Physics
Longitudinal Wave
A longitudinal wave is a type of wave in which the oscillations occur in the same direction as the wave's propagation. This means that the particles of the medium move parallel to the direction of the wave. Sound waves are a common example of longitudinal waves, where the air particles move back and forth in the same direction as the sound wave travels.
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12 Key excerpts on "Longitudinal Wave"
eBook - ePubOscillations and Waves
An Introduction, Second Edition
- Richard Fitzpatrick(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
CHAPTER 5Longitudinal Standing Waves
5.1 INTRODUCTIONThe aim of this chapter is to generalize the analysis of the previous chapter in order to deal with longitudinal standing waves. A standing wave is a disturbance in a physically continuous mechanical system that is periodic in space as well as in time, but which does not propagate. A Longitudinal Wave is one in which the direction of oscillation is parallel to the direction along which the phase of the waves varies sinusoidally.5.2 SPRING-COUPLED MASSESConsider a mechanical system consisting of a linear array of N identical masses m that are free to slide in one dimension over a frictionless horizontal surface. Suppose that the masses are coupled to their immediate neighbors via identical light springs of unstretched length a, and force constant K. (Here, we employ the symbol K to denote the spring force constant, rather than k, because k is already being used to denote wavenumber.) Let x measure distance along the array (from the left to the right). If the array is in its equilibrium state then the x-coordinate of the ith mass is xi = i a, for i = 1, N. Consider longitudinal oscillations of the masses. Namely, oscillations for which the x-coordinate of the ith mass is(5.1)x i= i a +ψ i( t ) ,where ψi (t) represents longitudinal displacement from equilibrium. It is assumed that all of the displacements are relatively small; that is, |ψi | ≪ a, for i = 1, N.Consider the equation of motion of the ith mass. See Figure 5.1 . The extensions of the springs to the immediate left and right of the mass are ψi − ψi− 1 and ψi +1 − ψi , respectively. Thus, the x-directed forces that these springs exert on the mass are −K (ψi − ψi− 1 ) and K (ψi +1 − ψi ), respectively. The mass’s equation of motion therefore becomes(5.2)=ψ ¨iω 0 2(ψ− 2i − 1ψ i+ψ) ,i + 1where. Because there is nothing special about the ith mass, the preceding equation is assumed to hold for all N masses; that is, for i = 1, N. Equation (5.2)ω 0=K / m
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 - ePubMechanical Wave Vibrations
Analysis and Control
- Chunhui Mei(Author)
- 2023(Publication Date)
- Wiley(Publisher)
2 Longitudinal Waves in BeamsSimilar to the propagation of acoustical waves and electro-magnetic waves through gaseous, liquid, and solid media, mechanical vibrations can be described as waves that propagate in a solid medium (Graff 1975; Cremer et. al. 1987; and Doyle 1989). From the wave standpoint, vibrations in a distributed or continuous structure can be viewed as waves that propagate along uniform waveguides and are reflected and transmitted at structural discontinuities (Mace 1984). The propagation relationships of waves are governed by the equations of motion of a beam for free vibration, and the reflection and transmission relationships are determined by the equilibrium and continuity at a structural discontinuity. Assembling these propagation, reflection, and transmission relationships provides a concise and systematic approach for vibration analysis of a distributed or continuous structure.In this chapter, fundamental concepts related to Longitudinal Waves are introduced, such as the propagation coefficient of longitudinal vibration waves along a uniform beam (the waveguide) and the reflection coefficient of longitudinal vibration waves at either a classical or non-classical boundary (the discontinuity). Natural frequencies, modeshapes, as well as steady state frequency responses, are obtained, with comparison to experimental results. MATLAB scripts for numerical simulations are provided.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
(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. The brown curve is a snapshot of the wave at t 5 0, and the blue curve is another snapshot at some later time t. x Equilibrium density Density a b Figure 13.25 (a) A Longitudinal Wave on a spring. (b) The crests of the waveform correspond to com- pressed regions of the spring, and the troughs correspond to stretched regions of the spring. Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202 444 TOPIC 13 | Vibrations and Waves Unless otherwise noted, all content on this page is © Cengage Learning. 13.8 Frequency, Amplitude, and Wavelength Figure 13.26 illustrates a method of producing a continuous wave or a steady stream of pulses on a very long string. One end of the string is connected to a blade that is set vibrating. As the blade oscillates vertically with simple harmonic motion, a trav- - eling wave moving to the right is set up in the string. Figure 13.26 shows the wave at intervals of one- quarter of a period. Note that each small segment of the string, such as P, oscillates vertically in the , oscillates vertically in the y - direction with simple harmonic motion. That must be the case because each segment follows the simple harmonic motion of the blade. Every segment of the string can therefore be treated as a simple harmonic oscillator vibrating with the same frequency as the blade that drives the string.
