{"id":910,"date":"2017-10-27T16:31:01","date_gmt":"2017-10-27T16:31:01","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/speed-of-sound-frequency-and-wavelength\/"},"modified":"2017-11-08T03:25:46","modified_gmt":"2017-11-08T03:25:46","slug":"speed-of-sound-frequency-and-wavelength","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/speed-of-sound-frequency-and-wavelength\/","title":{"raw":"Speed of Sound, Frequency, and Wavelength","rendered":"Speed of Sound, Frequency, and Wavelength"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define pitch.<\/li>\n<li>Describe the relationship between the speed of sound, its frequency, and its wavelength.<\/li>\n<li>Describe the effects on the speed of sound as it travels through various media.<\/li>\n<li>Describe the effects of temperature on the speed of sound.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">When a firework explodes, the light energy is perceived before the sound energy. Sound travels more slowly than light does. (credit: Dominic Alves, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2588880\" data-alt=\"A photograph of a fireworks display in the sky.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_01a.jpg\" data-media-type=\"image\/png\" alt=\"A photograph of a fireworks display in the sky.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3013063\">Sound, like all waves, travels at a certain speed and has the properties of frequency and wavelength. You can observe direct evidence of the speed of sound while watching a fireworks display. The flash of an explosion is seen well before its sound is heard, implying both that sound travels at a finite speed and that it is much slower than light. You can also directly sense the frequency of a sound. Perception of frequency is called <span data-type=\"term\" id=\"import-auto-id1588064\">pitch<\/span>. The wavelength of sound is not directly sensed, but indirect evidence is found in the correlation of the size of musical instruments with their pitch. Small instruments, such as a piccolo, typically make high-pitch sounds, while large instruments, such as a tuba, typically make low-pitch sounds. High pitch means small wavelength, and the size of a musical instrument is directly related to the wavelengths of sound it produces. So a small instrument creates short-wavelength sounds. Similar arguments hold that a large instrument creates long-wavelength sounds.<\/p>\n<p id=\"import-auto-id3154439\">The relationship of the speed of sound, its frequency, and wavelength is the same as for all waves:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{v}_{\\text{w}}=\\mathrm{f\\lambda ,}[\/latex]<\/div>\n<p id=\"import-auto-id2401892\">where [latex]{v}_{w}[\/latex] is the speed of sound, [latex]f[\/latex] is its frequency, and <em data-effect=\"italics\">[latex]\\lambda [\/latex]<\/em> is its wavelength. The wavelength of a sound is the distance between adjacent identical parts of a wave\u2014for example, between adjacent compressions as illustrated in <a href=\"#import-auto-id1538012\" class=\"autogenerated-content\">(Figure)<\/a>. The frequency is the same as that of the source and is the number of waves that pass a point per unit time.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1538012\">\n<div class=\"bc-figcaption figcaption\">A sound wave emanates from a source vibrating at a frequency [latex]f[\/latex], propagates at<br>\n[latex]{v}_{\\text{w}}[\/latex], and has a wavelength [latex]\\lambda [\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2655120\" data-alt=\"A picture of a vibrating tuning fork is shown. The sound wave compressions and rarefactions are shown to emanate from the fork on both the sides as semicircular arcs of alternate bold and dotted lines. The wavelength is marked as the distance between two successive bold arcs. The frequency of the vibrations is shown as f and velocity of the wave represented by v sub w.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"A picture of a vibrating tuning fork is shown. The sound wave compressions and rarefactions are shown to emanate from the fork on both the sides as semicircular arcs of alternate bold and dotted lines. The wavelength is marked as the distance between two successive bold arcs. The frequency of the vibrations is shown as f and velocity of the wave represented by v sub w.\" width=\"350\"><\/span><\/p><\/div>\n<p><a href=\"#import-auto-id3177545\" class=\"autogenerated-content\">(Figure)<\/a> makes it apparent that the speed of sound varies greatly in different media. The speed of sound in a medium is determined by a combination of the medium\u2019s rigidity (or compressibility in gases) and its density. The more rigid (or less compressible) the medium, the faster the speed of sound. This observation is analogous to the fact that the frequency of a simple harmonic motion is directly proportional to the stiffness of the oscillating object. The greater the density of a medium, the slower the speed of sound. This observation is analogous to the fact that the frequency of a simple harmonic motion is inversely proportional to the mass of the oscillating object. The speed of sound in air is low, because air is compressible. Because liquids and solids are relatively rigid and very difficult to compress, the speed of sound in such media is generally greater than in gases.<\/p>\n<table id=\"import-auto-id3177545\" summary=\"Two-column table listing various media for sound in the first column and their speeds of sound in the second column. The list of media is divided into three groups: gases, liquids, and solids.\">\n<caption><span data-type=\"title\">Speed of Sound in Various Media<\/span><\/caption>\n<thead>\n<tr>\n<th>Medium<\/th>\n<th><em data-effect=\"italics\">v<\/em><sub>w<\/sub>(m\/s)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td colspan=\"2\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Gases at [latex]\\text{0\u00baC}[\/latex]<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>Air<\/td>\n<td>331<\/td>\n<\/tr>\n<tr>\n<td>Carbon dioxide<\/td>\n<td>259<\/td>\n<\/tr>\n<tr>\n<td>Oxygen<\/td>\n<td>316<\/td>\n<\/tr>\n<tr>\n<td>Helium<\/td>\n<td>965<\/td>\n<\/tr>\n<tr>\n<td>Hydrogen<\/td>\n<td>1290<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Liquids at [latex]\\text{20\u00baC}[\/latex]<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>Ethanol<\/td>\n<td>1160<\/td>\n<\/tr>\n<tr>\n<td>Mercury<\/td>\n<td>1450<\/td>\n<\/tr>\n<tr>\n<td>Water, fresh<\/td>\n<td>1480<\/td>\n<\/tr>\n<tr>\n<td>Sea water<\/td>\n<td>1540<\/td>\n<\/tr>\n<tr>\n<td>Human tissue<\/td>\n<td>1540<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Solids (longitudinal or bulk)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>Vulcanized