern actually expands. This effect can limit our ability to see detail in small objects or to resolve two stars nearby to one another. The position of the first dark fringe is given by y = λx/w. This means the central bright fringe has a width ~ 2y and increases as w gets smaller 3/30/2016 Physics 214 Spring 2016 31
Polarization In a normal beam of light the electric field vector points in all directions. There are materials like polaroid that will only transmit light with the electric field vector along a specific direction. Light reflected by water also is polarized so polaroid sun glasses at the right orientation will block the reflected light. Transparent objects under stress can also cause transmitted light to be polarized. Car windshields are one example where patterns can be seen when wearing polaroid sun glasses 3/30/2016 Physics 214 Spring 2016 32
Summary of Chapter 16 f = c/λ c = 3 x 10 8 m/s Refractive index c = fλ vacuum and v = fλ glass then c/v = n and λ vacuum /λ glass = n and λ glass is smaller than λ vacuum Polarization Thin film interference 3/30/2016 Physics 214 Spring 2016 33
Interference Bright fringes are located at positions given by dy/x = nλ n is an integer Dark fringes occur when dy/x = nλ/2 where n is an odd integer Diffraction grating dy/x = mλ where m is an odd or even integer The position of the first dark fringe is given by y = λx/w. This means the central bright fringe has a width ~ 2y and increases as w gets smaller 3/30/2016 Physics 214 Spring 2016 34
Coatings for lenses As light passes from one transparent medium to another a few percent of the light will be reflected. This is a particular problem in optical systems like lenses where there may be many glass elements. For example if 96% of the light is transmitted at a surface after 8 surfaces only 72% of the light remains and the other 28% will be scattered everywhere. Thin coatings are put on glass surfaces so that for particular wavelengths the light reflected from the top surface is exactly cancelled by the light from the bottom surface. This is only true for a single wavelength and to reduce the reflections for a range of wavelengths requires multiple thin film layers very often just λ/4 thick. This is why camera lenses usually have a blue green color. Often binocular lenses are ruby red. 3/30/2016 Physics 214 Spring 2016 35
7B-11 Color in Thin Films Studying thin-film interference effects What kind of patterns appear on the screen? Why are the effects on white light and monochromatic light different? LIGHT REFLECTED FROM THE FRONT AND THE BACK OF A THIN SOAP FILM INTERFERES WITH ITSELF. INTERFERENCE OF MONOCHROMATIC LIGHT PRODUCES BRIGHT AND DARK FRINGES WHILE INTERFERENCE OF WHITE LIGHT PRODUCES COLORED BANDS (DIFFERENT FREQUENCIES OF LIGHT INTERFERE DIFFERENTLY). 3/30/2016 Physics 214 Spring 2016 36
7B-15 Line Spectrum Identifying the emission line spectrum of specific elements What do you expect to see with the diffraction grating? THE COLOR WE SEE IN THE TUBE (WITHOUT PASSING THROUGH THE GRATING) IS A COMPOSITE OF COLORS. THE COLORS ARE SEPARATED (DIFFERENT COLORS INTERFERE AT DIFFERENT POSITIONS) BY GRATING AND PROVIDE UNIQUE IDENTIFICATION OF ELEMENT. 3/30/2016 Physics 214 Spring 2016 37
Questions Chapter 16 Q3 Is it possible for an electromagnetic wave to travel through a vacuum? Yes Q11 Skylight is produced by scattering of light from the direct beam coming from the sun. Why is the color of the sky different from the color of the light of the sun itself? Blue light is scattered more strongly than red light 3/30/2016 Physics 214 Spring 2016 38
Q12 Two waves interfere to form fringes in Young s double-slit experiment. Do these two waves come from the same light source? Yes. They must be coherent Q13 If two waves start out in phase with one another, but one wave travels half a wavelength farther than the other before they come together, will the waves be in phase or out of phase when they combine? They will be exactly out of phase 3/30/2016 Physics 214 Spring 2016 39