- John van Opstal(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
Chapter 2The Nature of Sound
Abstract
This chapter summarizes the physical properties of sound waves that will be relevant in the later chapters. A simple elastic gas model can explain how the propagation speed of sound depends only on the bulk modulus and density of the air. We then introduce the homogeneous linear wave equation, from which the superposition of harmonic solutions (standing waves and traveling waves), and the concept of Fourier series are derived. As an example, we discuss the fundamental problem of spectral–temporal resolution. We then describe how inhomogeneities in the medium may change the wave equation and lead to nonharmonic spatial waves. We introduce the concept of acoustic impedance, and how this defines the conditions for transmission and reflection at boundaries between different media. We illustrate these ideas for the middle-ear transfer of acoustic energy from air to the fluid-filled cochlea. Finally, we discuss how the dispersion relation for homogeneous and inhomogeneous media relates to the phase and group velocities of the acoustic signals.Keywords
speed of sound acoustic impedance wave equation superposition Fourier series binaural beats transmission reflection decibel dispersion relation phase velocity group velocity2.1. Longitudinal pressure waves in a medium
Sound is a pressure perturbation in a medium like air or water, which is caused by a vibratory source that oscillates in the 50–20,000 Hz range (Serway and Jewett, 2013 ). The perturbation propagates as a longitudinal traveling wave through the medium, in which the molecular movements associated with the perturbation are along the same direction as the propagating traveling wave. Think of a vibrating membrane (a loudspeaker) at position x = 0, moving inward and outward in the x -direction at frequency f 0 Hz. When the membrane moves, it pushes or drags the air molecules at x = 0 in the same direction, thereby creating a local increase or decrease in the density of air molecules, which consequently increases or decreases the local pressure. These pressure changes propagate in the x -direction through the medium. The wave is thus described by a succession of compressions (regions of high pressure, c ) and rarefactions (regions of low pressure, r ), which propagate through the medium at velocity v sound . In air, at sea level at a temperature of 20°C, and at a mean pressure of 105 N/m2 (1 atm), the speed of sound is approximately v sound
eBook - PDF- P.F. Kelly(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Recalling Chapter 27, we consider a Longitudinal Wave propagating within a homogeneous medium, confined to a channel with uniform cross-sectional area A . For this analysis, the quantification of the wave disturbance is provided by the displacement of material elements away from their equilibrium positions . [ The medium itself does not experience bulk motion. ] A longitudinal harmonic wave is portrayed in both panels of Figure 28.1. On the left, regions of lower/higher density appear with darker/lighter shading. The sketch on the right illustrates the to-and-fro sloshing 1 of thin planar slabs of medium. By squinting carefully at Figure 28.1, it is possible to convince oneself of the two panels’ mutual consistency. c → c → PUSHED BKWDS PULLED FWDS depleted a64 a82 enhanced a64 a64 a73 FIGURE 28.1 Two Views of a Longitudinal Harmonic Wave The harmonic wave disturbance may be presumptively written 2 as S ( t,x ) = S 0 cos( kx − ωt ) . The LHS represents the forward–backward displacement of an infinitesimally thin slice of medium which would be at rest at x were the wave disturbance not present. With no wave 1 This is an essential point. The local motions appear to be just so much sloshing around. The wave arises through spatially coherent/correlated motions of swathes of the medium. 2 Phase factors which would otherwise appear may be set to zero by careful and consistent choices of the origins of the spatial and temporal coordinates. 28–167 28–168 Properties of Materials present, a material element [ with cross-sectional area A ] whose left edge is at x , while its rightmost edge is at x + Δ x , has an [ unperturbed ] volume of Δ V = A bracketleftbig ( x + Δ x ) − x bracketrightbig = A Δ x. When the harmonic disturbance is present, the left and right edges are shifted to x + S ( t,x ) and x + Δ x + S ( t,x + Δ x ), respectively.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
