普通物理 General Physics 16 - Transverse 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-16 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 16-1 What Is Physics? 16-2 Types of Waves 16-3 Transverse and Longitudinal Waves 16-4 Wavelength and Frequency 16-5 The Speed of a Traveling Wave 16-6 Wave Speed on a Stretched String 16-7 Energy and Power of a Wave Traveling Along a String 16-8 The Wave Equation 16-9 The Principle of Superposition for Waves 16-10 Interference of Waves 16-11 Phasors 16-12 Standing Waves 16-13 Standing Waves and Resonance 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-1 What Is Physics? A wave is defined as a disturbance that is self-sustained and propagates in space with a constant speed. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-2 Types of Waves Waves can be classified in the following three categories: Mechanical waves. These involve motions that are governed by Newton’s laws and can exist only within a material medium such as air, water, rock,… etc. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-2 Types of Waves Electromagnetic waves. These waves involve propagating disturbances in the electric and magnetic field governed by Maxwell’s equations. They do not require a material medium in which to propagate but they travel through vacuum. Matter waves. All microscopic particles such as electrons, protons, neutrons, atoms etc have a wave associated with them governed by Schrodinger’s equation. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-3 Transverse and Longitudinal Waves If the disturbance associated with a particular wave is perpendicular to the wave propagation velocity, this wave is called “transverse”. A wave in which the associated disturbance is parallel to the wave propagation velocity is known as a “longitudinal wave”. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-1 Seismic waves are waves that travel either through Earth’s interior or along the ground. Seismology stations are set up mainly to record seismic waves generated by earthquakes, but they also record seismic waves generated by any large release of energy near Earth’s surface, such as an explosion. As the seismic waves travel past a station, they oscillate a recording pen and the pen traces out a graph. Figure 16-3a is one of the graphs created by seismic waves from the sinking of the submarine Kursk. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-1 The first oscillations of the pen are marked with an arrow and were of small amplitude. Much stronger oscillations began about 134 s later. From this, analysis concludes that the first seismic waves were generated by an onboard explosion, possibly a torpedo that failed to launch when fired. The explosion presumably breached the hull, started a fire, and sank the submarine. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-1 The later, much stronger seismic waves were generated after the submarine was sunk and were possibly generated when the fire caused several of the powerful missiles on board to explode simultaneously. These stronger waves arrived at seismology stations as pulses separated by a time interval Δt of about 0.11 s. To what depth D did the submarine sink? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-1 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-4 Wavelength and Frequency Such a wave which is described by a sine (or a cosine) function is known as “harmonic wave”. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-4 Wavelength and Frequency 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-5 The Speed of a Traveling Wave 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-2 (a) A wave traveling along a string is described by y(x, t) = 0.00327 sin( 72.1x-2.72t ) in which the numerical constants in SI units ( 0.00327 m,72.1 rad/m, and 2.72 rad/s ). What is the amplitude of this wave? Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-2 (b) What are the wavelength period and frequency of this wave? Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-2 (c)、(d) (c) What is the velocity of this wave? (d) What is the displacement y at x=22.5cm and t=18.9s? Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-3 (a) In Example 16-2d, we showed that at t=18.9s the transverse displacement y of the element of the string at x=0.255 m due to the equation of y(x, t)=0.00327 sin(72.1x-2.72t) is 1.92mm. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-3 (a) What is u, the transverse velocity of the same element of the string, at that time? (This velocity, which is associated with the transverse oscillation of an element of the string, is in the y direction. Do not confuse it with v, the constant velocity at which the wave form travels along the x axis.) 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-3 (a) Key idea: The transverse velocity u is the rate at which the displacement y of the element is changing. In general, that displacement is given by y(x, t) = ym sin(kx - ωt). For an element at a certain location x, we find the rate of change of y by taking the derivative of y(x, t) = ym sin(kx - ωt) with respect to t while treating x as a constant. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-3 (a) Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-3 (b) What is the transverse acceleration ay of the same element at that time? Key idea: the transverse acceleration ay is the rate at which the transverse velocity of the element is changing. From , again treating x as a constant but allowing t to vary, we find . Comparison with y(x, t) = ym sin(kx - ωt), we can write this as . 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-3 (b) Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-6 Wave Speed on a Stretched String A string whose linear mass density is μ. The tension on the string is equal to τ. Dimensional Analysis: The goal here is to combine μ (dimension ML-1) τ (dimension MLT-2) in such a way as to generate v (dimension LT-1). C is a dimensionless constant that cannot be determined with dimensional analysis. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-6 Wave Speed on a Stretched String 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-4 Figure shows two strings have been tied together with a knot and then stretched between two rigid supports. The strings have linear densities kg/m and kg/m. Their lengths are L1=3.0m and L2=2.0m, and string 1 is under a tension of 400 N. Simultaneously, on each string a pulse is sent from the rigid support end, toward the knot. Which pulse reaches the knot first? