《物理双语教学课件》Chapter 10 Waves 波动

Chapter 10 Waves 10.1 Types of Waves 1. Mechanical waves: These waves are most familiar because we encounter them almost constantly; common examples include water waves, sound waves, and seismic waves. All these waves have certain central features: they are governed by Newton’s laws, and they can exist only within a material medium, such as water, air, and rock. 2. Electromagnetic waves: These waves are less familiar, but you use them constantly; common examples include visible and ultraviolet light, radio and television waves, microwaves, x-rays, and radar waves. These waves require no material medium to exist. Light waves from stars, for example, travel through the vacuum of space to reach us. All electromagnetic waves travel through a vacuum at the same speed c, given by c=299,792,458m/s. 3. Matter waves: Although these waves are commonly used in modern technology, their type is probably very unfamiliar to you. Electrons, protons, and other fundamental particles, and even atoms and molecules, travel as waves. Because we commonly think of these things as constituting matter, these waves are called matter waves. 4. Much of what we discuss in this chapter applies to waves of all kinds. However, for specific examples we shall refer to mechanical waves. 10.2 Transverse and Longitudinal Waves 1. Transverse wave (1). A wave sent along a stretched, taut string is the simplest mechanical wave. If you give one end of a stretched string a single up-and-down jerk, a wave in the form of a single pulse travels along the string, as in the figure. This pulse and its motion can occur because the string is under tension. When you pull your end of the string upward, it begins to pull upward on the adjacent section of the string via tension between the two sections. As the adjacent section moves upward, it begins to pull the next section upward, and so on. Meanwhile, you have pulled down on your end of the string. So, as each section moves upward in turn, it begins to be pulled back downward by neighboring sections that already on the way down. The net result is that a distortion in the string’s shape (the pulse) moves along the string at some velocity v. (2). If you move your hand up and down in continuous simple harmonic motion, a continuous wave travels along the string at velocity v. Because the motion of your hand is a sinusoidal function of time, the wave has a sinusoidal shape at any given instant, as in the figure (b). That is, the wave has the shape of a sine curve or a cosine curve. (3). We consider here only an “ideal” string, in which no friction-like forces within cause the wave to die out as it travels along the string. In addition, we assume that the string is so long that we need not consider a wave rebounding from the far end. (4). One way to study the waves of the figure is to monitor the wave’s form (shape of wave) as it moves to the right. Alternatively, we can monitor the motion of an element of the string as the element oscillates up and down while the wave passes through it. We would find that the displacement of every such oscillating string element is perpendicular to the direction of travel of the wave, as indicated in the figure. This motion is said to be transverse, and the wave is said to be a transverse wave. 2. Longitudinal wave: (1). The right figure shows how a sound wave can be produced by a piston in a long, air filled pipe. If you suddenly move the piston rightward and then leftward, you can send a pulse of sound along the pipe. The rightward motion of the piston moves the elements of air next to it rightward, changing the air pressure there. The increased air pressure then pushes rightward on the element of air somewhat farther along the pipe. Once they have moved rightward, the elements move back leftward. Thus the motion of the air and the change in air pressure travel rightward along the pipe as a pulse. (2). If you push and pull on the piston in simple harmonic motion, as is being done in the figure, a sinusoidal wave travels along the pipe. Because the motion of the elements of air is parallel to the direction of the wave’s travel. The motion is said to be longitudinal wave. 3. Both a transverse wave and a longitudinal wave are said to be traveling waves because the wave travels from one point to another, as from one end of the string to the other end or from one end of the pipe to the other end. Note that it is the wave that moves between the two points and not the material (string or air) through which the wave moves. 10.3 Wavelength and Frequency 1. Introduction (1). To completely describe a wave on a string, we need a function that gives the shape of the wave. This means that we need a relation in the form , in which y is the transverse displacement of any string element as a function h of the time t and the position x of the element along the string. In general, a sinusoidal shape like the wave can be described with h being either a sine function or a cosine function; both give the same general shape for the wave. In this chapter we use the sine function. (2). For a sinusoidal wave, traveling toward increasing values of x, the transverse displacement y of a string element at position x at time is given by , here is the amplitude of the wave; the subscript m stands for maximum, because the amplitude is the magnitude of the maximum displacement of the string element in either direction parallel to the y axis. The quantities k and are constants whose meanings are about to discuss. The quantity is called the phase of the wave. 2. Wavelength and angular Wave Number (1). The figure shows how the transverse displacement y varies with position x at an instant, arbitrarily called t=0. That is, the figure is a “snapshot” of the wave at that instant. With t=0, the wave equation becomes .  