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37953 results in Cambridge Textbooks

Page 1883 of 1898

  • Hints for exercises

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 483-507

  • Summary

    Chapter I

    1. Resolve in fig. I.1.

    2. Sketches need to be three-dimensional versions of fig. 1.2: you need either a length dz or a length corresponding to dλ and ds will no longer be in the plane of r and θ.

    Otherwise you need to find formulae corresponding to

    dx = dr cos θ − r sin θ dθ,

    and calculate ds2 from

    ds2 = dx2 + dy2 + dz2.

    3. The formulae for are in §2(c) You must differentiate these, and then express the result in terms of these unit vectors.

    4. Taylor's theorem in one-dimension is

    when (φ is smooth enough (twice differentiable with continuous second derivative is certainly enough), and where 0 < θ < 1. The last term on the right is less than some constant times h2 if φ″ is continuous on (x, x + h), because φ″ is then bounded in that interval.

    Remember that h2 = hihi = hkhk.

    5. (i) ∇ × (φA) has ith component

    (ii) When is ∂2u/∂x∂t = ∂2u/∂t∂x?

    (iii) ∇ · A = ∂Ai/∂xi, and here A = φ∇ψ, and ∇ψ has ith component ∂ψ/∂xi.

    (iv), (v) ε123 = + 1, ε213 = −1: look for terms which have these ε in them. Then use ∂2/∂x1∂x2 = ∂2/∂x2∂x1 for smooth functions.

    (vi) This is and use a theorem in §4(c).

    6. ∇ × ∇ = 0, and write what remains as

    ∇ × (A × B),

    which works out rather like Q5(vi). Finally put the proper value back for B.

    7. Use Q5(iii), and the divergence theorem from §5(b).

    8. For ∇ · A proceed as in §6(a), and use formulae for derivatives of from §2(c) and Q3. For ∇ × A evaluate the determinant in §6(c).

  • XII - Sound waves in fluids

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 248-288

  • Summary

    Background

    The basic facts of the propagation and generation of sound are well known. Sound is a pressure disturbance in air (or water) which travels at a rather high speed. Sound is generated in regions of fluid motion; sometimes a motion of solid boundaries is involved, as in a drum; sometimes not, as in a flute. Sound is received in our ears (or by a microphone) where the pressure oscillations are converted into electrical signals.

    The speed of sound is typically around 330–40m s−1 in still air near ground level, and around 1400–50ms−1 in water. Sound also travels through solid materials in the form of elastic waves; we cannot do the theory for such waves here, though it is of the same general type as what we shall do. Often the speed of sound in solids is much higher than the speed of sound in air.

    The pressure fluctuations in sound in air are typically between 10−4 and 1 N m−2, so that the ratio of pressure fluctuations to ambient pressure in the atmosphere lies between

    10−9 and 10−5.

    The smallness of this quantity is one of the basic facts in the simple theory of sound. It is quite possible to get much higher pressure fluctuations, in explosions or in jet engines, so that the ratio of pressures is not small: the simple theory of sound cannot apply to such cases.

    The frequencies of audible sound waves lie between about 20 Hz and 2 × 104 Hz. Below about 20 Hz, oscillations are felt by the body rather than heard by the ear, and so may not be classed as sound waves, though they are still described by the same mathematics – sometimes the word infrasound is used. Similarly, oscillations above 2 × 104 Hz cannot be heard by everyone; again, the mathematics is mainly the same, and the word ultrasound is used.

  • XIII - Water waves

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 289-324

  • Summary

    Background

    It is part of everyone's experience that waves and oscillations occur on the surface of water, and they indeed form a wide range of phenomena for any mathematical theory to deal with. Ocean waves from the tsunami – the ‘tidal wave’ which is generated by an earthquake and which may travel at high speed for huge distances – and the tides themselves, down to small ripples in a calm sea. Other types of disturbance occur on rivers, such as the tidal bore which travels up the River Severn, and the surface waves which can be caused by an irregularity on the bottom. A lake may have a ‘seiche’ in it, which is an oscillation to-and-fro of the surface of the fluid (very much like the same phenomenon in a bath). On a wet day you may see waves on thin sheets of water flowing down a road.

    Real waves have awkward properties: they may break, or have their tops blown off by the wind; they may be suppressed by a small amount of oil on their surfaces; or be running over water of non-uniform depth. We model the waves that you see on water in rather a drastic fashion, to give approachable mathematics, and get some very interesting results; but we cannot cope with too much reality, and so will not predict every visible effect.

    This chapter deals with the easy aspects of waves in oceans, lakes and smaller containers. We cannot deal here with such interesting topics as:

  • (i) the generation of waves by the wind;

  • (ii) the interactions of waves which meet;

  • (iii) the changes in waves as they approach the shore;

  • (iv) waves due to ships, or interaction of waves and ships;

  • (v) energy generation from waves.

