Vibrating String

6 Key excerpts on "Vibrating String"

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  eBook - PDF

  The Basics of Physics

  • Richard L. Myers(Author)
  • 2005(Publication Date)
  • Greenwood

    (Publisher)

  D Vibrations, Waves, and Sound Introduction A vibration is an oscillatory repetitive motion that occurs at a specific location over time. Common examples of vibrations are a pen- dulum swinging back and forth, a weight bobbing on a spring, or a plucked string on a musical instrument. Vibrations lead to the propagation of waves. A Vibrating String creates a sound wave, electromag- netic waves are produced by the vibrations of electric charge, and seismic waves come from vibrations from within the Earth. All waves can be described by common char- acteristics such as wavelength, frequency, and period. This chapter begins the study of vibrations by examining simple harmonic motion and follows this with an examina- tion of the general principles that apply to all waves. Different types of waves will be examined, concluding with a more compre- hensive examination of sound. Simple Harmonic Motion A simple model used to examine vibra- tions is that of a simple harmonic oscillator. A simple harmonic oscillator is a system consisting of a "to and fro" motion around an equilibrium position. A condition for simple harmonic motion is that the restoring force is proportional to the displacement from the equilibrium position. For example, con- sider a weight attached to a vertical spring (Figure 8.1). If the spring is stretched by pulling on the weight and then released, the weight will bob up and down around the initial equilibrium position. As the weight moves down past the equilibrium position, stretching the spring, it will decelerate in proportion to a restoring force acting upward that attempts to return the weight to its equi- librium position. Once the weight "bottoms out," it will then move upward through the equilibrium position, compressing the spring. In the absence of friction, this up and down motion would continue forever, but in actuality, it eventually damps out due to friction.

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  String Theory For Dummies

  • Andrew Zimmerman Jones, Alessandro Sfondrini, Alessandro Sfondrini(Authors)
  • 2022(Publication Date)
  • For Dummies

    (Publisher)

  As the system collapses, the other options are no longer available to the system. The Standard Model of particle physics, as well as string theory (which includes the Standard Model in an appropriate limit), predicts that some properties of the universe were once highly symmetrical but have undergone spontaneous symme- try breaking into the universe we observe now. All Shook Up: Waves and Vibrations In string theory, the most fundamental objects are tiny strings of energy that vibrate or oscillate in simple, regular patterns. In physics, such systems are called harmonic oscillators. Harmonic oscillators are the simplest (and in many ways most universal) physical system that you will ever encounter. Though the strings of string theory are different, understanding the vibrations of classical objects — like air, water, jump ropes, springs — can help you understand the behavior of these exotic little creatures when you encounter them. These clas- sical objects can carry what are called mechanical waves. Catching the wave Waves (as we usually think of them) move through some sort of medium. Like in the examples we discussed when talking about kinetic energy, tidal waves can move through the water, and sound waves through the air, with those materials acting as the medium for the motion. Similarly, if you flick the end of a jump rope or string, a wave moves along the rope or string. In classical physics, waves trans- port energy, but not matter, from one region to another. One set of water mole- cules transfers its energy to the nearby water molecules, which means that the 68 PART 2 The Physics Upon Which String Theory Is Built wave moves through the water, even though the actual water molecules don’t really travel all the way from the start of the wave to the end of the wave. This is even more obvious if we take the end of a jump rope and shake it, causing a wave to travel along its length. Clearly, the molecules at our end of the jump rope aren’t traveling along it.

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  Invitation to Partial Differential Equations

  • Mikhail Shubin, Maxim Braverman, Robert McOwen, Peter Topalov, Maxim Braverman, Robert McOwen, Peter Topalov(Authors)
  • 2020(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 2 One-dimensional wave equation 2.1. Vibrating String equation Let us derive an equation describing small vibrations of a string. Note right away that our derivation will not be mathematical but rather physical or mechanical. However, its understanding is essential to grasp the physical meaning of, first, the wave equation itself and, second, but not less impor-tantly, the initial and boundary conditions. A knowledge of the derivation and physical meaning helps also to find various mathematical tools in the study of this equation (the energy integral, standing waves, etc.). Generally, derivation of equations corresponding to different physical and mechanical problems is important for understanding mathematical physics and is essen-tially a part of it. So, let us derive the equation of small vibrations of a string. We consider the vibrations of a taut string such that each point moves in a direction perpendicular to the direction of the string in its equilibriuim. We assume here that all the forces appearing in the string are negligible compared to the tension directed along the string (we assume the string to be perfectly flexible, i.e., nonresistant to bending). First of all, choose the variables describing the form of the string. Sup-pose that in equilibrium the taut string is stretched along the x -axis. To start, we will consider the inner points of the string disregarding its ends. Suppose that the vibrations keep the string in the ( x, y )-plane so that each point of the string is only shifted parallel to the y -axis and this shift at time t equals u ( t, x ); see Figure 2.1. 15 16 2. One-dimensional wave equation x y x u ( t, x ) Figure 2.1. Vibrating String. Therefore, at a fixed t the graph of u ( t, x ), as a function of x , is the shape of the string at the moment t , and for a fixed x the function u ( t, x ) describes the motion of one point of the string.

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  eBook - ePub

  Fourier Series and Orthogonal Functions

  • Harry F. Davis(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 6

  WAVES AND VIBRATIONS; HARMONIC ANALYSIS

  6.1 • THE Vibrating String

  Our study of harmonic analysis begins with an investigation of the vibrations of a stretched elastic string. We have in mind a piano string, a violin string, or a long flexible chain.

