ࡱ>  Bw~ bjbj .jjl:::::::3333T5 rl7NNNSoToToT$އ H:oToToToToT\::NS,\\\oT:N:S\oT\\]nyy::|N7 3oTz |\ B0rzFk\"F|\N::::Surface Waves The waves on the ocean are surface waves: They occur at the interface of water and air, extending down into the water and up into the air at the expense of becoming exponentially reduced in amplitude. They are neither transverse nor longitudinal. The water both at and below the surface travels in vertical circles, with exponentially smaller radius as a function of depth. Both empirical measurements and calculations beyond the scope of introductory physics give the propagation speed of water waves as  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v%2C20%2C3D%2C20%2C5C-_mYtF0lH81UBhiHQxV3nRQ.gif" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g%2C3D9%2C2E8%2C5C%2C-85KD4u5B69qWwxOWEFkFpA.gif" \* MERGEFORMATINET is the magnitude of the acceleration due to gravity and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET is the wavenumber. This relationship applies only when the following three conditions hold: The water is several times deeper than the wavelength. The wavelength is large enough that the surface tension of the waves can be neglected. The ratio of wave height to wavelength is small. The restoring force (analogous to the tension in a string) that restores the water surface to flatness is due to gravity, which explains why these waves are often called "gravity waves." Find the speed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET of water waves in terms of the wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET . Express the speed in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET =sqrt(g*lambda/(2*pi)) Find the speed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET of a wave of wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda%2C20%2C-Ns1cpHdjbLoF7N0P5dZWiQ.gif" \* MERGEFORMATINET . Give your answer in meters per second to a precision of two significant figures.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v%2C28%2C5Clambda-w3g1BtDDvOGhYKTPh1AhPw.gif" \* MERGEFORMATINET =3.5 (+/- 2%) m/sFind the period  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T.gif" \* MERGEFORMATINET for a wave of wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET . Express the period in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T.gif" \* MERGEFORMATINET =sqrt(2*pi*lambda/g) On the East Coast of the United States, the National Weather Service frequently reports waves with a period of 4.0 s. Find the wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET and speed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET of these waves. Express your answers numerically as an ordered pair separated by a comma. Give an accuracy of two significant figures.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET =25 (+/- 3%) , 6.25 (+/- 1.5%) m, m/s INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET On the West Coast of the United States, the National Weather Service frequently reports waves (really swells) with a period of 15 s. Find the wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET and speed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET of these waves. Express your answers numerically as an ordered pair separated by a comma. Give an accuracy of two significant figures.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET =350 (+/- 2%) , 23 (+/- 4%) m, m/s[ HYPERLINK "javascript:window.print();" Print] Standard Expression for a Traveling Wave Learning Goal: To understand the standard formula for a sinusoidal traveling wave. One formula for a wave with a y displacement (e.g., of a string) traveling in the x direction is  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C28x%2C2Ct%2C29%2C-5J%2BHx2FYFyaqD6sbr9kzJQ.gif" \* MERGEFORMATINET . All the questions in this problem refer to this formula and to the wave it describes. Which of the following are independent variables?  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET Which of the following are parameters that determine the characteristics of the wave?  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET only  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET What is the phase  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cphi%2C5Cleft%2C28x%2C2Ct%2C5Cright%2C29.gif" \* MERGEFORMATINET of the wave? Express the phase in terms of one or more given variables ( INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET ) and any needed constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cphi%2C5Cleft%2C28x%2C2Ct%2C5Cright%2C29.gif" \* MERGEFORMATINET =k*x-omega*t What is the wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET of the wave? Express the wavelength in terms of one or more given variables ( INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET ) and any needed constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET =2*pi/k What is the period  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T.gif" \* MERGEFORMATINET of this wave? Express the period in terms of one or more given variables ( INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET ) and any needed constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T.gif" \* MERGEFORMATINET =2*pi/omega What is the speed of propagation  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET of this wave? Express the