rubber<\/td>\n<td>54<\/td>\n<\/tr>\n<tr>\n<td>Polyethylene<\/td>\n<td>920<\/td>\n<\/tr>\n<tr>\n<td>Marble<\/td>\n<td>3810<\/td>\n<\/tr>\n<tr>\n<td>Glass, Pyrex<\/td>\n<td>5640<\/td>\n<\/tr>\n<tr>\n<td>Lead<\/td>\n<td>1960<\/td>\n<\/tr>\n<tr>\n<td>Aluminum<\/td>\n<td>5120<\/td>\n<\/tr>\n<tr>\n<td>Steel<\/td>\n<td>5960<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id1441562\">Earthquakes, essentially sound waves in Earth\u2019s crust, are an interesting example of how the speed of sound depends on the rigidity of the medium. Earthquakes have both longitudinal and transverse components, and these travel at different speeds. The bulk modulus of granite is greater than its shear modulus. For that reason, the speed of longitudinal or pressure waves (P-waves) in earthquakes in granite is significantly higher than the speed of transverse or shear waves (S-waves). Both components of earthquakes travel slower in less rigid material, such as sediments. P-waves have speeds of 4 to 7 km\/s, and S-waves correspondingly range in speed from 2 to 5 km\/s, both being faster in more rigid material. The P-wave gets progressively farther ahead of the S-wave as they travel through Earth\u2019s crust. The time between the P- and S-waves is routinely used to determine the distance to their source, the epicenter of the earthquake.<\/p>\n<p id=\"import-auto-id3246097\">The speed of sound is affected by temperature in a given medium. For air at sea level, the speed of sound is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-330\">[latex]{v}_{\\text{w}}=\\left(\\text{331}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}\\right)\\sqrt{\\frac{T}{\\text{273}\\phantom{\\rule{0.25em}{0ex}}\\text{K}}},[\/latex]<\/div>\n<p id=\"import-auto-id1431577\">where the temperature (denoted as  [latex]T[\/latex]) is in units of kelvin.  The speed of sound in gases is related to the average speed of particles in the gas, [latex]{v}_{\\text{rms}}[\/latex], and that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{v}_{\\text{rms}}=\\sqrt{\\frac{\\text{3}\\mathrm{kT}}{m}},[\/latex]<\/div>\n<p id=\"import-auto-id2447377\">where  [latex]k[\/latex] is the Boltzmann constant ([latex]1.38\u00d7{\\text{10}}^{\\text{\u221223}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/K}[\/latex]) and  [latex]m[\/latex] is the mass of each (identical) particle in the gas. So, it is reasonable that the speed of sound in air and other gases should depend on the square root of temperature. While not negligible, this is not a strong dependence. At<br>\n[latex]\\text{0\u00baC}[\/latex], the speed of sound is 331 m\/s, whereas at<br>\n[latex]\\text{20.0\u00baC}[\/latex] it is 343 m\/s, less than a 4% increase. <a href=\"#import-auto-id1578485\" class=\"autogenerated-content\">(Figure)<\/a> shows a use of the speed of sound by a bat to sense distances. Echoes are also used in medical imaging.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">A bat uses sound echoes to find its way about and to catch prey. The time for the echo to return is directly proportional to the distance.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1931095\" data-alt=\"The picture is of a bat trying to catch its prey an insect using sound echoes. The incident sound and sound reflected from the bat are shown as semicircular arcs.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The picture is of a bat trying to catch its prey an insect using sound echoes. The incident sound and sound reflected from the bat are shown as semicircular arcs.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2444602\">One of the more important properties of sound is that its speed is nearly independent of frequency. This independence is certainly true in open air for sounds in the audible range of 20 to 20,000 Hz. If this independence were not true, you would certainly notice it for music played by a marching band in a football stadium, for example. Suppose that high-frequency sounds traveled faster\u2014then the farther you were from the band, the more the sound from the low-pitch instruments would lag that from the high-pitch ones. But the music from all instruments arrives in cadence independent of distance, and so all frequencies must travel at nearly the same speed. Recall that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{v}_{\\text{w}}=\\mathrm{f\\lambda .}[\/latex]<\/div>\n<p>In a given medium under fixed conditions,<br>\n[latex]{v}_{\\text{w}}[\/latex]<br>\nis constant, so that there is a relationship between [latex]f[\/latex] and <em data-effect=\"italics\">[latex]\\lambda [\/latex]<\/em>; the higher the frequency, the smaller the wavelength. See <a href=\"#import-auto-id1593942\" class=\"autogenerated-content\">(Figure)<\/a> and consider the following example.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1593942\">\n<div class=\"bc-figcaption figcaption\">Because they travel at the same speed in a given medium, low-frequency sounds must have a greater wavelength than high-frequency sounds. Here, the lower-frequency sounds are emitted by the large speaker, called a woofer, while the higher-frequency sounds are emitted by the small speaker, called a tweeter.<\/div>\n<p><span data-type=\"media\" data-alt=\"Picture of a speaker having a woofer and a tweeter. High frequency sound coming out of the woofer shown as small circles closely spaced. Low frequency sound coming out of tweeter are shown as larger circles distantly spaced.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Picture of a speaker having a woofer and a tweeter. High frequency sound coming out of the woofer shown as small circles closely spaced. Low frequency sound coming out of tweeter are shown as larger circles distantly spaced.\" width=\"200\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2001560\">\n<div data-type=\"title\" class=\"title\">Calculating Wavelengths: What Are the Wavelengths of Audible Sounds?<\/div>\n<p id=\"import-auto-id1477887\">Calculate the wavelengths of sounds at the extremes of the audible range, 20 and 20,000 Hz, in<br>\n[latex]\\text{30.0\u00baC}[\/latex] air. (Assume that the frequency values are accurate to two significant figures.)<\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3396364\">To find wavelength from frequency, we can use [latex]{v}_{\\text{w}}=\\mathrm{f\\lambda }[\/latex].<\/p>\n<p id=\"import-auto-id2381606\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id2409442\" data-number-style=\"arabic\">\n<li>Identify knowns. The value for [latex]{v}_{\\text{w}}[\/latex], is given by\n<div data-type=\"equation\" class=\"equation\">[latex]{v}_{\\text{w}}=\\left(\\text{331}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}\\right)\\sqrt{\\frac{T}{\\text{273}\\phantom{\\rule{0.25em}{0ex}}\\text{K}}}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1815176\">Convert the temperature into kelvin and then enter the temperature into the equation\n<div data-type=\"equation\" class=\"equation\" id=\"eip-575\">[latex]{v}_{\\text{w}}=\\left(\\text{331}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}\\right)\\sqrt{\\frac{\\text{303 K}}{\\text{273}\\phantom{\\rule{0.25em}{0ex}}\\text{K}}}=\\text{348}\\text{.