The Figure (a) is a plot of this equation; it shows the shape of the actual wave at time t=0. (2). The wavelength of a wave is the distance between repetitions of the wave shape. A typical wavelength is marked in figure (a). By definition, the displacement y is the same at both ends of this wavelength, that is, at and . Thus . (3). A sine function begins to repeat itself when its angle is increased by rad; so we have . We call k the angular wave number of the wave; its SI unit is the radian per meter. 3. Period, angular frequency, and frequency: (1). The figure (b) shows how the displacement y varies with time t at a fixed position, taken to be x=0.  If you were to monitor the string, you would see that the single element of the string at that position moves up and down in simple harmonic motion with x=0: . The figure (b) is a plot of this equation; it does not show the shape of the wave. (2). We define the period of oscillations T of a wave to be the time interval between repetitions of the motion of an oscillating string element. A typical period is marked in the figure (b). We have . (3). This can be true only if . We call the angular frequency of the wave; its SI unit is the radian per second. (4). The frequency f of the wave is defined as 1/T and is related to the angular frequency by . This frequency f is a number of oscillations per unit time-made by a string element as the wave moves through it, and f is usually measured in hertz or its multiples. 10.4 The Speed of a Traveling wave 1. The figure shows two snapshots of the wave taken a small time interval apart. The wave is traveling in the direction of increasing x, the entire wave pattern moving a distance in that direction during the interval . The ratio (or, in the differential limit, dx/dt) is the wave speed v. How can we find its value? 2. As the wave moves, each point of the moving wave form retains its displacement y. For each such point, the argument of the sine function must be a constant: . 3. To find wave speed v, we take the derivative of the equation, get . The equation tells us that the wave speed is one wavelength per period. 4. The wave equation describes a wave moving in the direction of increasing x. (1). We can find the equation of a wave traveling in the opposite direction by replacing t with –t. (2). This corresponds to the condition . (3). Thus a wave traveling toward decreasing x is described by the equation . (4). Its velocity is . 5. Consider now a wave of generalized shape, given by , where h represents any function, the sine function being one possibility. Our analysis above shows that all waves in which the variables x and t enter in the combination are traveling waves. Further more, all traveling waves must be the form above. Thus represents a possible traveling wave. The function , on the other hand, does not represent a traveling wave. 6. Wave Speed on a Stretched String (1). The speed of a wave is related to the wave’s wavelength and frequency, but it is set by the medium. If a wave is through a medium such as water, air, steel, or a stretched string, it must cause the particles of that medium to oscillate as it passes. For that happen, the medium must possess both inertia and elasticity. These two properties determine how fast the wave can travel in the medium. And conversely, it should be possible to calculate the speed of the wave through the medium in terms of these properties. (2). We can derive the speed from Newton’s second law as , where is the linear density of the string, and the tension in the string. (3). The equation tells us that the speed of a wave along a stretched ideal string depends only on the characteristics of the string and not on the frequency of the wave. 10.5 Energy and Power of a Traveling String Wave When we set up a wave on a stretched string, we provide energy for the motion of the string. As the wave moves away from us, it transports that energy as both kinetic energy and elastic potential energy. Let us consider each form in turn. 1. Kinetic energy: An element of the string mass dm, oscillating transversely in simple harmonic motion as the wave passes through it, has kinetic energy associated with its transverse velocity u: . (1). So when the element is rushing through its y=0 position, its transverse velocity-and thus its kinetic energy-is a maximum. (2). When the element is at its extreme position y=ym, its transverse velocity-and thus again its kinetic energy-is zero. 2. Elastic potential energy: To send a sinusoidal wave along a previously straight string, the wave must necessarily stretch the string. As a string element of length dx oscillates transversely, its length must increase and decrease in a periodic way if the string element is to fit the sinusoidal wave’s form. Elastic potential energy is associated with these length changes, just as for a spring. (1). When the string element is at its y=ym position, its length has its normal undisturbed value dx, so its elastic potential energy is zero. (2). However, when the element is rushing through its y=0 position, it is stretched to its maximum extent, and its elastic potential energy then is a maximum. 3. In the snapshot of the right figure, the regions of the string at maximum displacement have no energy, and the regions at zero displacement have maximum energy. 