Physics 2A (Fall 2012) Chapters 15: Traeling Waes and Sound and 16: Superposition and Standing Waes There is only one corner of the unierse that you can be certain of improing, and that s your own. Aldous Huxley We either make ourseles miserable, or we make ourseles strong. The amount of work is the same Carlos Castaneda Too many people oeralue what they are not and underalue what they are. Margaret B. Runbeck Reading: sections 15.1 15.4, 15.7 Outline: introduction to waes (PowerPoint) Mechanical waes EM waes and Matter Waes transerse and longitudinal waes graphical description of waes amplitude, waelength, period, frequency, and wae speed traeling waes waes on a string sound waes sound and light waes (PowerPoint) the Doppler effect moing source, moing obserer, and general case shock waes Reading: sections 16.1 16.3, 16.6 16.7 Outline: the principle of superposition constructie and destructie interference interference of waes from two sources standing waes standing waes on a string beats (PowerPoint)
Problem Soling Many of the problems inoling waes on a string deal with the relationships = λ f = λ /T, where is the wae speed, λ is the waelength, f is the frequency, and T is the period. Typical problems might gie you the waelength and frequency, then ask for the wae speed, or might gie you the wae speed and period, then ask for the waelength. Sometimes the quantities are gien by describing the motion. For example, a problem might tell you that the string at one point takes a certain time to go from its equilibrium position to maximum displacement. This, of course, is one-fourth the period. In other problems, you may be asked how long it takes a particle on a string to moe through a total distance. You must then recognize that a particle on the string moes through a distance a 4A (where A is the amplitude) during a time equal to the period. Some problems deal with the wae speed. For waes on a string, the fundamental equation is = Ts / μ, where T s is the tension in the string and μ is the linear mass density. The tension may not be gien directly but, if the problem asks for the wae speed, sufficient information will be gien to calculate it. o 1± Nearly all Doppler shift problems can be soled using f = f0 s 1 where is speed of sound, S is the speed of the source, o is the speed of the obserer, f 0 is the frequency of the source, and f is the frequency detected by the obserer. The upper sign in the numerator refers to a situation in which the obserer is moing toward the source; the lower sign refers to a situation in which the obserer is moing away from the source. The upper sign in the denominator refers to a situation in which the source is moing toward the obserer; the lower sign refers to a situation in which the source is moing away from the obserer. Some problems deal with the production of beats by two sound waes with nearly the same frequency. You may be gien the frequency f 1 of one of the waes and the beat frequency f beat, then asked for the frequency f 2 of the other wae. Since fbeat = f 1 f 2, it is gien by f 2= f 1± fbeat. You require more information to determine which sign to use in this equation. One way to gie this information is to tell you what happens to the beat frequency if f 1 is increased (or decreased). If the beat frequency increases when f 1 increases, then f 1 must be greater than f 2 and f 2 = f 1 - f beat. If the beat frequency decreases, then f 1 must be less than f 2 and f 2 = f 1 + f beat. Some problems deal with standing waes on a string. If you are told the distance between successie nodes or successie antinodes, double the distance to find the waelength. If you are told the distance between a node and a neighboring antinode, multiply it by 4 to find the waelength. If a standing wae is generated in a string with both ends fixed, the wae pattern must hae a node at each end of the string. This means the length L of the string and the waelength λ of the traeling waes must be related by L = nλ/2, where n is an integer.
SUMMARY The goal of Chapter 15 has been to learn the basic properties of traeling waes. GENERAL PRINCIPLES The Wae Model This model is based on the idea of a traeling wae, which is an organized disturbance traeling at a well-defined wae speed. In transerse waes the particles of the medium moe perpendicular to the direction in which the wae traels. In longitudinal waes the particles of the medium moe parallel to the direction in which the wae traels. A wae transfers energy, but there is no material or substance transferred. Mechanical waes require a material medium. The speed of the wae is a property of the medium, not the wae. The speed does not depend on the size or shape of the wae. For a wae on a string, the string is the medium. T s Asound wae is a wae of compressions and rarefactions of a medium such as air. m 5 m L sound In a gas: sound = A grt M Electromagnetic waes are waes of the electromagnetic field. They do not require a medium. All electromagnetic waes trael at the same speed in a acuum, c = 3.00 * 10 8 m/s. string = A T s m IMPORTANT CONCEPTS Graphical representation of waes A snapshot graph is a picture of a wae at one instant in time. For a periodic wae, the waelength l is the distance between crests. Fixed t: y A 0 2A l r x Mathematical representation of waes Sinusoidal waes are produced by a source moing with simple harmonic motion. The equation for a sinusoidal wae is a function of position and time: y1x, t2 = A cosa2pa x l t T bb The intensity of a wae is the ratio of the power to the area: I = P A For a spherical wae the power decreases with the surface area of the spherical wae fronts: A history graph is a graph of the displacement of one point in a medium ersus time. For a periodic wae, the period T is the time between crests. Fixed x: y A 0 2A T t +: wae traels to left -: wae traels to right For sinusoidal and other periodic waes: T = 1 f = fl P source l r I = P source 4pr 2 l Wae fronts APPLICATIONS The loudness of a sound is gien by the sound intensity leel. This is a logarithmic function of intensity and is in units of decibels. The usual reference leel is the quietest sound that can be heard: I 0 = 1.0 * 10-12 W/m 2 The Doppler effect is a shift in frequency when there is relatie motion of a wae source (frequency f 0, wae speed ) and an obserer. Moing source, stationary obserer: Receding source: ƒ - = ƒ 0 1 + s / 0 1 2 3 r s Approaching source: f + = f 0 1 - s / The sound intensity leel in db is computed relatie to this alue: b = 110 db2 log 10 a I I 0 b A sound at the reference leel corresponds to 0 db. Moing obserer, stationary source: Approaching the source: Moing away from the source: f + = a1 + o b f 0 f - = a1 - o b f 0 Reflection from a moing object: o For o V, ƒ = 2ƒ 0 When an object moes faster than the wae speed in a medium, a shock wae is formed.