}7\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2682073\">Solve the relationship between speed and wavelength for <em data-effect=\"italics\">[latex]\\lambda [\/latex]<\/em>:\n<div data-type=\"equation\" class=\"equation\">[latex]\\lambda =\\frac{{v}_{w}}{f}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id963388\">Enter the speed and the minimum frequency to give the maximum wavelength:\n<div data-type=\"equation\" class=\"equation\">[latex]{\\lambda }_{\\text{max}}=\\frac{\\text{348}\\text{.}7\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}}{\\text{20 Hz}}=\\text{17}\\phantom{\\rule{0.25em}{0ex}}\\text{m}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1816494\">Enter the speed and the maximum frequency to give the minimum wavelength:\n<div data-type=\"equation\" class=\"equation\">[latex]{\\lambda }_{\\text{min}}=\\frac{\\text{348}\\text{.}7\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}}{\\text{20}\\text{,}\\text{000 Hz}}=0\\text{.}\\text{017}\\phantom{\\rule{0.25em}{0ex}}\\text{m}=1\\text{.}\\text{7 cm}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2991817\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id3250053\">Because the product of [latex]f[\/latex] multiplied by <em data-effect=\"italics\">[latex]\\lambda [\/latex]<\/em> equals a constant, the smaller [latex]f[\/latex] is, the larger <em data-effect=\"italics\">[latex]\\lambda [\/latex]<\/em> must be, and vice versa.<\/p>\n<\/div>\n<p id=\"import-auto-id3024041\">The speed of sound can change when sound travels from one medium to another. However, the frequency usually remains the same because it is like a driven oscillation and has the frequency of the original source. If <\/p>\n[latex]{v}_{\\text{w}}[\/latex]\n<p>changes and [latex]f[\/latex] remains the same, then the wavelength <\/p>\n[latex]\\lambda [\/latex]\n<p>must change. That is, because <\/p>\n<p>[latex]{v}_{\\text{w}}=\\mathrm{f\\lambda }[\/latex], <\/p>\n<p>the higher the speed of a sound, the greater its wavelength for a given frequency.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2383870\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Investigation\u2014Voice as a Sound Wave<\/div>\n<p id=\"import-auto-id1427729\">Suspend a sheet of paper so that the top edge of the paper is fixed and the bottom edge is free to move. You could tape the top edge of the paper to the edge of a table. Gently blow near the edge of the bottom of the sheet and note how the sheet moves. Speak softly and then louder such that the sounds hit the edge of the bottom of the paper, and note how the sheet moves. Explain the effects.<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1437960\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2722326\">\n<p id=\"fs-id1622890\">\nImagine you observe two fireworks explode. You hear the explosion of one as soon as you see it. However, you see the other firework for several milliseconds before you hear the explosion. Explain why this is so.\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1447062\" data-print-placement=\"here\">\n<p id=\"fs-id1421271\">\nSound and light both travel at definite speeds. The speed of sound is slower than the speed of light. The first firework is probably very close by, so the speed difference is not noticeable. The second firework is farther away, so the light arrives at your eyes noticeably sooner than the sound wave arrives at your ears.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1526208\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2737920\">\n<p id=\"fs-id2205803\">\nYou observe two musical instruments that you cannot identify. One plays high-pitch sounds and the other plays low-pitch sounds. How could you determine which is which without hearing either of them play?\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1122763\" data-print-placement=\"here\">\n<p id=\"fs-id2195046\">\nCompare their sizes. High-pitch instruments are generally smaller than low-pitch instruments because they generate a smaller wavelength.\n<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1931189\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<p id=\"import-auto-id2600539\">The relationship of the speed of sound [latex]{v}_{w}[\/latex], its frequency [latex]f[\/latex], and its wavelength <em data-effect=\"italics\">[latex]\\lambda [\/latex]<\/em> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{v}_{\\text{w}}=\\mathrm{f\\lambda },[\/latex]<\/div>\n<p id=\"import-auto-id3021013\">which is the same relationship given for all waves.<\/p>\n<p>In air, the speed of sound is related to air temperature <em data-effect=\"italics\">[latex]T[\/latex]<\/em> by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{v}_{\\text{w}}=\\left(\\text{331}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}\\right)\\sqrt{\\frac{T}{\\text{273}\\phantom{\\rule{0.25em}{0ex}}\\text{K}}}.[\/latex]<\/div>\n<p id=\"import-auto-id2962616\">[latex]{v}_{\\text{w}}[\/latex] is the same for all frequencies and wavelengths.<\/p>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1386961\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1375143\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1954277\">\n<p id=\"import-auto-id1410916\">How do sound vibrations of atoms differ from thermal motion?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3008692\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2992665\">\n<p id=\"import-auto-id3065090\">When sound passes from one medium to another where its propagation speed is different, does its frequency or wavelength change? Explain your answer briefly.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2591417\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3451972\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2931366\">\n<p id=\"import-auto-id2399660\">When poked by a spear, an operatic soprano lets out a 1200-Hz shriek. What is its wavelength if the speed of sound is 345 m\/s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1328160\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3175746\">0.288 m<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3110312\">What frequency sound has a 0.10-m wavelength when the speed of sound is 340 m\/s?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3043771\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1945477\">\n<p id=\"import-auto-id1870708\">Calculate the speed of sound on a day when a 1500 Hz frequency has a wavelength of 0.221 m.