speed of propagation in terms of one or more given variables ( INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET ) and any needed constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET =omega/k [ HYPERLINK "javascript:window.print();" Print] Ant on a Tightrope A large ant is standing on the middle of a circus tightrope that is stretched with tension  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T_%2C7B%2C5Crm%2C20s%2C7D.gif" \* MERGEFORMATINET . The rope has mass per unit length  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/m/%2C5Cmu.gif" \* MERGEFORMATINET . Wanting to shake the ant off the rope, a tightrope walker moves her foot up and down near the end of the tightrope, generating a sinusoidal transverse wave of wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET and amplitude  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET . Assume that the magnitude of the acceleration due to gravity is  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET . What is the minimum wave amplitude  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A_%2C7B%2C5Crm%2C20min%2C7D.gif" \* MERGEFORMATINET such that the ant will become momentarily weightless at some point as the wave passes underneath it? Assume that the mass of the ant is too small to have any effect on the wave propagation. Express the minimum wave amplitude in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T_%2C7B%2C5Crm%2C20s%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/m/%2C5Cmu.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A_%2C7B%2C5Crm%2C20min%2C7D.gif" \* MERGEFORMATINET =mu*(lambda/(2*pi))^2*g/T_s [ HYPERLINK "javascript:window.print();" Print] Breaking Storm Waves Large waves on the deep ocean propagate at the speed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v%2C20%2C3D%2C20%2C5C-_mYtF0lH81UBhiHQxV3nRQ.gif" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET is the magnitude of the acceleration due to gravity and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET is the wavenumber. Seafaring mariners report that in great storms when the average peak-to-peak wave height becomes about 1/7 of the wavelength, the tops of the largest ocean waves can become separated from the rest of the wave. They claim that the wind and the wave's forward velocity cause huge "hunks" of water to tumble down the face of the wave. Some are reportedly large enough to damage or capsize small vessels. The reason these "rogue waves" appear is that the amplitude of the water waves becomes so large that the acceleration of the water in the top of the wave would have to be greater than  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET for the wave to stay in one piece. Because gravity is the only significant vertical force on the water, the acceleration cannot exceed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET , so instead the water at the top of the wave breaks off and is blown down the side of the wave. In this problem, you will compute the ratio of amplitude to wavelength of a rogue wave. The analytic expression for the vertical displacement of the water surface when an ocean wave of amplitude  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET is propagating in the +x direction is  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/z/z%2C28x%2C2Ct%2C29%2C-5pgKn5XR3AxKWwnvG7BJAA.gif" \* MERGEFORMATINET . Find the angular frequency  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET of water waves. Express the angular frequency in terms of the wavenumber  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET and constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET =sqrt(g*k)  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET Find the vertical acceleration  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/a/a_%2C7B%2C5Cmit%2C2-Yb67oY%2Bmn6QESMeJV_bJ5g.gif" \* MERGEFORMATINET of this wave at position  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET and time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET . Express the vertical acceleration in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET , and the independent variables  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/a/a_%2C7B%2C5Cmit%2C2-Yb67oY%2Bmn6QESMeJV_bJ5g.gif" \* MERGEFORMATINET =-A*omega^2*cos(k*x-omega*t)  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET Now find the critical ratio of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET to  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET such that this wave will have a maximum acceleration at the top of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/%2C2Dg.gif" \* MERGEFORMATINET . Then the water at the top would have to accelerate faster than  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET to stay connected with the wave, which is clearly not possible! Express  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A%2C2F%2C5Clambda.gif" \* MERGEFORMATINET as a function of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET and constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/g/g.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET .  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/%2C5Cfrac%2C7BA%2C7D%2C7B%2C5Clambda%2C7D.gif" \* MERGEFORMATINET =1/(2*pi) [ HYPERLINK "javascript:window.print();" Print] Why the Highest Piano Notes Have Short Strings The steel used for piano wire has a breaking (tensile) strength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/p_%2C7B%2C5Cmit%2C20T%2C7D.gif" \* MERGEFORMATINET of about  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/3/3%2C20%2C5Ctimes%2C-_qZMihO9yJdegTYxK1A74Q.gif" \* MERGEFORMATINET and a density  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/r/%2C5Crho.gif" \* MERGEFORMATINET of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/7/7800%2C5C%2C3B%2C7B%2C5Crm%2C20kg%2C2Fm%2C5E3%2C7D.gif" \* MERGEFORMATINET . What is the speed  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET of a wave traveling down such a wire if the wire is stretched to its breaking point? Express the speed of the wave numerically, in meters per second, to the nearest integer.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET =620 (+/- 0.1%) m/sImagine that the wire described in the problem introduction is used for the highest C on a piano ( INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/r/%2C5Crm%2C20C8%2C20-M36zjbXxLQNJUrqjzjwoSg.gif" \* MERGEFORMATINET ). If the wire is in tune when stretched to its breaking point, what must the vibrating length of the wire be? Express the length numerically, in centimeters, using three significant figures.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/L/L.gif" \* MERGEFORMATINET =7.75 (+/- 0.1%) cm[ HYPERLINK "javascript:window.print();" Print] Creating a Standing Wave Learning Goal: To see how two traveling waves of the same frequency create a standing wave. Consider a traveling wave described by the formula  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_1%2C28x%2C2Ct%2C2-GuSknLieH1VCfZckjBWS7g.gif" \* MERGEFORMATINET . This function might represent the lateral displacement of a string, the local electric field, the position of the surface of the water, or many other physical manifestations of waves. Which one of the following statements about this wave is correct?  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling in the +x direction.  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling in the -x direction.  HTMLCONTROL Forms.HTML:Option.1 This wave is oscillating but not traveling.  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling but not oscillating. What is the mathematical expression  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-2qLAGVC5SAUWd4Y5B2qMPw.gif" \* MERGEFORMATINET for a wave of the same amplitude that is traveling in the opposite direction? At time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t%2C3D0.gif" \* MERGEFORMATINET this new wave should have the same displacement as  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-GxHTBV01swISFp6nSSRmrg.gif" \* MERGEFORMATINET , the wave described in the problem introduction.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-2qLAGVC5SAUWd4Y5B2qMPw.gif" \* MERGEFORMATINET = HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A%2C20%2C5Ccos%2C20-urBT4swhkL028QIK16qOSA.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A%2C20%2C5Ccos%2C20-1XHrRwy3UYH0bHb9daf35w.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A%2C20%2C5Csin%2C20-zo9z_Kc_76RUgQky9Se9Bw.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A%2C20%2C5Csin%2C20-IvHpOls7wRCQi_q0nsqr5g.gif" \* MERGEFORMATINET The principle of superposition states that if two functions each separately satisfy the wave equation then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave. Consider the sum of these two waves  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_1%2C28x%2C2Ct%2C29%2C2By_2%2C28x%2C2Ct%2C29.gif" \* MERGEFORMATINET . These waves have been chosen so that their sum can be written as follows:  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-z94Daexqxv3Jjd841R99lw.gif" \* MERGEFORMATINET . This form is significant because  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-MYgHGFYBHtq8QFP7A1UlhQ.gif" \* MERGEFORMATINET --called the envelope--depends only on position, and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-5nsZfVv9slRxVgy9pNXVHQ.gif" \* MERGEFORMATINET depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-MYgHGFYBHtq8QFP7A1UlhQ.gif" \* MERGEFORMATINET . Find  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-MYgHGFYBHtq8QFP7A1UlhQ.gif" \* MERGEFORMATINET and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-5nsZfVv9slRxVgy9pNXVHQ.gif" \* MERGEFORMATINET . Remember that  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-5nsZfVv9slRxVgy9pNXVHQ.gif" \* MERGEFORMATINET should be a trigonometric function of unit amplitude. Express your answers in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET . Separate the two functions with a comma.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-MYgHGFYBHtq8QFP7A1UlhQ.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-5nsZfVv9slRxVgy9pNXVHQ.gif" \* MERGEFORMATINET =2*A*sin(k*x) , cos(omega*t) Which one of the following statements about the superposition wave  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-SgY1WfAgsfSyD0NteYLUdw.gif" \* MERGEFORMATINET is correct?  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling in the +x direction.  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling in the -x direction.  HTMLCONTROL Forms.HTML:Option.1 This wave is oscillating but not traveling.  