4. The rate of energy transmission: (1). The rate at which kinetic energy is carried along by the wave can be got from the equation as . (2). The average rate at which kinetic energy is transported is , where an overhead bar means an average value of the quantity. (3). Elastic potential energy is also carried along with the wave, and at the same average rate given by above equation. 5. The average power, which is the average rate at which energy of both kinds is transmitted by the wave, is then . The dependence of the average power of a wave on the square of its amplitude and also on the square of its angular frequency is a general result, true for waves of all types. 10.6 The Principle of Superposition for Waves It often happens that two or more waves pass simultaneously through the same region. When we listen to a concert, for example, sounds from many instruments fall simultaneously on our eardrums. 1. Suppose that two waves travel simultaneously along the same stretched string. Let and be the displacements that the string would experience if each wave acted alone. The displacement of the string when both waves act is then , the sum being an algebraic sum. This summation of displacements along the string means: Overlapping waves algebraically add to produce a resultant wave. This is another example of the principle of superposition, which says that when several effects occur simultaneously, their net effect is the sum of the individual effects. 2. The right figure shows a sequence of snapshots of two pulses traveling in opposite directions on the same stretched string. When the pulses overlap, the resultant pulse is their sum. Moreover, each pulse moves through the other, as if the other were not present: Overlapping waves do not in any way alter the travel of each other. 3. Fourier analysis (1) French mathematician Jean Baptiste Fourier (1786-1830) explained how the principle of superposition can be used to analyze non-sinusoidal wave forms. He showed that any wave’s form can be represented as the sum of a large number of sinusoidal waves, of carefully chosen frequencies and amplitudes. (2) English physicist Sir James Jean expressed it well: [Fourier’s] theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic [sinusoidal] curves-in brief, every curve can be built up by piling up waves. (3) The figure shows an example of a Fourier series, as such sums are called. The saw-tooth curve in figure (a) shows the variation with time (at position x=0) of the wave we wish to represent. The Fourier series that represents it can be shown to be , in which , where T is the period of the saw-tooth curve. The green curve of figure (a), which represents the sum of the first six terms of above equation, matches the saw-tooth curve rather well. Figure (b) shows these six terms separately. By adding more terms, we can approximate the saw-tooth curve as closely as we wish. (4) This is why we spent so much time analyzing the behavior of a sinusoidal wave. When we understand that, Fourier’s theorem will open the door to all other wave shape. 10.7 Interference of Waves Suppose we send two sinusoidal waves of the same wavelength and amplitude in the same direction along a stretched string. The superposition principle applies. What resultant wave does it predict for the string? 1. The resultant wave depends on the extend to which the waves are in phase with respect to each other, that is, how much one wave form is shifted from the other wave form. (1). If the waves are exactly in phase (so that the peaks and valleys of one are exactly aligned with those of the other without any shift), they combine to double the displacement of either wave acting alone. (2). If they are exactly out of phase (the peaks of one are exactly aligned with the valleys of the other), they cancel everywhere and the string remains straight. (3). We call this phenomenon of combining and canceling of waves interference, and the waves are said to interfere. 2. Let one wave traveling a long a stretched string be given by and another, shifted from the first, by . The waves have the same angular frequency , the same angular wave number k, and the same amplitude ym, they travel in the same direction, that increasing x, with the same speed. They differ only by a constant angle , which call the phase constant. These waves are said to be out of phase by or have a phase difference of , or one wave is said to be phase-shifted from the other by . 3. From the principle of superposition, the combined wave has displacement: The resultant wave is thus also a sinusoidal wave traveling in the direction of increasing x with its phase constant being and its amplitude being . It means if two sinusoidal waves of the same amplitude and wavelength travel in the same direction along a stretched string, they interfere to produce a resultant sinusoidal wave traveling in that direction. (1). If rad, the two combining waves are exactly in phase. Then . So the amplitude of the resultant wave is twice the amplitude of either combining wave. Interference that produces the greatest possible amplitude is called fully constructive interference. (2). If , the combining waves are exactly out of phase. We then have for all values of x and t, . Now, although we sent two waves along the string, we see no motion of the string at all. This type of interference is called fully destructive interference. (3). When interference is neither fully constructive nor fully destructive, it is called intermediate interference. The