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3028783\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id760991\">332 m\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2443672\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1549327\">(a) What is the speed of sound in a medium where a 100-kHz frequency produces a 5.96-cm wavelength? (b) Which substance in <a href=\"#import-auto-id3177545\" class=\"autogenerated-content\">(Figure)<\/a> is this likely to be?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1587092\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2681777\">\n<p id=\"import-auto-id2051396\">Show that the speed of sound in<br>\n[latex]\\text{20.0\u00baC}[\/latex]<br>\n air is 343 m\/s, as claimed in the text.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3397939\" data-element-type=\"problems-exercises\">\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3089366\">[latex]\\begin{array}{lll}{v}_{\\text{w}}&amp; =&amp; \\left(\\text{331 m\/s}\\right)\\sqrt{\\frac{T}{\\text{273 K}}}=\\left(\\text{331 m\/s}\\right)\\sqrt{\\frac{\\text{293 K}}{\\text{273 K}}}\\\\ &amp; =&amp; \\text{343 m\/s}\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2666191\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3116303\">\n<p id=\"import-auto-id3062563\">Air temperature in the Sahara Desert can reach<br>\n[latex]\\text{56.0\u00baC}[\/latex]<br>\n (about<br>\n[latex]\\text{134\u00baF}[\/latex]). What is the speed of sound in air at that temperature?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2032287\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3055563\">\n<p>Dolphins make sounds in air and water. What is the ratio of the wavelength of a sound in air to its wavelength in seawater? Assume air temperature is<br>\n[latex]\\text{20.0\u00baC}[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1448362\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id2399752\">0.223<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1560729\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2597921\">A sonar echo returns to a submarine 1.20 s after being emitted. What is the distance to the object creating the echo? (Assume that the submarine is in the ocean, not in fresh water.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1986379\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3234387\">\n<p id=\"import-auto-id2667613\">(a) If a submarine\u2019s sonar can measure echo times with a precision of 0.0100 s, what is the smallest difference in distances it can detect? (Assume that the submarine is in the ocean, not in fresh water.)<\/p>\n<p id=\"eip-id1517636\">(b) Discuss the limits this time resolution imposes on the ability of the sonar system to detect the size and shape of the object creating the echo.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2663384\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3027585\">(a) 7.70 m<\/p>\n<p id=\"import-auto-id2669394\">(b) This means that sonar is good for spotting and locating large objects, but it isn\u2019t able to resolve smaller objects, or detect the detailed shapes of objects. Objects like ships or large pieces of airplanes can be found by sonar, while smaller pieces must be found by other means.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1381531\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1849553\">\n<p id=\"import-auto-id2032403\">A physicist at a fireworks display times the lag between seeing an explosion and hearing its sound, and finds it to be 0.400 s. (a) How far away is the explosion if air temperature is [latex]\\text{24.0\u00baC}[\/latex] and if you neglect the time taken for light to reach the physicist? (b) Calculate the distance to the explosion taking the speed of light into account. Note that this distance is negligibly greater.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1386574\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2403356\">\n<p id=\"import-auto-id3455423\">Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See <a href=\"#import-auto-id1578485\" class=\"autogenerated-content\">(Figure)<\/a>.) (a) Calculate the echo times for temperatures of<br>\n[latex]\\text{5.00\u00baC}[\/latex] and<br>\n[latex]\\text{35.0\u00baC}[\/latex]. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty and whether it could cause difficulties for the bat. (In practice, the bat continues to use sound as it closes in, eliminating most of any difficulties imposed by this and other effects, such as motion of the prey.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2903564\">\n<p id=\"eip-id2732260\">(a) 18.0 ms, 17.1 ms<\/p>\n<p id=\"eip-id1592361\">(b) 5.00%<\/p>\n<p id=\"eip-id2908620\">(c) This uncertainty could definitely cause difficulties for the bat, if it didn\u2019t continue to use sound as it closed in on its prey. A 5% uncertainty could be the difference between catching the prey around the neck or around the chest, which means that it could miss grabbing its prey.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2442201\">\n<dt>pitch<\/dt>\n<dd id=\"fs-id1414345\">the perception of the frequency of a sound<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define pitch.<\/li>\n<li>Describe the relationship between the speed of sound, its frequency, and its wavelength.<\/li>\n<li>Describe the effects on the speed of sound as it travels through various media.<\/li>\n<li>Describe the effects of temperature on the speed of sound.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">When a firework explodes, the light energy is perceived before the sound energy. Sound travels more slowly than light does. (credit: Dominic Alves, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2588880\" data-alt=\"A photograph of a fireworks display in the sky.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_01a.jpg\" data-media-type=\"image\/png\" alt=\"A photograph of a fireworks display in the sky.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3013063\">Sound, like all waves, travels at a certain speed and has the properties of frequency and wavelength. You can observe direct evidence of the speed of sound while watching a fireworks display. The flash of an explosion is seen well before its sound is heard, implying both that sound travels at a finite speed and that it is much slower than light. You can also directly sense the frequency of a sound. Perception of frequency is called <span data-type=\"term\" id=\"import-auto-id1588064\">pitch<\/span>. The wavelength of sound is not directly sensed, but indirect evidence is found in the correlation of the size of musical instruments with their pitch. Small instruments, such as a piccolo, typically make high-pitch sounds, while large instruments, such as a tuba, typically make low-pitch sounds. High pitch means small wavelength, and the size of a musical instrument is directly related to the wavelengths of sound it produces. So a small instrument creates short-wavelength sounds. Similar arguments hold that a large instrument creates long-wavelength sounds.