普通物理 General Physics 17 - Longitudinal Waves 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Outline 17-1 What Is Physics? 17-2 Sound Waves 17-3 The Speed of Sound 17-4 Traveling Sound Waves 17-5 Interference 17-6 Intensity and Sound Level 17-7 Sources of Musical Sound 17-8 Beats 17-9 The Doppler Effect 17-10 Supersonic Speeds, Shock Waves 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-1 What Is Physics? Some acoustic engineers are concerned with improving the acoustics of cathedrals and concert halls, with reducing noise near freeways and road construction, and with reproducing music by speaker systems. Some medical researchers are concerned with how noises produced by the heart and lungs can signal a medical problem in a patient. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-2 Sound Waves Sound waves are mechanical longitudinal waves that propagate in solids, liquids, and gases. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-2 Sound Waves The locus of the points of a sound wave that has the same displacement is called a “wavefront”. Lines perpendicular to the wavefronts are called “rays”. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-3 The Speed of Sound Bulk modulus: SI unit: the Pascal ∆V/V is the fractional change in volume produced by a change in pressure ∆p Note: The negative sign denotes the decrease in volume when ∆p is positive. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-3 The Speed of Sound The speed of sound: (m/s) Note 1: Bulk modulus is smaller for more compressible media. Note 2: Denser materials (higher ρ) have lower speed of sound. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-1 When a sound pulse, as from a handclap, is produced at the foot of the stairs at the Mayan pyramid shown in the chapter’s opening photograph, the sound waves reflect from the steps in succession, the closest (lowest) one first and the farthest (highest) one last. The depth and height of the steps are d = 0.263 m, and the speed of sound is 343 m/s. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-1 The paths taken by the sound waves to and from the steps near the bottom of the stairs are approximately horizontal. The slanted paths taken sound waves to and from the steps near the top are approximately 45° to the horizontal. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-1 At what frequency fbot do the echo pulses arrive at the listener from the bottom step? At what frequency ftop do they arrive from the top steps a short time later? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-1 Key Idea: The frequency f at which the pulses return to the listener is the inverse of the time ∆t between successive pulses. The time interval ∆t required by sound to travel a given distance L is related to the speed of sound v by v = L/∆t. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-1 Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-4 Traveling Sound Waves The two amplitudes are connected by the equation : Note: The displacement and the pressure variation have a phase difference of . 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-2 The maximum pressure amplitude ∆pm that the human ear can tolerate in loud sounds is about 28 Pa (which is very much less than the normal air pressure of about 105 Pa). What is the displacement amplitude sm for such a sound in air of density ρ= 1.21 kg/m3, at a frequency of 1000 Hz and a speed of 343 m/s? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-2 Key Idea: The displacement amplitude sm of a sound wave is related to the pressure amplitude ∆pm of the wave according ∆pm = (vρω)sm. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-2 Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-5 Interference At time t the phase of sound wave arriving from S at point P is: S1: S2: The two waves at P have a phase difference: is known as the path length difference λ is the wavelength of the two waves. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-5 Interference 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-5 Interference Constructive interference: Destructive interference: ∆L equal to an integral multiple of λ→constructive interference ∆L equal to a half-integral multiple of λ→destructive 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-3 (a) Two point sources S1 and S2, which are in phase and separated by distance D = 1.5λ, emit identical sound waves of wavelength . 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-3 (a) What is the path length difference of the waves from S1 and S2 at point P1, which lies on the perpendicular bisector of distance D, at a distance greater than D from the sources? What type of interference occurs at P1? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-3 (b) What are the path length difference and type of interference at point P2 ? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-3 (c) A circle with a radius much greater than D, centered on the midpoint between sources S1 and S2.What is the number of points N around this circle at which the interference is fully constructive? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-3 (a) (b) (c) Solutions: →full constructive →full destructive 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-6 Intensity and Sound Level We define at the wave intensity I the ratio P/A → SI unit: W/m2 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-6 Intensity and Sound Level 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-6 Intensity and Sound Level The sound intensity at a distance γ from S is : The intensity of a sound wave for a point sources is proportional to : 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-6 Intensity and Sound Level The figure indicates three small patches 1, 2, and 3 that lie on the surfaces of two imaginary spheres; the spheres are centered on an isotropic point source S of sound. The rates at which energy is transmitted through the three patches by the sound waves are equal. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-6 Intensity and Sound Level Rank the patches according to (a) the intensity of the sound on them and (b) their area, greatest first. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-6 Intensity and Sound Level The decibel: β is defined in such a way as to mimic the response of the human ear. β is expressed in decibels (dB) Note 1: For I = I0 , we have: β = 10 log1= 0. Note 2: β increases by 10 decibels every time I increase by a factor of 10. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-4 (a) An electric spark jumps along a straight line of length L = 10 m, emitting a pulse of sound that travels radially outward from the spark. (The spark is said to be a line source of sound.) The power of the emission is Ps = 1.6 × 104 W. What is the intensity I of the sound when it reaches a distance r = 12 m from the spark? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-4 (b) At what time rate Pd is sound energy intercepted by an acoustic detector of area Ad = 2.0 cm2, aimed at the spark and located a distance r = 12 m from the spark? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-4 (a) Key Idea: The intensity I at the cylindrical surface is the ratio P/A, where P is the time rate at which sound energy passes through the surface and A is the surface area. The principle of conservation of energy applies to the sound energy. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-4 (a) This means that the rate P at which energy is transferred through the cylinder must equal the rate Ps at which energy is emitted by the source. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-4 (b) Key Idea: Applying the first of part (a), we know that the intensity of sound at the detector is the ratio of the energy transfer rate Pd there to the detector’s area Ad 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-4 (a) (b) Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-5 Many veteran rockers suffer from acute hearing damage because of the high sound levels they endured for years while playing music near loudspeakers or listening to music on headphones. Some, like Ted Nugent, can no longer hear in a damaged ear. Others, like Peter Townshend of the Who, have a continuous ringing sensation (tinnitus). 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-5 Recently, many rockers, such as Lars Ulrich of Metallica, began wearing special earplugs to protect their hearing during performances. If an earplug decreases the sound level of the sound waves by 20 dB, what is the ratio of the final intensity If of the waves to their initial intensity Ii ? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-5 Key Idea: For both the final and initial waves, the sound level is related to the intensity by the definition of sound level in β= (10 dB) log (I/I0). 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-5 Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-7 Sources of Musical Sound The standing wave of Fig. (b) is known as the fundamental mode or first harmonic of the tube. Note: Antinodes in the displacement amplitude correspond to nodes in the pressure amplitude. This is because sm and Δpm are out of phase. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-7 Sources of Musical Sound Standing waves in tubes open at both ends: The integer n is known as the harmonic number. The corresponding frequencies: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-7 Sources of Musical Sound Standing waves in tubes open at one end and closed at the other: The corresponding frequencies: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-6 (a) Weak background noises from a room set up the fundamental standing wave in a cardboard tube of length L = 67.0 cm with two open ends. Assume that the speed of sound in the air within the tube is 343 m/s. What frequency do you hear from the tube? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-6 (b) If you jam your ear against one end of the tube, what fundamental frequency do you hear from the tube? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-6 (a) Key Idea: with both pipe ends open, we have a symmetric situation in which the standing wave has an antinode at each end of the tube. The standing wave pattern (in string wave style). 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-6 (b) Key Idea: With your ear effectively closing one end of the tube, we have an asymmetric situation — an antinode still exists at the open end, but a node is now at the other (closed) end. The standing wave pattern is the top one. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-6 (a) (b) Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-8 Beats Using: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-8 Beats 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-8 Beats where and Thus the angular frequency of the beats: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-8 Beats The frequency of the beats: Tbeat T' 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-7 Most birds vocalize by using only one side of their two-sided vocal organ, called the syrinx. Emperor penguins, however, vocalize by using both sides simultaneously. Each side sets up acoustic standing waves in the bird’s throat and mouth, much like in a pipe with two open ends. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-7 Suppose that the frequency of the first harmonic produced by side A is fA1 = 432 Hz and the frequency of the first harmonic produced by side B is fB1 = 371 Hz. What is the beat frequency between those two first- harmonic frequencies and between the two second-harmonic frequencies? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-7 Key Idea: The beat frequency between two frequencies is their difference, as given by fbeat = f1 - f2. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-7 Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-9 The Doppler Effect The Doppler effect: Consider the source and the detector of sound waves shown in the figure. We assume that the frequency of the source is equal to f. The frequency is given by the equation: Here vs and vD are the speeds of the source and detector with respect to air. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-9 The Doppler Effect Detector Moving, Source Stationary: D moves to the left: D moves away from the source: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-9 The Doppler Effect Source Moving, Detector Stationary: In the direction opposite that taken by S: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 17-9 The Doppler Effect 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-8 Bats navigate and search out prey by emitting, and then detecting reflections of, ultrasonic waves, which are sound waves with frequencies greater than can be heard by a human. Suppose a bat emits ultrasound at frequency fbe = 82.52 kHz while flying with velocity = (9.00 m/s) as it chases a moth that flies with velocity = (8.00 m/s) . 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-8 What frequency fmd does the moth detect? What frequency fbd does the bat detect in the returning echo from the moth? 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-8 Key Idea: The frequency is shifted by the relative motion of the bat and moth. Because they move along a single axis, the shifted frequency is given by the general Doppler effect. Motion toward tends to shift the frequency up, and motion away tends to shift the frequency down. 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 Example 17-8 Solutions: 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

17-10 Supersonic Speeds, Shock Waves θ: Mach cone angle 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授

普通物理講義-17/國立彰化師範大學物理系/郭艷光教授 End of chapter 17! 2018/11/29 普通物理講義-17/國立彰化師範大學物理系/郭艷光教授