    • Some aspects of waves on rivers will be covered in Chapter XV, so we assume in this chapter that the water is at rest apart from near its disturbed surface.

  • XI - Potential theory

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 205-247

  • Summary

    The velocity potential and Laplace's equation

    (a) Occurrence of irrotational flows

    If a closed curve in an in viscid fluid for which ρ = f(p) has at some time no circulation round it, then by Kelvin's theorem there is never any circulation round the curve. Of course there are no such ‘ideal’ fluids around, but we have seen that at high Reynolds number and away from boundaries and other awkward regions, a fluid will behave in a near enough ideal fashion. Hence we expect that there can be large regions of flow which have no circulation round any circuit, and hence no vorticity. Thus it is well worth discussing irrotational flows, which have

    ∇ × v = 0.

    Naturally we must not try to use irrotational flow theory in those regions where we have already seen vorticity to be inevitably developed from the no-slip condition and the diffusive action of viscosity, such as in boundary layers, wakes, eddies and enclosed regions. But away from these regions we can use irrotational flow theory provided that the flow is started, or arrives, with no vorticity. For example, the following flows are closely irrotational.

  • (i) Flow of air round a streamlined aeroplane wing or body. The aircraft flies into air that is effectively at rest, and the boundary layers and wake are thin enough to be neglected at a first approximation. Such flows will be discussed in detail in Chapter XVII.

  • (ii) Waves on the surface of reasonably deep water. The boundary condition at the surface does not bring in a noticeable boundary layer because the air is so much less dense than the water. Water waves are dealt with in Chapter XIII.

  • (iii) Sound waves in air (or water) are of such short time scale that diffusive effects have no time to act, and hence irrotational flow is an appropriate model in many cases. Sound waves are considered in Chapter XII.

  • (iv) In certain cases irrotational flow theory is useful for some regions of the flow of a uniform stream past a blunt body.

  • VII - Thermodynamics

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 111-126

  • Summary

    Basic ideas and equations of state

    (a) Functions of state

    In our discussions so far we have introduced five functions to describe the motion of a fluid:

  • (i) density ρ;

  • (ii) velocity v, three components;

  • (iii) pressure p.

    • So far there are only four equations in sight for these functions, the equation of mass-conservation, and the equation of momentum change (three components). We clearly need something like an energy equation, which must include energy of compression of a gas; and it is well known that compressing a gas heats it (try pumping a bicycle tyre), so that temperature will come in as well, as a related quantity to heat energy. In order to discuss these ideas clearly, we must set up some, but not too much, of the theory of thermodynamics. By the end of the chapter we will have three standard mathematical models to work with, one for liquids and two for gases under reasonable conditions; but we should also have a good idea of when and why these models are adequate – a model that is not understood is a model that will be used in the wrong way.

    A new area of physical theory will have new observables, and the mathematical theory will bring in new functions which are not observable and yet which are the best ones for framing the theory. Our early stages in fluid dynamics are helped by the fact that density and velocity are common measurable concepts in other forms of mechanics; and the idea of a stress tensor, which though not directly observable is extremely useful, is not too unlikely a generalisation from a force vector. But thermodynamics brings in ideas which may be quite new to those without much background in physics, so the early stage of thermodynamics needs careful attention, to note where axioms based on experiments are being brought in, and where new definitions are being made.

  • Preface

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp ix-x

  • Summary

    An author makes his excuses in a preface, so here are mine. For a number of years at Bristol University we tried to find a suitable text to introduce fluid dynamics to second year mathematics students, and failed. The modern texts with the ‘right’ attitude to the subject were too hard for a first course, the older texts were dominated by potential theory and unrealistic examples. This text has been tried in draft form for several years on our students, and has been judged ‘hard, but interesting’. New work in mathematics is always hard, but I believe that the level chosen here is a suitable one.

    I apologise to my colleagues for the gross over-simplification of their work and their subject which is committed in this book. And also for the errors and misapprehensions – students, beware! all texts have mistakes in them. I thank my colleagues for helpful comments and discussions over many years; I also thank a succession of seminar speakers for maintaining my awareness of the full range of fluid dynamics.

  • 3 - Learning how to talk in Trackton

  • from PART I - ETHNOGRAPHER LEARNING
  • Shirley Brice Heath, Stanford University, California
  • Book:
    Ways with Words
    Published online:
    12 September 2018
    Print publication:
    07 July 1983, pp 73-112

  • Summary

    The first year

    Babies born to Trackton residents are brought home from the local hospital in the rectangular bassinet box it provides. A small supply of disposable diapers and plastic bottles also comes home with the baby, for Trackton mothers do not expect to nurse their babies, and they depend primarily on disposable diapers, accepting these as necessary since their homes have no laundry facilities. Relatively few provisions are made for the baby beforehand; a few shirts, blankets, and wrappers may be new, and shortly after the arrival home, the father, whether present in the household or not, is expected to send diapers and a supply of milk-formula. He and his family come to visit the child and arrange a time for “keepin”im” sometimes every weekend, or a week or so out of each month. If the child is born to a young mother remaining in the home of her family, and the father remains with his family, each family buys independently for the child, especially during the preschool years. The paternal grandparents may buy a snowsuit one week, and the maternal grandparents may buy a heavy sweater the next. The provision of a planned wardrobe to consist of a certain number of particular items is not a goal; the emphasis is on the buying of items for the child. In each household of both the mother's and father's family, the child will have at least a minimal set of provisions kept there at all times.