<\/p>\n<p id=\"import-auto-id3154439\">The relationship of the speed of sound, its frequency, and wavelength is the same as for all waves:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6478c427739d0b24c836d38c48b8d5b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#102;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2401892\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7e6b5b5e8c87e834f914a77abdfcb817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/> is the speed of sound, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is its frequency, and <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em> is its wavelength. The wavelength of a sound is the distance between adjacent identical parts of a wave\u2014for example, between adjacent compressions as illustrated in <a href=\"#import-auto-id1538012\" class=\"autogenerated-content\">(Figure)<\/a>. The frequency is the same as that of the source and is the number of waves that pass a point per unit time.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1538012\">\n<div class=\"bc-figcaption figcaption\">A sound wave emanates from a source vibrating at a frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/>, propagates at<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcac2ff523ad63d66e25cbbd723009cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/>, and has a wavelength <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2655120\" data-alt=\"A picture of a vibrating tuning fork is shown. The sound wave compressions and rarefactions are shown to emanate from the fork on both the sides as semicircular arcs of alternate bold and dotted lines. The wavelength is marked as the distance between two successive bold arcs. The frequency of the vibrations is shown as f and velocity of the wave represented by v sub w.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"A picture of a vibrating tuning fork is shown. The sound wave compressions and rarefactions are shown to emanate from the fork on both the sides as semicircular arcs of alternate bold and dotted lines. The wavelength is marked as the distance between two successive bold arcs. The frequency of the vibrations is shown as f and velocity of the wave represented by v sub w.\" width=\"350\" \/><\/span><\/p>\n<\/div>\n<p><a href=\"#import-auto-id3177545\" class=\"autogenerated-content\">(Figure)<\/a> makes it apparent that the speed of sound varies greatly in different media. The speed of sound in a medium is determined by a combination of the medium\u2019s rigidity (or compressibility in gases) and its density. The more rigid (or less compressible) the medium, the faster the speed of sound. This observation is analogous to the fact that the frequency of a simple harmonic motion is directly proportional to the stiffness of the oscillating object. The greater the density of a medium, the slower the speed of sound. This observation is analogous to the fact that the frequency of a simple harmonic motion is inversely proportional to the mass of the oscillating object. The speed of sound in air is low, because air is compressible. Because liquids and solids are relatively rigid and very difficult to compress, the speed of sound in such media is generally greater than in gases.<\/p>\n<table id=\"import-auto-id3177545\" summary=\"Two-column table listing various media for sound in the first column and their speeds of sound in the second column. The list of media is divided into three groups: gases, liquids, and solids.\">\n<caption><span data-type=\"title\">Speed of Sound in Various Media<\/span><\/caption>\n<thead>\n<tr>\n<th>Medium<\/th>\n<th><em data-effect=\"italics\">v<\/em><sub>w<\/sub>(m\/s)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td colspan=\"2\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Gases at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e7ab0b9e576b6ece65f08badc1eb428f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>Air<\/td>\n<td>331<\/td>\n<\/tr>\n<tr>\n<td>Carbon dioxide<\/td>\n<td>259<\/td>\n<\/tr>\n<tr>\n<td>Oxygen<\/td>\n<td>316<\/td>\n<\/tr>\n<tr>\n<td>Helium<\/td>\n<td>965<\/td>\n<\/tr>\n<tr>\n<td>Hydrogen<\/td>\n<td>1290<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Liquids at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c5021f856100c9924f694724df9862a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>Ethanol<\/td>\n<td>1160<\/td>\n<\/tr>\n<tr>\n<td>Mercury<\/td>\n<td>1450<\/td>\n<\/tr>\n<tr>\n<td>Water, fresh<\/td>\n<td>1480<\/td>\n<\/tr>\n<tr>\n<td>Sea water<\/td>\n<td>1540<\/td>\n<\/tr>\n<tr>\n<td>Human tissue<\/td>\n<td>1540<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><strong data-effect=\"bold\"><em data-effect=\"italics\">Solids (longitudinal or bulk)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>Vulcanized rubber<\/td>\n<td>54<\/td>\n<\/tr>\n<tr>\n<td>Polyethylene<\/td>\n<td>920<\/td>\n<\/tr>\n<tr>\n<td>Marble<\/td>\n<td>3810<\/td>\n<\/tr>\n<tr>\n<td>Glass, Pyrex<\/td>\n<td>5640<\/td>\n<\/tr>\n<tr>\n<td>Lead<\/td>\n<td>1960<\/td>\n<\/tr>\n<tr>\n<td>Aluminum<\/td>\n<td>5120<\/td>\n<\/tr>\n<tr>\n<td>Steel<\/td>\n<td>5960<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id1441562\">Earthquakes, essentially sound waves in Earth\u2019s crust, are an interesting example of how the speed of sound depends on the rigidity of the medium. Earthquakes have both longitudinal and transverse components, and these travel at different speeds. The bulk modulus of granite is greater than its shear modulus. For that reason, the speed of longitudinal or pressure waves (P-waves) in earthquakes in granite is significantly higher than the speed of transverse or shear waves (S-waves). Both components of earthquakes travel slower in less rigid material, such as sediments. P-waves have speeds of 4 to 7 km\/s, and S-waves correspondingly range in speed from 2 to 5 km\/s, both being faster in more rigid material. The P-wave gets progressively farther ahead of the S-wave as they travel through Earth\u2019s crust. The time between the P- and S-waves is routinely used to determine the distance to their source, the epicenter of the earthquake.<\/p>\n<p id=\"import-auto-id3246097\">The speed of sound is affected by temperature in a given medium. For air at sea level, the speed of sound is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-330\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2363d4feff8f479140dd92d62e59909a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#84;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"183\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id1431577\">where the temperature (denoted as  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>) is in units of kelvin.  