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-4 Key idea: The time t taken by a pulse to travel a length L is t = L/v, where v is the constant speed of the pulse. The speed of a pulse on a stretched string depends on the string’s tension τ and linear density μ, and is given by . Because the two strings are stretched together, they must both be under the same tension τ (= 400 N). 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-4 Solutions: Wave 2 reaches first 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-7 Energy and Power of a Wave Traveling Along a String 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-7 Energy and Power of a Wave Traveling Along a String 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-5 A stretched string has linear density μ=525 g/m and is under tension τ=45 N. We send a sinusoidal wave with frequency f=120 Hz and amplitude ym=8.5 mm along the string. At what average rate does the wave transport energy? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-5 Key idea: The average rate of energy transport is the average power Pavg as given by: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-5 Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-5 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-8 The Wave Equation θ2 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-8 The Wave Equation 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-8 The Wave Equation 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-9 The Principle of Superposition for Waves Overlapping waves algebraically add to produce a result wave (or net wave). Note: Overlapping waves do not in any way alter the travel of each other. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-10 Interference of Waves 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-10 Interference of Waves Constructive interference It has its maximum value if ψ= 0. This phenomenon is known as fully constructive interference. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-10 Interference of Waves Destructive interference It has its minimum value if ψ= π. This phenomenon is known as fully destructive interference. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-10 Interference of Waves Intermediate interference The amplitude of two interefering waves is given by: When interference is neither fully constructive nor fully destructive, it is called: intermediate interference. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-10 Interference of Waves An example is given in the figure for . In this case, y’m=ym. The displacement of the resulting wave is: Note: sometimes the phase difference is expressed as a difference in wave lengh λ. In this case, remember that: 2πradians 1λ 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-10 Interference of Waves 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-6 (a) Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude ym of each wave is 9.8 mm, and the phase difference ψ between them is 100°. What is the amplitude ym of the resultant wave due to the interference, and what is the type of this interference? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-6 (a) Key idea: These are identical sinusoidal waves traveling in the same direction along a string, so they interfere to produce a sinusoidal traveling wave. Because they are identical, they have the same amplitude. Thus, the amplitude y’m of the resultant wave is given by: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-6 (a) Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-6 (b) What phase difference, in radians and wavelengths, will give the resultant wave an amplitude of 4.9 mm? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-6 (b) Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-11 Phasors 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-7 Two sinusoidal waves y1(x, t) and y2(x, t) have the same wavelength and travel together in the same direction along a string. Their amplitudes are ym1=4.0 mm and ym2=3.0 mm, and their phase constants are 0 and π/3, respectively. What are the amplitude y’m and phase constant of the resultant wave? Write the resultant wave in the form of y(x, t) = y’m sin(kx - ωt + β). 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-7 Key idea: The two waves have a number of properties in common: Because they travel along the same string, they must have the same speed v, as set by the tension and linear density of the string according to . With the same wavelength λ, they must have the same angular wave number k (= 2π/λ). Also, with the same wave number k and speed v, they must have the same angular frequency ω (= kv). 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-7 Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-7 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-12 Standing Waves 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-12 Standing Waves 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-12 Standing Waves The displacement of a standing wave is given by the equation: The position dependant amplitude is equal to 2ymsinkx. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-12 Standing Waves Nodes: These are defined as positions where the standing wave amplitude vanishes. They occur when 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 16-12 Standing Waves Antinodes: These are defined as positions where the standing wave amplitude is maximum. They occur when Note1: The distance between ajacent nodes and antinodes is λ/2 Note2: The distance between a node and an ajacent antinode is λ/4 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-13 Standing Waves and Resonance 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

16-13 Standing Waves and Resonance The resonant frequencies that correspond to these wavelengths. The lowest frequency is called the fundamental mode or the first harmonic (n=1). The second harmonic is the oscillation mode with n = 2. The third harmonic is that with n = 3, and so on. 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-8 Figure shows a pattern of resonant oscillation of a string of mass m=2.500 g and length L=0.800 m and that is under tension τ=325.0 N. What is the wavelength λ of the transverse waves producing the standing-wave pattern, and what is the harmonic number n ? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-8 What is the frequency f of the transverse waves and of the oscillation of the moving string elements? What is the maximum magnitude of the transverse velocity um of the element oscillation at coordinate x=0.180 m (note the x axis in the figure)? At what point during the element’s oscillation is the transverse velocity maximum? 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-8 Solutions: 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 Example 16-8 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 16! 2018/11/27 普通物理講義-16 / 國立彰化師範大學物理系/ 郭艷光教授