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling but not oscillating.  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET At the position  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x%2C20%2C3D%2C200.gif" \* MERGEFORMATINET , what is the displacement of the string (assuming that the standing wave  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-SgY1WfAgsfSyD0NteYLUdw.gif" \* MERGEFORMATINET is present)? Express your answer in terms of parameters given in the introduction.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y_%2C7B%2C5Crm%2C20-jF2XsZ6yJMJym0JxE5PE9A.gif" \* MERGEFORMATINET =0 At certain times, the string will be perfectly straight. Find the first time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t%2C3E0.gif" \* MERGEFORMATINET when this is true. Express  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t_%2C7B%2C5Crm%2C201%2C7D.gif" \* MERGEFORMATINET in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET , and necessary constants.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t_%2C7B%2C5Crm%2C201%2C7D.gif" \* MERGEFORMATINET =pi/(2*omega)  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET From the previous part we know that the string is perfectly straight at time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t%2C3D%2C5Cfrac%2C7-nxA_8tCysdyRltELZgNjDQ.gif" \* MERGEFORMATINET . What does this imply about the energy stored in the string? There is no energy stored in the string--the string will remain straight for all subsequent time. Energy will flow into the string, causing the standing wave to form at a later time. Although the string is straight at time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t%2C3D0.gif" \* MERGEFORMATINET , parts of the string have nonzero velocity. Therefore, there is energy stored in the string. The total mechanical energy in the string oscillates but is constant if averaged over a complete cycle.  HTMLCONTROL Forms.HTML:Option.1 a  HTMLCONTROL Forms.HTML:Option.1 b  HTMLCONTROL Forms.HTML:Option.1 c  HTMLCONTROL Forms.HTML:Option.1 d [ HYPERLINK "javascript:window.print();" Print] Nodes of a Standing Wave (Cosine) Learning Goal: To understand the concept of nodes of a standing wave. The nodes of a standing wave are points where the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move). Consider a standing wave, where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y.gif" \* MERGEFORMATINET represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave:  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C28x%2C2Ct%2C29%2C-4trgvtT_kCmaNuHhI1cLnQ.gif" \* MERGEFORMATINET  where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET is the wavenumber,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/o/%2C5Comega.gif" \* MERGEFORMATINET is the angular frequency of the wave, and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t.gif" \* MERGEFORMATINET is time. Which one of the following statements about wave  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C5Cleft%2C28x%2C2Ct%2C5Cright%2C29.gif" \* MERGEFORMATINET is correct?  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling toward +x.  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling toward -x.  HTMLCONTROL Forms.HTML:Option.1 This wave is oscillating but not traveling.  HTMLCONTROL Forms.HTML:Option.1 This wave is traveling but not oscillating. At time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/t/t%2C3D0.gif" \* MERGEFORMATINET , what is the displacement of the string  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C5Cleft%2C28x%2C2C0%2C5Cright%2C29.gif" \* MERGEFORMATINET ? Express your answer in terms of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/A/A.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/k/k.gif" \* MERGEFORMATINET , and other previously introduced quantities.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C5Cleft%2C28x%2C2C0%2C5Cright%2C29.gif" \* MERGEFORMATINET =0 What is the displacement of the string as a function of  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x.gif" \* MERGEFORMATINET at time  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T%2C2F4.gif" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/T/T.gif" \* MERGEFORMATINET is the period of oscillation of the string? Express the displacement in terms of the given variables.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C5Cleft%2C28x%2C-JsI1JIjzFVmWJCu6w0cwzw.gif" \* MERGEFORMATINET =A*cos(k*x) At which three points  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x_%2C7B%2C5Crm%2C201%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x_%2C7B%2C5Crm%2C202%2C7D.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x_%2C7B%2C5Crm%2C203%2C7D.gif" \* MERGEFORMATINET closest to  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x%2C3D0.gif" \* MERGEFORMATINET but with  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x%2C3E0.gif" \* MERGEFORMATINET will the displacement of the string  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/y/y%2C5Cleft%2C28x%2C2Ct%2C5Cright%2C29.gif" \* MERGEFORMATINET be zero for all times? These are the first three nodal points. Express the first three nonzero nodal points as multiples of the wavelength  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET , using constants like  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/p/%2C5Cpi.gif" \* MERGEFORMATINET . List them in increasing order, separated by commas.