The speed of sound in gases is related to the average speed of particles in the gas, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-457599ab17c21c27d86746a542c7f8c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#109;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: -4px;\" \/>, and that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-30cbe688185811c17058857756767573_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#109;&#115;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#107;&#84;&#125;&#125;&#123;&#109;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"105\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id2447377\">where  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the Boltzmann constant (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18a0d640eab5774815b215b183b609b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#51;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/>) and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is the mass of each (identical) particle in the gas. So, it is reasonable that the speed of sound in air and other gases should depend on the square root of temperature. While not negligible, this is not a strong dependence. At<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e7ab0b9e576b6ece65f08badc1eb428f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>, the speed of sound is 331 m\/s, whereas at<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab5d8026ce8291cd12b58cb16b7b20d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/> it is 343 m\/s, less than a 4% increase. <a href=\"#import-auto-id1578485\" class=\"autogenerated-content\">(Figure)<\/a> shows a use of the speed of sound by a bat to sense distances. Echoes are also used in medical imaging.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">A bat uses sound echoes to find its way about and to catch prey. The time for the echo to return is directly proportional to the distance.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1931095\" data-alt=\"The picture is of a bat trying to catch its prey an insect using sound echoes. The incident sound and sound reflected from the bat are shown as semicircular arcs.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The picture is of a bat trying to catch its prey an insect using sound echoes. The incident sound and sound reflected from the bat are shown as semicircular arcs.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2444602\">One of the more important properties of sound is that its speed is nearly independent of frequency. This independence is certainly true in open air for sounds in the audible range of 20 to 20,000 Hz. If this independence were not true, you would certainly notice it for music played by a marching band in a football stadium, for example. Suppose that high-frequency sounds traveled faster\u2014then the farther you were from the band, the more the sound from the low-pitch instruments would lag that from the high-pitch ones. But the music from all instruments arrives in cadence independent of distance, and so all frequencies must travel at nearly the same speed. Recall that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce9c63634e2203fe87ae694753ae3419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#102;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"64\" style=\"vertical-align: -3px;\" \/><\/div>\n<p>In a given medium under fixed conditions,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcac2ff523ad63d66e25cbbd723009cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/><br \/>\nis constant, so that there is a relationship between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em>; the higher the frequency, the smaller the wavelength. See <a href=\"#import-auto-id1593942\" class=\"autogenerated-content\">(Figure)<\/a> and consider the following example.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1593942\">\n<div class=\"bc-figcaption figcaption\">Because they travel at the same speed in a given medium, low-frequency sounds must have a greater wavelength than high-frequency sounds. Here, the lower-frequency sounds are emitted by the large speaker, called a woofer, while the higher-frequency sounds are emitted by the small speaker, called a tweeter.<\/div>\n<p><span data-type=\"media\" data-alt=\"Picture of a speaker having a woofer and a tweeter. High frequency sound coming out of the woofer shown as small circles closely spaced. Low frequency sound coming out of tweeter are shown as larger circles distantly spaced.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_18_02_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Picture of a speaker having a woofer and a tweeter. High frequency sound coming out of the woofer shown as small circles closely spaced. Low frequency sound coming out of tweeter are shown as larger circles distantly spaced.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2001560\">\n<div data-type=\"title\" class=\"title\">Calculating Wavelengths: What Are the Wavelengths of Audible Sounds?<\/div>\n<p id=\"import-auto-id1477887\">Calculate the wavelengths of sounds at the extremes of the audible range, 20 and 20,000 Hz, in<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e63bba1ea3da2cd8931cbe3ebc54107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/> air. (Assume that the frequency values are accurate to two significant figures.)<\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3396364\">To find wavelength from frequency, we can use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2132c6dcf886c46b01c05f312109e33f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#102;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"import-auto-id2381606\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id2409442\" data-number-style=\"arabic\">\n<li>Identify knowns. The value for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcac2ff523ad63d66e25cbbd723009cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/>, is given by\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-90556ad7858ff9156b9e1837cbf64ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#84;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"183\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1815176\">Convert the temperature into kelvin and then enter the temperature into the equation\n<div data-type=\"equation\" class=\"equation\" id=\"eip-575\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c4667085026b02426e9eb09f75194d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#51;&#32;&#75;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"282\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2682073\">Solve the relationship between speed and wavelength for <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em>:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62e61e67aee61ef60b145eeb00156a4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#119;&#125;&#125;&#123;&#102;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id963388\">Enter the speed