Q15 When light is reflected from a thin film of oil on a water puddle, the colors we see are produced by interference. What two waves are interfering in this situation? One ray is reflected by the top of the oil, the second from the oil/water interface 3/30/2016 Physics 214 Spring 2016 40
Ch 16 E4 X-rays often have wavelength of about =10-10 m. What is the frequency of such waves? f = c/ = 3 1018 Hz 3/30/2016 Physics 214 Spring 2016 41
Ch 16 E 8 Light of 500 nm is reflected from a thin film of air between two glass plates. The thickness of film: d=1 m=1000 nm. a) How much farther does light reflected from bottom plate travel than that reflected from top plate? b) How many wavelengths of light does this represent? top bottom a) Path difference = 2d = 2000 nm b) 2000 nm/ = 4 4 wavelengths of light 3/30/2016 Physics 214 Spring 2016 42 d
Ch 16 E 12 Diffraction grating has 1000 slits ruled in space of 1.4 cm. What is the distance d between adjacent slits? d d d d 1.4 cm 1000 d = 1.4 cm (see picture) d = 1.4 cm/1000 = 0.014 m/1000 = 1.4 10-5 m 3/30/2016 Physics 214 Spring 2016 43
Ch 16 E14 When passed through a diffraction grating of slit spacing 4 10-6 m, the first-order fringe, for light of a single wavelength, lies 29.0 cm from center of screen 2.0 m away from grating. What is? y = 0.24 m grating X = 2.0 m d = 4 10-6 m d y/x = m, m=1 because this is first order fringe = dy/x = (4 10-6 )(0.29)/2.0 = 5.8 10-7 m = 580 nm 3/30/2016 Physics 214 Spring 2016 44
Ch 16 CP 2 Light of wavelength =600 nm passes through a double slit with d = 0.03 mm. The resulting fringe pattern is observed on a screen 1.2 m from the double slit. a) How far from the center of the screen is the first bright fringe? b) Second bright fringe? c) First dark fringe? d) Sketch a picture of the central seven bright fringes (central fringe & 3 on either side). Clearly indicate the distance from each fringe to the center of the screen. y 1 y z 1 - first dark fringe d = 0.03 mm x = 1.2 m 2 z 1 y 1, y 2 - first and second bright fringes 3/30/2016 Physics 214 Spring 2016 45
Ch 16 CP 2 cont. a) d y/x=n or y n = n x/d = n (1.2)/(0.03 10-3 ) (600 10-9 ) y n =n(0.024)m y 1 = 1(0.024) = 0.024m = 2.4 cm b) y 2 = 2 (0.024) m = 4.8 cm c) z is spacing n th dark fringe dz/x = (n-1/2) or z n = (n-1/2) x/d z n = (n-1/2) (0.024m) z 1 = (1-1/2)(0.024m) = 1.2 cm d) y 1 = 2.4 cm y 2 = 4.8 cm y 3 = 3(0.024m) = 7.2 cm y 3 = + 7.2 cm y 2 = 4.8 cm y 1 = + 2.4 cm y = 0 y -1 = - 2.4 cm y -2 = - 4.8 cm y -3 = - 7.2 + 4-4 3/30/2016 Physics 214 Spring 2016 46
Ch 16 CP 4 Soap film has index of refraction n=1.333. This means that the wavelength of light in the film is reduced by a factor of 1/n. Index of refraction for air = 1.0. a) If light has = 600 nm in air enters the soap film, what is the new? b) Film is 900 nm thick. How many wavelengths farther does light reflected off bottom of film travel than that reflected off top? c) Would you be surprised to find the thickness produces destructive interference for reflected light? 3/30/2016 Physics 214 Spring 2016 47
CH 16 CP4 cont a) f = 600 nm/1.333 = 450 nm air n = 1.0 top bottom film n = 1.333 b) Light reflected from bottom surface travels 2 (900 nm) farther. 2(900 nm)/ f = 4. = 600 nm f = 450 nm In the film this is a 4 full wavelengths. 900 nm c) The above analysis determines that the reflected light should constructively interfere. That is, the light reflected from the bottom surface is in-phase with that reflected from the top. However the actual reflection process introduces a phase change between the two waves so the waves are in fact out of phase. 3/30/2016 Physics 214 Spring 2016 48