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x_%2C7B%2C5Crm%2C201%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x_%2C7B%2C5Crm%2C202%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/x/x_%2C7B%2C5Crm%2C203%2C7D.gif" \* MERGEFORMATINET =1/4 , 3/4 , 5/4  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clambda.gif" \* MERGEFORMATINET [ HYPERLINK "javascript:window.print();" Print] The Decibel Scale Learning Goal: To understand the decibel scale. The decibel scale is a logarithmic scale for measuring the sound intensity level. Because the decibel scale is logarithmic, it changes by an additive constant when the intensity as measured in  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/r/%2C5Crm%2C7BW%2C2Fm%2C5E2%2C7D.gif" \* MERGEFORMATINET changes by a multiplicative factor. The number of decibels increases by 10 for a factor of 10 increase in intensity. The general formula for the sound intensity level, in decibels, corresponding to intensity  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I.gif" \* MERGEFORMATINET is  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta%2C20%2C3D-z8G6%2B2zSFNrdlKfzX7eZfw.gif" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I_%2C7B%2C5Crm%2C200%2C7D.gif" \* MERGEFORMATINET is a reference intensity. For sound waves,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I_%2C7B%2C5Crm%2C200%2C7D.gif" \* MERGEFORMATINET is taken to be  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/1/10%2C5E%2C7B%2C2D12-rPaExuqCSXee%2BjyhWdReuQ.gif" \* MERGEFORMATINET . Note that  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/l/%2C5Clog.gif" \* MERGEFORMATINET refers to the logarithm to the base 10. What is the sound intensity level  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET , in decibels, of a sound wave whose intensity is 10 times the reference intensity (i.e.,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I%2C20%2C3D%2C2010I_0.gif" \* MERGEFORMATINET )? Express the sound intensity numerically to the nearest integer.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET =10 dBWhat is the sound intensity level  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET , in decibels, of a sound wave whose intensity is 100 times the reference intensity (i.e.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I%2C20%2C3D%2C20100I_0.gif" \* MERGEFORMATINET )? Express the sound intensity numerically to the nearest integer.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET =20 dBOne often needs to compute the change in decibels corresponding to a change in the physical intensity measured in units of power per unit area. Take  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f.gif" \* MERGEFORMATINET to be the factor of increase of the physical intensity. Calculate the change in decibels ( INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/%2C5CDelta%2C20%2C5Cbeta_%2C7B%2C5Crm%2C202%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/%2C5CDelta%2C20%2C5Cbeta_%2C7B%2C5Crm%2C204%2C7D.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/%2C5CDelta%2C20%2C5Cbeta_%2C7B%2C5Crm%2C208%2C7D.gif" \* MERGEFORMATINET ) corresponding to  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f%2C20%2C3D%2C202.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f%2C20%2C3D%2C204.gif" \* MERGEFORMATINET , and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f%2C20%2C3D%2C208.gif" \* MERGEFORMATINET . Give your answers, separated by commas, to the nearest integer--this will give an accuracy of 20%, which is good enough for sound.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/%2C5CDelta%2C20%2C5Cbeta_%2C7B%2C5Crm%2C202%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/%2C5CDelta%2C20%2C5Cbeta_%2C7B%2C5Crm%2C204%2C7D.gif" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/%2C5CDelta%2C20%2C5Cbeta_%2C7B%2C5Crm%2C208%2C7D.gif" \* MERGEFORMATINET =3 , 6 , 9 dBEar Damage from a Small Firecracker Learning Goal: To understand how to convert between different sound intensity scales and how the decibel intensity of a sound changes with distance. The decibel scale is logarithmic in intensity:  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta%2C20%2C3D-1AMId0wvvPpP6OVrzsJ__w.gif" \* MERGEFORMATINET . In this formula,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I_%2C7B%2C5Crm%2C200%2C7D.gif" \* MERGEFORMATINET is a reference intensity, which, for sound waves, is taken to be  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/1/10%2C5E%2C7B%2C2D12-aJF%2BjEYxcz_u9DWs28Te4g.gif" \* MERGEFORMATINET . This constant must be used to convert a particular physical intensity into a sound intensity level measured in decibels. Once we know the sound intensity level (in decibels) at a certain reference distance from a sound source, the  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/1/1%2C2Fr%2C5E2.gif" \* MERGEFORMATINET decrease of intensity with distance can be accounted for by subtracting the decibel value appropriate to the ratio of the new distance to the reference distance. In this problem you will use the decibel scale to analyze a small firecracker that emits 1200 W of peak power. To avoid confusion, intensities denoted by  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/I/I.gif" \* MERGEFORMATINET are in units of watts per meter squared; intensities denoted by  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET are in units of decibels. What is the peak intensity  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET in decibels at a distance of 1 m from the firecracker? Express  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta.gif" \* MERGEFORMATINET in decibels to the nearest integer.