and the minimum frequency to give the maximum wavelength:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1604930c8cfb182150ead91c0f71a48c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#120;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#72;&#122;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"188\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1816494\">Enter the speed and the maximum frequency to give the minimum wavelength:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43d68c3cec5ded0182a511411a7cea7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&#72;&#122;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#32;&#99;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"283\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2991817\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id3250053\">Because the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> multiplied by <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em> equals a constant, the smaller <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is, the larger <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em> must be, and vice versa.<\/p>\n<\/div>\n<p id=\"import-auto-id3024041\">The speed of sound can change when sound travels from one medium to another. However, the frequency usually remains the same because it is like a driven oscillation and has the frequency of the original source. If <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcac2ff523ad63d66e25cbbd723009cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/><\/p>\n<p>changes and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> remains the same, then the wavelength <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<p>must change. That is, because <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2132c6dcf886c46b01c05f312109e33f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#102;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -3px;\" \/>, <\/p>\n<p>the higher the speed of a sound, the greater its wavelength for a given frequency.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2383870\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Investigation\u2014Voice as a Sound Wave<\/div>\n<p id=\"import-auto-id1427729\">Suspend a sheet of paper so that the top edge of the paper is fixed and the bottom edge is free to move. You could tape the top edge of the paper to the edge of a table. Gently blow near the edge of the bottom of the sheet and note how the sheet moves. Speak softly and then louder such that the sounds hit the edge of the bottom of the paper, and note how the sheet moves. Explain the effects.<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1437960\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2722326\">\n<p id=\"fs-id1622890\">\nImagine you observe two fireworks explode. You hear the explosion of one as soon as you see it. However, you see the other firework for several milliseconds before you hear the explosion. Explain why this is so.\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1447062\" data-print-placement=\"here\">\n<p id=\"fs-id1421271\">\nSound and light both travel at definite speeds. The speed of sound is slower than the speed of light. The first firework is probably very close by, so the speed difference is not noticeable. The second firework is farther away, so the light arrives at your eyes noticeably sooner than the sound wave arrives at your ears.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1526208\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2737920\">\n<p id=\"fs-id2205803\">\nYou observe two musical instruments that you cannot identify. One plays high-pitch sounds and the other plays low-pitch sounds. How could you determine which is which without hearing either of them play?\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1122763\" data-print-placement=\"here\">\n<p id=\"fs-id2195046\">\nCompare their sizes. High-pitch instruments are generally smaller than low-pitch instruments because they generate a smaller wavelength.\n<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1931189\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<p id=\"import-auto-id2600539\">The relationship of the speed of sound <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7e6b5b5e8c87e834f914a77abdfcb817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/>, its frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/>, and its wavelength <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fc522cac61be5297a3a2cdf1b94a0b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#102;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id3021013\">which is the same relationship given for all waves.<\/p>\n<p>In air, the speed of sound is related to air temperature <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em> by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-90556ad7858ff9156b9e1837cbf64ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#84;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"183\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id2962616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcac2ff523ad63d66e25cbbd723009cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/> is the same for all frequencies and wavelengths.<\/p>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1386961\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1375143\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1954277\">\n<p id=\"import-auto-id1410916\">How do sound vibrations of atoms differ from thermal motion?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3008692\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2992665\">\n<p id=\"import-auto-id3065090\">When sound passes from one medium to another where its propagation speed is different, does its frequency or wavelength change? Explain your answer briefly.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2591417\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3451972\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2931366\">\n<p id=\"import-auto-id2399660\">When poked by a spear, an operatic soprano lets out a 1200-Hz shriek. What is its wavelength if the speed of sound is 345 m\/s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1328160\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3175746\">0.288 m<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3110312\">What frequency sound has a 0.10-m wavelength when the speed of sound is 340 m\/s?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3043771\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1945477\">\n<p id=\"import-auto-id1870708\">Calculate the speed of sound on a day when a 1500 Hz frequency has a wavelength of 0.221 m.