Physics
Longitudinal and Transverse Waves
Longitudinal waves are waves in which the particles of the medium move parallel to the direction of the wave. Examples include sound waves. Transverse waves are waves in which the particles of the medium move perpendicular to the direction of the wave. Examples include light waves and water waves.
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12 Key excerpts on "Longitudinal and Transverse Waves"
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
KEY IDEAS • Mechanical waves can exist only in material media and are governed by Newton’s laws. Transverse mechanical waves, like those on a stretched string, are waves in which the particles of the medium oscillate perpendicular to the wave’s direction of travel. Waves in which the particles of the medium oscillate parallel to the wave’s direction of travel are longitudinal waves. • A sinusoidal wave moving in the positive direction of an x axis has the mathematical form y (x, t) = y m sin(kx − t), where y m is the amplitude (magnitude of the maximum displacement) of the wave, k is the angular wave number, is the angular frequency, and kx − t is the phase. The wavelength is related to k by k = 2 . Pdf_Folio:313 • The period T and frequency f of the wave are related to by = 2f = 2 T . • The wave speed v (the speed of the wave along the string) is related to these other parameters by v = k = T = f. • Any function of the form y (x, t) = h (kx ± t) can represent a travelling wave with a wave speed as given above and a wave shape given by the mathematical form of h. The plus sign denotes a wave travelling in the negative direction of the x axis, and the minus sign a wave travelling in the positive direction. Why study physics? The Australian whipcracking championships, at the Sydney Royal Easter Show, demonstrate how a whip is designed to take advantage of the relationship between the speed of the wave travelling along a whip and the mass per unit length of the whip. Seismic waves in New Zealand give rise to longitudinal waves (compressional, primary or P-waves) and transverse waves (shear wave, secondary or S-waves), which enable the location of the earthquake’s epicentre to be determined. 1 This chapter focuses on waves travelling along a stretched string, such as on a guitar. The next chapter focuses on sound waves, such as those produced by a guitar string being played.
- Christopher Lavers(Author)
- 2017(Publication Date)
- Thomas Reed(Publisher)
Initially, particles are undisturbed (a), but then one is disturbed by a vertical motion (b) (figure 2.2). This first particle vibrates about the equilibrium position, as time progresses, out to a maximum extension (c). But this first particle is physically connected or joined to the second by molecular forces holding the material together and so a restoring force starts to act upon the first molecule, bringing it back from its position of maximum extension. Because of the molecular forces holding the material together, the second particle is forced to repeat the motion of the first but slightly later in time (d), and so on. Similar effects occur for all other particles in the chain. The disturbance thus travels through the medium until the disturbance has passed completely by the first particle, which once more returns to its undisturbed equilibrium position (i)- through a variety of loss or dampening mechanisms. The particles thus move in a direction at right angles (orthogonal, or at 90 degrees) to the wave propagation direction, and is said to be a transverse wave . This type of wave motion can be demonstrated by giving a taunt string a flick at right angles to the direction of the string. Types of Waves 25 25 Propagation Distance Displacement abo ve and belo w the undisturbed postion Time Increasing i h g e d c b a f Figure 2.4: Transverse wave generated along a string, showing individual molecular motion developing over time. As we have already seen, sound waves are longitudinal, while waves on a string are transverse. Many other forms of wave motion can be built up from these two fundamental modes. For example, the waves on the surface of the sea, which have a complicated form, may simply be considered as combinations of both longitudinal and transverse wave motion out of step, since if a particle moves forwards, then downwards, backwards and upwards, it will actually describe a rotational motion.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
381 16-1 TRANSVERSE WAVES What Is Physics? One of the primary subjects of physics is waves. To see how important waves are in the modern world, just consider the music industry. Every piece of music you hear, from some retro-punk band playing in a campus dive to the most elo- quent concerto playing on the web, depends on performers producing waves and your detecting those waves. In between production and detection, the infor- mation carried by the waves might need to be transmitted (as in a live perfor- mance on the web) or recorded and then reproduced (as with CDs, DVDs, or the other devices currently being developed in engineering labs worldwide). The financial importance of controlling music waves is staggering, and the rewards to engineers who develop new control techniques can be rich. This chapter focuses on waves traveling along a stretched string, such as on a guitar. The next chapter focuses on sound waves, such as those produced by a guitar string being played. Before we do all this, though, our first job is to classify the countless waves of the everyday world into basic types. Types of Waves Waves are of three main types: 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 two 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 = 299 792 458 m/s.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