    If the baby is not the child of a young girl whose family will be taking primary responsibility for childcare, the baby does not have the benefits of moving from one home to another. Instead, if the mother has entered a somewhat stable relationship or marriage, usually by her early twenties, the baby is brought into the home of the mother and father. There he enters a household in which any other children born to the mother before this marriage will take some responsibility for the child. The father's family will also contribute, especially if this is either his first child or the first child of this union, but subsequent children of this union may not receive a particularly favored status or spend as much time in the home of the father's family.

  • 8 - Teachers as learners

  • from PART II - ETHNOGRAPHER DOING
  • Shirley Brice Heath, Stanford University, California
  • Book:
    Ways with Words
    Published online:
    12 September 2018
    Print publication:
    07 July 1983, pp 265-314

  • Summary

    A student again

    The first chapters of this book have detailed the ways children of Trackton, Roadville, and the townspeople learn to use oral and written language. The ways of the black and white townspeople are different from the ways of either Trackton or Roadville. In businesses, mills, and schools, Roadville and Trackton residents interact with the townspeople, who are the managers of these commercial, workplace, and educational settings. For the children of Roadville and Trackton, school is the first place in which they meet on an extended basis the townspeople's ways of using oral and written language. They have come to school, bearing the high hopes of their parents and believing school can make a difference in whether or not they learn enough to enable them to move beyond their parents' workplaces. Their physical transition from home to kindergarten and the primary grades will eventually be followed by a larger transition to the commercial establishments and institutions of employment controlled by the townspeople. Intuitively, they and their parents feel language is power, and though they may not articulate precisely their reasons for needing to learn to read, write, and speak in the ways the school teaches, they believe that such learning has something to do with moving them up and out of Trackton and Roadville.

    Of what use might the detailed ethnographies of communication in Roadville and Trackton be in enabling teachers and students to bridge their different ways? The answer to this question depends on finding ways to make accessible to teachers an understanding of the differences in language and culture their students bring to their classrooms. In the 1970s, the townspeople who studied in graduate courses I taught learned to become ethnographers of their own and others' interactions and to put to use knowledge about the different ways of learning and using language which existed in the communities of their region. These teachers, in taking social science courses, examined their own habits at home and learned to recognize that they carried these home habits into the classroom just as did their students from other communities.

  • 1 - The Piedmont: Textile mills and times of change

  • from PART I - ETHNOGRAPHER LEARNING
  • Shirley Brice Heath, Stanford University, California
  • Book:
    Ways with Words
    Published online:
    12 September 2018
    Print publication:
    07 July 1983, pp 19-29

  • Summary

    A quiet early morning fog shrouds rolling hills blanketed by pine-green stands of timber, patched with fields of red clay. As the sun rises and burns off the fog, the blue sky is feathered with smoke let go from chimney stacks of textile mills: this is the Piedmont of the Carolinas.

    The Piedmont, a term used by outsiders and newcomers to the region, but never by local residents, is the “foot of the mountain” which sprawls east of the Appalachians from the southwestern hills of Virginia through North and South Carolina into Georgia and Alabama. Its textile mills are at home along the rivers and railways of the numerous towns and cities which dot the rolling terrain. From Danville, Virginia to Birmingham, Alabama, the Southern Railway cuts through the Piedmont, mingling the smoke of its locomotives with that of the textile mills along its track.

    The Piedmont includes the area west of the Atlantic Coastal Plain and east of the Southern Appalachian Mountains. Separating the central Piedmont of the Carolinas from the Coastal Plain is a strip of sand hills with no vegetation except jack pines and scrub oak. Along this strip is a geological fault called the “fall line.” River rapids and shoals mark this point at which the rushing waters of rivers falling from the Appalachians across the foothills enter into the Coastal Plain. Above this line is “the up-country,” below it the “low-country.” Through the Coastal Plain, the rivers slow their pace to meander through pine-dotted valleys to the Atlantic Ocean. Along these rivers, cities such as Columbia and Camden in South Carolina, and Raleigh in North Carolina, mark the historical limits of inland navigation.

    Here, in the eighteenth century, slaves, manning barges loaded with goods brought from the coastal port of Charleston, S.C., had to remove the goods for distribution either in these city markets or by wagon through the region above the fall line. In this back country, wagons moved along roads which after a rain could entrap wagon and horses or oxen in a mire of red or grayish clay; rivers too swift to be forded discouraged traveling merchants.

Page 1883 of 1898