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3028783\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id760991\">332 m\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2443672\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1549327\">(a) What is the speed of sound in a medium where a 100-kHz frequency produces a 5.96-cm wavelength? (b) Which substance in <a href=\"#import-auto-id3177545\" class=\"autogenerated-content\">(Figure)<\/a> is this likely to be?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1587092\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2681777\">\n<p id=\"import-auto-id2051396\">Show that the speed of sound in<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab5d8026ce8291cd12b58cb16b7b20d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/><br \/>\n air is 343 m\/s, as claimed in the text.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3397939\" data-element-type=\"problems-exercises\">\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3089366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b4b7af61f261adf00a130de09fa7da1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#49;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#84;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#32;&#75;&#125;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#49;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#51;&#32;&#75;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#32;&#75;&#125;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#51;&#32;&#109;&#47;&#115;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"364\" style=\"vertical-align: -20px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2666191\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3116303\">\n<p id=\"import-auto-id3062563\">Air temperature in the Sahara Desert can reach<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a879eb4e743e44b6ce35ac2fec8ce5b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#54;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><br \/>\n (about<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-feaa94c5d9c6b7819a2f56ba87d48a3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#52;&ordm;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: -1px;\" \/>). What is the speed of sound in air at that temperature?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2032287\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3055563\">\n<p>Dolphins make sounds in air and water. What is the ratio of the wavelength of a sound in air to its wavelength in seawater? Assume air temperature is<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab5d8026ce8291cd12b58cb16b7b20d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1448362\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id2399752\">0.223<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1560729\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2597921\">A sonar echo returns to a submarine 1.20 s after being emitted. What is the distance to the object creating the echo? (Assume that the submarine is in the ocean, not in fresh water.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1986379\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3234387\">\n<p id=\"import-auto-id2667613\">(a) If a submarine\u2019s sonar can measure echo times with a precision of 0.0100 s, what is the smallest difference in distances it can detect? (Assume that the submarine is in the ocean, not in fresh water.)<\/p>\n<p id=\"eip-id1517636\">(b) Discuss the limits this time resolution imposes on the ability of the sonar system to detect the size and shape of the object creating the echo.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2663384\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id3027585\">(a) 7.70 m<\/p>\n<p id=\"import-auto-id2669394\">(b) This means that sonar is good for spotting and locating large objects, but it isn\u2019t able to resolve smaller objects, or detect the detailed shapes of objects. Objects like ships or large pieces of airplanes can be found by sonar, while smaller pieces must be found by other means.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1381531\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1849553\">\n<p id=\"import-auto-id2032403\">A physicist at a fireworks display times the lag between seeing an explosion and hearing its sound, and finds it to be 0.400 s. (a) How far away is the explosion if air temperature is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d82bc99aab1fefb46e96b6e5b2c47c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: -1px;\" \/> and if you neglect the time taken for light to reach the physicist? (b) Calculate the distance to the explosion taking the speed of light into account. Note that this distance is negligibly greater.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1386574\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2403356\">\n<p id=\"import-auto-id3455423\">Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See <a href=\"#import-auto-id1578485\" class=\"autogenerated-content\">(Figure)<\/a>.) (a) Calculate the echo times for temperatures of<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-abe2d73a0e00484fcda63d46c1d78dcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4499761d4033c4801e916601304a70e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/>. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty and whether it could cause difficulties for the bat. (In practice, the bat continues to use sound as it closes in, eliminating most of any difficulties imposed by this and other effects, such as motion of the prey.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2903564\">\n<p id=\"eip-id2732260\">(a) 18.0 ms, 17.1 ms<\/p>\n<p id=\"eip-id1592361\">(b) 5.00%<\/p>\n<p id=\"eip-id2908620\">(c) This uncertainty could definitely cause difficulties for the bat, if it didn\u2019t continue to use sound as it closed in on its prey. A 5% uncertainty could be the difference between catching the prey around the neck or around the chest, which means that it could miss grabbing its prey.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2442201\">\n<dt>pitch<\/dt>\n<dd id=\"fs-id1414345\">the perception of the frequency of a sound<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-910","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":895,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/910","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/910\/revisions"}],"predecessor-version":[{"id":911,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/910\/revisions\/911"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/895"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/910\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=910"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=910"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=910"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}