C H A P T E R 16 After reading this module, you should be able to . . . 16.1.1 Identify the four main types of waves. 16.1.2 Distinguish between transverse waves and longitudinal waves. 16.1.3 Given a displacement function for a transverse wave, determine amplitude y m , angular wave number k, angular frequency ω, phase constant ϕ, and direction of travel, and calculate the phase kx ± ωt + ϕ and the displacement at any given time and position. 16.1.4 Given a displacement function for a transverse wave, calculate the time between two given displacements. 16.1.5 Sketch a graph of a transverse wave as a function of position, identifying amplitude y m , wavelength λ, where the slope is greatest, where it is zero, and where the string elements have positive velocity, negative velocity, and zero velocity. 16.1 TRANSVERSE WAVES KEY IDEAS 1. Mechanical waves can exist only in material media and are governed by Newton’s laws. Transverse mechanical waves, like those on a stretched string, are waves in which the particles of the medium oscillate perpendicular to the wave’s direction of travel. Waves in which the particles of the medium oscillate parallel to the wave’s direction of travel are longitudinal waves. 2. A sinusoidal wave moving in the positive direction of an x axis has the math- ematical form y(x, t) = y m sin(kx − ωt), where y m is the amplitude (magnitude of the maximum displacement) of the wave, k is the angular wave number, ω is the angular frequency, and kx − ωt is the phase. The wavelength λ is related to k by k = 2π _ λ . 3. The period T and frequency f of the wave are related to ω by ω _ 2π = f = 1 _ T . 4. The wave speed v (the speed of the wave along the string) is related to these other parameters by v = ω _ k = λ _ T = λf. 5. Any function of the form y(x, t) = h(kx ± ωt) can represent a traveling wave with a wave speed as given above and a wave shape given by the mathematical form of h.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
C H A P T E R 1 6 469 16.1 TRANSVERSE WAVES What Is Physics? One of the primary subjects of physics is waves. To see how important waves are in the modern world, just consider the music industry. Every piece of music you hear, from some retro-punk band playing in a campus dive to the most elo- quent concerto playing on the Web, depends on performers producing waves and your detecting those waves. In between production and detection, the infor- mation carried by the waves might need to be transmitted (as in a live perfor- mance on the Web) or recorded and then reproduced (as with CDs, DVDs, or the other devices currently being developed in engineering labs worldwide). The financial importance of controlling music waves is staggering, and the rewards to engineers who develop new control techniques can be rich. This chapter focuses on waves traveling along a stretched string, such as on a guitar. The next chapter focuses on sound waves, such as those produced by a guitar string being played. Before we do all this, though, our first job is to classify the countless waves of the everyday world into basic types. Types of Waves Waves are of three main types: 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 two 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 = 299 792 458 m/s.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
427 SOUND WAVES I n Chapter 18 we studied transverse mechanical waves, in particular the vibrations of a stretched string. Now we turn our attention to longitudinal mechani- cal waves, in particular sound waves. What we call sound is a longitudinal mechanical vibration with fre- quencies from about 20 Hz to about 20,000 Hz, which is the typical range of human hearing. Longitudinal waves of higher frequency, which are called ultrasonic waves, are used in locating underwater objects and in medical imaging. Longitudinal (and transverse) mechanical waves of lower frequency, called infrasonic, occur as seismic waves in earthquakes. In this chapter we discuss the properties of sound waves, their propagation, and their production by vi- brating systems. 19-1 PROPERTIES OF SOUND WAVES Like the transverse wave on the string, sound is a mechani- cal wave, meaning that the disturbance propagates due to the mechanical (elastic) forces between particles in the medium. Mechanical waves can travel through any material medium (solid, liquid, or gas). In solids, mechanical waves can be longitudinal or transverse, but in fluids (which can- not support shearing forces) the waves are only longitudi- nal, which means that the particles of the medium oscillate along the same direction that the wave is traveling. When we discuss sound waves, we normally mean lon- gitudinal waves in the frequency range 20 Hz to 20,000 Hz, the normal range of human hearing. However, the branch of physics and engineering that deals with the study of sound waves, called acoustics, generally includes the study of me- chanical waves of all frequencies, with transverse as well as longitudinal vibrations in the case of solids. In this chapter we consider mainly sound waves in air, which are strictly longitudinal. Although a small source of sound in an open area emits waves that are three-dimensional, we will simplify the prob- lem by considering one-dimensional waves.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
The brown curve can be thought of as a snapshot of a traveling wave taken at some instant of time, say, t 5 0; the blue curve is a snapshot of the same traveling wave at a later time. This picture can also be used to represent a wave on water. In such a case, a high point would correspond to the crest of the wave and a low point to the of the wave and a low point to the trough of of the wave. The same waveform can be used to describe a longitudinal wave, even though no up- and- down motion is taking place. Consider a longitudinal wave traveling on a spring. Figure 13.25a is a snapshot of this wave at some instant, and Figure 13.25b shows the sinusoidal curve that represents the wave. Points where the coils of the spring are compressed correspond to the crests of the waveform, and stretched regions correspond to troughs. The type of wave represented by the curve in Figure 13.25b is often called a density wave or e or e pressure wave, because the crests, where the spring coils are com- pressed, are regions of high density, and the troughs, where the coils are stretched, are regions of low density. Sound waves are longitudinal waves, propagating as a series of high- and low- density regions. P P P Any element P (black dot) on (black dot) on the rope moves in a direction perpendicular to the direction of propagation of the wave motion (red arrows). Figure 13.22 A pulse traveling on a stretched string is a transverse wave. Figure 13.23 (a) A transverse wave is set up in a spring by moving one end of the spring perpendicular to its length. (b) A longitudinal wave along a stretched spring. Compressed Compressed Stretched Stretched Longitudinal wave Transverse wave Transverse wave T As the hand pumps back and forth, compressed regions alternate stretched regions both in space and time. a b t = 0 t y x vt v S Figure 13.24 A one- dimensional sinusoidal wave traveling to the right with a speed v.