3. Matter waves. Although these waves are commonly used in modern technol- ogy, they are probably very unfamiliar to you. These waves are associated with electrons, protons, and other fundamental particles, and even atoms and molecules. Because we commonly think of these particles as constituting matter, such waves are called matter waves. 4. Gravitational waves. In 1916, Albert Einstein predicted that when any mass accelerates, it sends out gravitational waves that are oscillations of space itself (more precisely, spacetime). In normal circumstances, the oscillations are so small as to be undetectable. The first direct detection of the waves came in 2015 when a detector based on the design of Rainer Weiss of MIT recorded the waves due to the merger of two distant black holes. The oscillations were much less than the radius of a proton. Much of what we discuss in this chapter applies to waves of all kinds. However, for specific examples we shall refer to mechanical waves. Transverse and Longitudinal Waves 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. 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. As each section moves upward in turn, it begins to be pulled back downward by neighboring sections that are already on the way down. The net result is that a distortion in the string’s shape (a pulse, as in Fig. 16.1.1a) moves along the string at some velocity v → . If you move your hand up and down in continuous simple harmonic motion, a continuous wave travels along the string at velocity v → .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
However, the speed of sound varies from substance to substance. Sound travels faster in liquids and non-porous solids than it does in air. It travels about 4.3 times faster in water (1,484 m/s), and nearly 15 times as fast in iron (5,120 m/s), than in air at 20 degrees Celsius. In solids, sound waves propagate as two different types. A Longitudinal Wave is ass-ociated with compression and decompression in the direction of travel, which is the same process as all sound waves in gases and liquids. A transverse wave, often called shear wave, is due to elastic deformation of the medium perpendicular to the direction of wave ________________________ WORLD TECHNOLOGIES ________________________ travel, The direction of deformation is called the polarization of the wave. In general, transverse waves occur as a pair of orthogonal polarizations. These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an earthquake, where sharp compression waves arrive first, and rocking transverse waves seconds later. The speed of an elastic wave in any medium is determined by the medium's com-pressibility and density. The speed of shear waves, which can occur only in solids, is determined by the solid material's stiffness, compressibility and density. Basic concept U.S. Navy F/A-18 breaking the sound barrier. The white halo consists of condensed water droplets formed by the sudden drop in air pressure behind the shock cone around the aircraft. The transmission of sound can be illustrated by using a toy model consisting of an array of balls interconnected by springs. For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on.
- 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 - PDFSound and Structural Vibration
Radiation, Transmission and Response
- Frank J. Fahy(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
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
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.
eBook - PDF- Jurjen A. Battjes, Robert Jan Labeur(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
2 Classification and Analysis of Long Waves The category of long waves encompasses different wave types, each with a different origin and with different dynamics, in the sense that the relative importance of the various physical processes, as expressed by the different terms in the equation of motion, can vary. In this chapter we will first give an overview of the various types of long wave and their generation, and we will verify formally that they can indeed be classified as ‘long’. Next, we will estimate the relative magnitudes of terms in the equation of motion, which provides an ordering of wave types based on the importance of inertia relative to the resistance. 2.1 Types of Long Waves In general, one can say that the faster the flow varies, the more important will be the inertia relative to the resistance, and the more it will be in balance with the net driving force. Before dealing with these dynamics we give short descriptions of the origin and typical characteristics of the different types of long waves: • translatory waves (transient variations in discharge and water level, usually caused by operation of controls) • tsunamis (sea waves generated by subsea earthquakes, volcanic eruptions etc.) • seiches (standing oscillations in lakes, bays, harbours etc.) • tides in oceans, shelf seas, estuaries and lowland rivers • flood waves in rivers 2.1.1 Translatory Waves As a result of manipulation (or breakdown!) of pumps or valves in the operation of locks, weirs, evacuation sluices, hydropower plants, etc., variations in discharge (δQ) can occur. These are accompanied by variations in water surface elevation (δh). Such disturbances travel as so-called translatory waves into the adjacent reaches of the conduit. The passage of such wave induces a rise in elevation in the case of an increase in discharge, and a lowering in the case of a decrease in discharge; see Figure 2.1.
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