amplitude of the resultant wave is then intermediate between 0 and 2ym. 10.8 Phasors 1. We can represent a string wave vectorially with a phasor. In essence, a phasor is a vector that has a magnitude equal to the amplitude of the wave and that rotates around an origin; the angular speed of the phasor is equal to the angular frequency of the wave. For example, the wave is represented by the phasor in figure (a). The magnitude of the phasor is the amplitude of the wave. As the phasor rotates around the origin at angular speed , its projection on the vertical axis varies sinusoidally, from a maximum of through zero to a minimum of . This variation corresponds to the sinusoidal variation in the displacement of any point along the string as the wave passes through it. 2. When two waves travel along the same string in the same direction, we can represent them and their resultant wave in a phasor diagram. The phasor in figure (b) represent the wave given by and a second wave given by . The angle between the phasors in the figure (b) is equal to the phase constant . 3. Because wave and have the same angular wave number k and angular frequency , we know that their resultant is of the form , where is the amplitude of the resultant wave and is its phase constant. To find the value of and , we vectorially add the two phasors at any instant during their rotation, as in figure ?. The magnitude of the vector sum equals the amplitude . The angle between the vector sum and the phasor for equals the phase constant . 10.9 Standing Waves We have discussed two sinusoidal waves of the same wavelength and amplitude traveling in the same direction along a stretched string. What if they travel in opposite direction? We can again find the resultant wave by applying the superposition principle. 1. Above figure suggests the situation graphically. It shows the two combining waves, one traveling to the left in figure (a), the other to the right in figure (b). Figure (c) shows their sum, obtained by applying the superposition principle graphically. The outstanding feature of the resultant wave is that there are places along the string, called nodes, where the string is permanently at rest. Four such nodes are marked by dots in the figure (c). Halfway between adjacent nodes are anti-nodes, where the amplitude of the resultant wave is a maximum. Wave patterns such as that of the figure (c) are called standing waves because the wave pattern do not move left or right: the locations of the maxima and minima do not change. If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave. 2. To analyze a standing wave, we represent the two combining waves with equations and . The principle of superposition gives, for the combined wave, . This is not traveling wave but a standing wave. 3. The quantity in the brackets of the equation can be viewed as the amplitude of oscillation of the string element that is located at position x. However, since an amplitude is always positive and can be negative, we take the absolute value of the quantity to be the amplitude at x. 4. In a traveling sinusoidal wave, the amplitude of the wave is the same for all string elements. That is not true for a standing wave, in which the amplitude varies with position. (1). The amplitude is zero for values of kx that give . Those values are . Substituting , we get , as the positions of zero amplitude-the nodes-for the standing wave. Note that adjacent nodes are separated by , half a wavelength. (2). The amplitude of the standing wave has a maximum value of , which occurs for values of kx that give . Those values are . In other word, , as the position of maximum amplitude-the anti-nodes-of the standing wave. The anti-nodes are one-half wavelength apart and are located halfway between pairs of nodes. 5. Reflections at a Boundary: See the figure. 10.11 Standing Waves and Resonance 1. If the left end of a stretched string is oscillated sinusoidally with the other end fixed, the oscillation sends a continuous traveling wave rightward along the string. The frequency of the wave is that of the oscillation. The wave reflects at the fixed end and travels leftward back through itself. The right-going wave and the left-going wave then interfere with each other. 2. For certain frequencies, the interference produces a standing wave pattern (or oscillation mode) with nodes and large anti-nodes like those in the figure . Such a standing wave is said to be produced at resonance, and the string is said to resonate at these certain frequencies, called resonant frequencies. 3. If the string is oscillated at some frequency other than a resonance frequency, a standing wave is not set up. Then the interference of the right-going wave and the left-going wave results in only small oscillations of the string. 4. Suppose the length of the string is L, and the string is somehow made to oscillate at a resonance frequency to set up a standing wave pattern. Since each end of the string is fixed, there must be a node at each end. The patterns that meet this requirement are that , and the resonance frequencies that correspond to these wavelengths are , here v is the speed of the traveling waves on the string. 5. Above equation tells us that the resonant frequencies are integer multiples of the lowest resonant frequency, , which corresponds to . The oscillation mode with that lowest frequency is called the fundamental mode or the first harmonic. The collection of all possible oscillation modes is called the harmonic series, and n is called the harmonic number of nth harmonic.