eBook - PDF- W. K. Nowacki(Author)
- 2018(Publication Date)
- Pergamon(Publisher)
CHAPTER V PLASTIC LONGITUDINAL-TRANSVERSE WAVES In this chapter we shall discuss first of all the solution of problems involving the propagation of simple waves. We shall analyse the case of a two-parameter loading of the boundary of the body under consideration. We shall consider in turn bodies whose properties are governed by the equations of the theory of plastic flow as exemplified by the constitutive equations due to Grigorian for soil dynamics and by the equations of the bilinear theory of plasticity. Following this we shall present solutions of problems concerning the propagation of longitudinal-transverse waves in homogeneous elastic/viscoplastic bodies (both plane waves and radial cylindrical waves). At the present time many solutions exist to problems of wave propagation involving complex stress states (for a single spatial variable and for two-parameter loading). Early papers in this field were confined to the solutions of self-similar problems [13] — [15], [127], [145], [146]. Thus the solutions were restricted to that class of the boundary conditions which are such that it was possible to make the stress state, the strain state, and the particle velocities of the medium dependent upon one independent variable only. This then reduced the system of partial differential equations describing the motion of the medium to ordinary differential equations. On account of the character of the external loading assumed in the papers quoted, the problem of the generation of plastic wave fronts, which appear due to the interaction of the Longitudinal and Transverse Waves did not arise. Problems of unloading wave generation also did not occur. These non-self-similar problems were discussed in references [59] —[62], [163], [164] where the problem of the propagation of longitudinal—transverse waves in an elastic/ viscoplastic medium was considered for arbitrary, time-dependent external loads.
eBook - PDF- John Kimball(Author)
- 2015(Publication Date)
- CRC Press(Publisher)
147 5 W AVES 5.1 Introduction Waves are everywhere. Sound waves, radio waves, and light waves are as much a part of the world we know as solids, liquids, and gasses. After an overview of general wave properties with some examples, the descriptions emphasize sound and light waves because without these waves we would be lost. Two of our five senses, vision and hearing, interpret light and sound waves and tell us almost every-thing we know. 5.2 Common Features of Waves The geometries of sound and light waves are suggested by the water waves (gravity waves) that can be seen on the surface of a lake or a bathtub. But, the analogy should be approached with caution. In many ways, sound and light are simpler than both water waves and the wave function of quantum mechanics (Chapter 6). The water waves in Figure 5.1 spread out in circles. A long way from the center, the wave peaks and valleys are nearly straight lines. The waves become nearly “plane waves” with a shape that varies only in the direction pointed away from the wave source. Far from the source, light and sound waves also approach plane wave shape. 5.2.1 Wavelength, Frequency, Speed, Amplitude, and Energy The simplest wave geometry is the plane wave. The simplest plane wave shape is the “sine wave” shown in Figure 5.2. Any wave shape can be constructed by adding together various sine waves, so the sine wave building blocks of all waves deserve special attention. 148 PHYSICS CURIOSITIES, ODDITIES, AND NOVELTIES Sine waves are characterized by three quantities: wavelength, fre-quency, and amplitude. The wavelength is the distance between wave peaks. The frequency is the number of times a wave oscillates up and down each second. The amplitude is the height of the wave. The speed of a wave is the distance one of the wave peaks moves in 1 second. It is related to wavelength and frequency by an impor-tant equation.