  We assume the tension in the string is sufficiently great that we can neglect the effects of air resistance, and (in the case of the piano string or violin string) can neglect forces due to gravity. We assume the string moves in a single plane and that the vibrations are transverse, i.e., that the points of the string move along straight lines perpendicular to the line of equilibrium of the string. We assume that the vibrations are quite small, so that the tension in the string does not depend on the displacement from equilibrium, although in the case of a vertically hanging chain the tension will presumably vary with position along the chain (being greater near the top of the chain, since the weight of the chain cannot be neglected in this case). We also assume that the vibrations are small, so the angle between the string and the line of equilibrium, at any instant of time, can be approximated by its tangent or by its cosine. (From the engineer’s viewpoint, these are the small angles that one reads on the combined sine-tangent scale of a sliderule.)

  It should be noted that an applied mathematician interested in the theory of elasticity would not make these assumptions initially but would set up the differential equation without them, and carefully analyze the maximum error each of these assumptions can produce under reasonable conditions. Such an analysis is far beyond the scope of this book.

  We let the x axis be the line of equilibrium. By virtue of the assumptions made, we can take y (x ,t ) to be the displacement from equilibrium, at time t , of a point on the string whose equilibrium position is the point (x ,0) in the xy plane. In other words, at any time t , this particle is located at the point (x ,y [x,t

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  Fundamentals of Noise and Vibration Analysis for Engineers

  • M. P. Norton, D. G. Karczub(Authors)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  1.4 Introductory concepts on natural frequencies, modes of vibration, forced vibrations and resonance Natural frequencies, modes of vibration, forced vibrations and resonance can be de-scribed both from an elastic continuum and a macroscopic viewpoint. The existence of natural frequencies and modes of vibration relates to the fact that all real physical systems are bounded in space. A mode of vibration (and the natural frequency associ-ated with it) on a taut, fixed string can be interpreted as being composed of two waves of equal amplitude and wavelength travelling in opposite directions between the two bounded ends. Alternatively, it can be interpreted as being a standing wave, i.e. the string oscillates with a spatially varying amplitude within the confines of a specific stationary waveform. The first interpretation of a mode of vibration relates to the wave model, and the second to the macroscopic model. Both describe the same physical motion and are mathematically equivalent – this will be illustrated in section 1.9. The concepts discussed above can be illustrated by means of a simple example. Let us consider a piece of string which is stretched and clamped at its ends, as illustrated in Figure 1.7(a). The string is plucked at some arbitrary point and allowed to vibrate freely. At the instant that the string is plucked, a travelling wave is generated in each direction (i.e. towards each clamped end of the string). It is important to recognise that, at this instant, the shape of the travelling wave is not that of a mode of vibration (Figure 1.7b) since a standing wave pattern has yet to be established. The travelling waves move along the string until they meet the clamped ends, at which point they are reflected. After these initial reflections (one from each clamped end) there is a further 11 1.4 Natural frequencies and resonance Fig. 1.7. Schematic illustration of travelling and standing waves for a stretched string.

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  An Introduction to Acoustics

  • Robert H. Randall(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Fig. 7–14 ). The four strings, sometimes of gut and sometimes of metal, are all of the same length (measured between the bridge and the upper clamping point) but are of different linear mass density and are under different tensions. As a result, the fundamental modes differ. The “open” string resonances cover a range of somewhat over an octave. The fundamental modes can be raised by shortening the string, i.e., by “stopping” the string with the finger. The range of the instrument may in this way be extended to about four octaves.

  It is important to recognize that the sound waves produced by a violin originate almost exclusively with the vibrations of the body of the instrument, not with the vibrations of the strings themselves. A Vibrating String, as mentioned in Chapter 3, is a linear array of double sources whose dipole components are almost coincident. The rate of dissipation of the vibrational energy by the radiation of sound waves is practically zero. Most of the energy supplied by the bowing action is transmitted to the body of the instrument through the bridge, the latter being set into motion by the periodic shortening and lengthening of the string associated with the stationary wave pattern. The amplitude of the motion at the top of the bridge is longitudinal and is smaller than that of the average transverse displacement of the string. The ends of the string can therefore still be considered fixed as far as stationary waves are concerned. Vibrations are transmitted to the bottom of the instrument by a wooden rod called the “sounding post,” which extends from the belly downwards (SP in Fig. 7–14 ). The sound waves originate largely with the vibration of the top and the bottom of the violin. These areas are enormously larger than the surface areas of the strings and the dipole components are separated in space by a much greater acoustical distance than are those of the strings. For these reasons the body of the instrument is an efficient radiator.

  FIG. 7–14 . Cross section of the body of a violin.

  The harmonic content of a note played upon a violin is a complex function of a number of different factors. The most obvious are the manner of excitation (bowing or plucking); the position on the string where the excitation occurs (although this is apparently of minor importance); the complex modes of vibration of the various sides of the instrument body, which are impossible to predict on any analytical basis; and the nature and shape of the air cavity within the instrument, which may sometimes act as a Helmholtz resonator and sometimes, for the higher frequencies, may act in the manner of a pipe closed at one end. The wooden body possesses internal dissipation qualities due partly to the presence of joints and partly to the nature of the wood itself. Such dissipation is greater for the higher frequencies than for the lower, and so contributes in an important way to the strength or weakness of the upper harmonics.

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