SUMMARY The goal of Chapter 16 has been to use the idea of superposition to understand the phenomena of interference and standing waes. GENERAL PRINCIPLES Principle of Superposition The displacement of a medium when more than one wae is present is the sum of the displacements due to each indiidual wae. Interference In general, the superposition of two or more waes into a single wae is called interference. Constructie interference occurs when Destructie interference occurs when crests crests are aligned with crests and troughs with are aligned with troughs. We say the waes troughs. We say the waes are in phase. It are out of phase. It occurs when the pathlength occurs when the path-length difference d is difference d is a whole number of a whole number of waelengths. waelengths plus half a waelength. Dp 2 1 Dd=l Dp 1 Dd= 2 l 0 x 0 x IMPORTANT CONCEPTS Standing Waes Two identical traeling waes moing in opposite directions create a standing wae. Antinodes A standing wae on a string has a node at each end. Possible modes: m 1 m 2 A standing sound wae in a tube can hae different boundary conditions: open-open, closed-closed, or open-closed. Open-open f m = ma Dp in tube 2L b m 5 1 m = 1, 2, 3, Á Nodes 1 Node spacing is l. The boundary conditions determine which standing-wae frequencies and waelengths are allowed. The allowed standing waes are modes of the system. 2 m 3 L l m = 2L m f m = ma 2L b = mf 1 m = 1, 2, 3, Á Closed-closed f m = ma 2L b m = 1, 2, 3, Á Open-closed f m = ma 4L b m = 1, 3, 5, Á L Dp in tube m 5 1 Dp in tube m 5 1 APPLICATIONS Beats (loud-soft-loud-soft modulations of intensity) are produced when two waes of slightly different frequencies are superimposed. Standing waes are multiples of a fundamental frequency, the frequency of the lowest mode. The higher modes are the higher harmonics. Fundamental frequency t For sound, the fundamental frequency determines the perceied pitch; the higher harmonics determine the tone quality. Relatie intensity Higher harmonics Loud Soft Loud Soft Loud Soft f beat = ƒ f 1 - f 2 ƒ Loud Our ocal cords create a range of harmonics. The mix of higher harmonics is changed by our ocal tract to create different owel sounds. 262 524 786 1048 1572 2096 f (Hz)
Questions and Example Problems from Chapters 15 and 16 Question 1 Two cars, one behind the other, are traeling in the same direction at the same speed. Does either drier hear the other s horn at a frequency that is different from that heard when both cars are at rest? Question 2 Refer to the figure below. As you walk along a line that is perpendicular to the line between the speakers and passes through the oerlap point, you do not obsere the loudness to change from loud to faint to loud. Howeer, as you walk along a line through the oerlap point and parallel to the line between the speakers, you do obsere the loudness to alternate between faint and loud. Explain why your obserations are different in the two cases.
Problem 1 A person lying on an air mattress in the ocean rises and falls through one complete cycle eery fie seconds. The crests of the wae causing the motion are 20.0 m apart. Determine (a) the frequency and (b) the speed of the wae. Problem 2 The linear density of the A string on a iolin is 7.8 10-4 kg/m. A wae on the string has a frequency of 440 Hz and a waelength of 65 cm. What is the tension in the string?
Problem 3 The middle C string on a piano is under a tension of 944 N. The period and waelength of a wae on this string are 3.82 ms and 1.26 m, respectiely. Find the linear density of the string. Problem 4 Two submarines are underwater and approaching each other head-on. Sub A has a speed of 12 m/s and sub B has a speed of 8 m/s. Sub A sends out a 1550 Hz sonar wae that traels at a speed of 1522 m/s. (a) What is the frequency detected by sub B? (b) Part of the sonar wae is reflected from B and returns to A. What frequency does A detect for this reflected wae?
Problem 5 The security alarm on a parked car goes off and produces a frequency of 960 Hz. The speed of sound is 343 m/s. As you drie toward this parked car, pass it, and drie away, you obsere the frequency to change by 95 Hz. At what speed are you driing? Problem 6 Two loudspeakers emit sound waes along the x-axis. The sound has maximum intensity when the speakers are 20 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 30 cm. (a) What is the waelength of the sound? (b) If the distance between the speakers continues to increase, at what separation will the sound intensity again be a maximum?
Problem 7 A pair of in-phase stereo speakers are placed next to each other, 0.60 m apart. You stand directly in front of one of the speakers, 1.0 m from the speaker. What is the lowest frequency that will produce constructie interference at your location? Problem 8 Two out-of-tune flutes play the same note. One produces a tone that has a frequency of 262 Hz, while the other produces 266 Hz. When a tuning fork is sounded together with the 262-Hz tone, a beat frequency of 1 Hz is produced. When the same tuning fork is sounded together with the 266 Hz tone, a beat frequency of 3 Hz is produced. What is the frequency of the tuning fork?
Problem 9 A string of length 0.28 m is fixed at both ends. The string is plucked and a standing wae is set up that is ibrating at its second harmonic. The traeling waes that make up the standing waes hae a speed of 140 m/s. What is the frequency of ibration? Problem 10 On a cello, the string with the largest linear density (1.56 10-2 kg/m) is the C string. The string produces a fundamental frequency of 65.4 Hz and has a length of 0.800 m between the two fixed ends. Find the tension in the string. Problem 11 The figure shows a standing wae oscillating at 100 Hz on a string. What is the wae speed?