eBook - PDFStructural Acoustics
Deterministic and Random Phenomena
- Joshua E. Greenspon(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
Sound does not propagate in free space. It must have a dense * This chapter was published in Encyclopedia of Physical Science and Technology , vol. 1, Greenspon, J. E., 114–152. Copyright Elsevier (1992). 10 Structural Acoustics: Deterministic and Random Phenomena © 2011 by Taylor and Francis Group, LLC medium in which to propagate. Thus, for example, when a sound wave is produced by a voice, the air particles in front of the mouth are vibrated, and this vibration, in turn, produces a disturbance in the adjacent air particles, and so on. If the wave travels in the same direction in which the particles are being moved, it is called a longitudinal wave. This same phenomenon occurs whether the medium is air, water, or a solid. If the wave is moving perpen-dicularly to the moving particles, it is called a transverse wave. The rate at which a sound wave thins out, or attenuates, depends to a large extent on the medium through which it is propagating. For example, sound attenuates more rapidly in air than in water, which is the reason that sonar is used more extensively underwater than in air. Conversely, radar (electro-magnetic energy) attenuates much less in air than in water, so it is more use-ful in air as a communication tool. Sound waves travel in solid or fluid materials by elastic deformation of the material, which is called an elastic wave. In air (below a frequency of 20 kHz) and in water, a sound wave travels at constant speed without its shape being distorted. In solid material, the velocity of the wave changes, and the distur-bance changes shape as it travels. This phenomenon in solids is called dis-persion. Air and water are, for the most part, nondispersive media, whereas most solids are dispersive media. 2.2.2 Reflection, Refraction, Diffraction, Interference, and Scattering Sound propagates undisturbed in a nondispersive medium until it reaches some obstacle.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
In a solid medium, however, sound waves can be transverse. In this case, the polarization is associated with the direction of the shear stress in the plane perpendicular to the propagation direction. This is important in seismology. Polarization is significant in areas of science and technology dealing with wave propagation, such as optics, seismology, telecommunications and radar science. The polarization of light can be measured with a polarimeter. ________________________ WORLD TECHNOLOGIES ________________________ Theory Basics: plane waves The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation of most light waves (a plane wave is a wave with infinitely long and wide wavefronts). For plane waves Maxwell's equations, specifically Gauss's laws, impose the transversality requirement that the electric and magnetic field be per-pendicular to the direction of propagation and to each other. Conventionally, when considering polarization, the electric field vector is described and the magnetic field is ignored since it is perpendicular to the electric field and proportional to it. The electric field vector of a plane wave may be arbitrarily divided into two perpendicular com-ponents labeled x and y (with z indicating the direction of travel). For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner in time, the two components have exactly the same frequency. However, these components have two other defining characteristics that can differ. First, the two components may not have the same amplitude. Second, the two components may not have the same phase, that is they may not reach their maxima and minima at the same time. Mathematically, the electric field of a plane wave can be written as, or alternatively, where A x and A y are the amplitudes of the x and y directions and φ is the relative phase between the two components.
eBook - PDFSound and Structural Vibration
Radiation, Transmission and Response
- Frank J. Fahy(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Hence the transverse normal vibration velocity of a surface determines the value at the surface of the pressure gradient in the direction normal to the surface; e.g., from (1.5b), (dp/dy) yssQ = —jcop 0 (v) yss0 , where the surface is assumed to lie in the x,z plane. This relationship is frequently used as a boundary condition in the analysis of sound radiation and scattering by vibrating surfaces. As-sumption of a plane wave field yields, from (1.7), 7 ) -» -or ( / --озро/ку. (1.9) This ratio has the form of a specific acoustic impedance, which is defined to be the ratio of the complex amplitudes of pressure and particle velocity at a single frequency. One of the applications of the impedance concept is in the evaluation of sound power radiated into a fluid by a vibrating surface. The time-averaged sound power radiated per unit area is given formally by P = j T 0 p{№)dt. For single-frequency normal surface vibration of the form v(t) = v expO'co t), which produces a local acoustic pressure p(t) = p exp( jcot we may write the normal specific acoustic impedance of the fluid at the surface as i = p/v. Hence P -1 Re(pfc) = Щ 2 Re(z) = Щ г Re(l/z), (1.10) where * indicates complex conjugate and | | indicates magnitude (real ampli-tude). This equation is valid for all types of acoustic field. For the special case of plane-wave radiation assumed above, z is given by (1.9): later we shall see that k y is not always real! 1.3 Longitudinal Waves in Solids In pure longitudinal wave motion the direction of particle displacement is purely in the direction of wave propagation; such waves can propagate in large volumes of solids. Two parallel planes in an undisturbed solid elastic medium, which are separated by a small distance 5x, may be moved by dif- 1.3 Longitudinal Waves in Solids 9 * + 0$/dx>5x 5x a + (d
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