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/b/%2C5Cbeta%2C5Cleft%2C281%2C5Cright%2C29.gif" \* MERGEFORMATINET =140 dB INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET It takes a sound intensity of about 160dB to rupture the human eardrum. How close must the firecracker described in the introduction be to the ear to rupture the eardrum? Express the distance  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/D_%2C7B%2C5Crm%2C20rupture%2C7D.gif" \* MERGEFORMATINET , in meters, to one decimal place.  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/D/D_%2C7B%2C5Crm%2C20rupture%2C7D.gif" \* MERGEFORMATINET =0.1 (+/- 9%) m INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET Will this firecracker produce temporary loss of hearing in someone who sets if off and stands 3 m away from the explosion? Momentary sounds above 120dB produce such loss.  HTMLCONTROL Forms.HTML:Option.1 yes  HTMLCONTROL Forms.HTML:Option.1 no [ HYPERLINK "javascript:window.print();" Print] Doppler Shift Learning Goal: To understand the terms in the Doppler shift formula. The Doppler shift formula gives the frequency  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f_%2C7B%2C5Crm%2C20L%2C7D.gif" \* MERGEFORMATINET at which a listener L hears the sound emitted by a source S at frequency  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f_%2C7B%2C5Crm%2C20S%2C7D.gif" \* MERGEFORMATINET :  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f_%2C7B%2C5Crm%2C20-aGWFngtfDwUdqiz7IBBMXg.gif" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v.gif" \* MERGEFORMATINET is the speed of sound in the medium,  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v_%2C7B%2C5Crm%2C20L%2C7D.gif" \* MERGEFORMATINET is the velocity of the listener, and  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/v/v_%2C7B%2C5Crm%2C20S%2C7D.gif" \* MERGEFORMATINET is the velocity of source. The velocity of the source is positive if the source is ______________.  HTMLCONTROL Forms.HTML:Option.1 traveling in the +x direction  HTMLCONTROL Forms.HTML:Option.1 traveling toward the listener  HTMLCONTROL Forms.HTML:Option.1 traveling away from the listener  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET The velocity of the source is measured with respect to the ________.  HTMLCONTROL Forms.HTML:Option.1 medium (such as air or water)  HTMLCONTROL Forms.HTML:Option.1 listener  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET The velocity of the listener is positive if the listener is _____________.  HTMLCONTROL Forms.HTML:Option.1 traveling in the +x direction  HTMLCONTROL Forms.HTML:Option.1 traveling toward the source  HTMLCONTROL Forms.HTML:Option.1 traveling away from the source  INCLUDEPICTURE "http://session.masteringphysics.com/images/dwIconPreviousFinished.gif" \* MERGEFORMATINET The velocity of the listener is measured with respect to the ________.  HTMLCONTROL Forms.HTML:Option.1 source  HTMLCONTROL Forms.HTML:Option.1 medium Imagine that the source is to the right of the listener, so that the positive reference direction (from the listener to the source) is in the  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/h/%2C2B%2C5Chat%2C7Bx%2C7D.gif" \* MERGEFORMATINET direction. If the listener is stationary, what value does  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f_%2C7B%2C5Crm%2C20L%2C7D.gif" \* MERGEFORMATINET approach as the source's speed approaches the speed of sound moving to the right?  HTMLCONTROL Forms.HTML:Option.1 0  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/%2C5Cfrac%2C7B1%2C7-DdG%2BHg4VesOQjBSzWWyJxg.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/2/2f_%2C7B%2C5Crm%2C20S%2C7D.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1 It approaches infinity. Now, imagine that the source is to the left of the listener, so that the positive reference direction is in the  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/h/%2C2D%2C5Chat%2C7Bx%2C7D.gif" \* MERGEFORMATINET direction. If the source is stationary, what value does  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/f_%2C7B%2C5Crm%2C20L%2C7D.gif" \* MERGEFORMATINET approach as the listener's speed (moving in the  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/h/%2C2B%2C5Chat%2C7Bx%2C7D.gif" \* MERGEFORMATINET direction) approaches the speed of sound?  HTMLCONTROL Forms.HTML:Option.1 0  HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/f/%2C5Cfrac%2C7B1%2C7-DdG%2BHg4VesOQjBSzWWyJxg.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1  INCLUDEPICTURE "http://session.masteringphysics.com/latexgif/2/2f_%2C7B%2C5Crm%2C20S%2C7D.gif" \* MERGEFORMATINET   HTMLCONTROL Forms.HTML:Option.1 It approaches infinity. In this last case, imagine that the listener is stationary and the source is moving toward the listener at the speed of sound. (Note that it is irrelevant whether the source is moving to the right or to the left.) 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