Physics
Linear Wave
A linear wave is a type of wave that follows the principle of superposition, meaning that the overall wave is the sum of the individual wave components. In a linear wave, the amplitude and frequency remain constant, and the wave propagates without distortion. This type of wave is commonly studied in physics and has applications in various fields such as acoustics and optics.
Written by Perlego with AI-assistance
Related key terms
1 of 5
8 Key excerpts on "Linear Wave"
eBook - PDF- Mitsuhiro Tanaka(Author)
- 2022(Publication Date)
- Springer(Publisher)
This is because the degree of penetration of motion in the direction toward the bottom changes depending on the wavelength, and also because the ratio of kinetic energy in the hori- 11 For a general theory of small oscillation around an equilibrium, see, for example, Chapter 6 of Goldstein [2]. 56 3. BASICS OF LINEAR WATER WAVES zontal direction to that in the vertical direction changes depending on the wavelength. It can be said that this wavelength dependence of “inertia” produces the wavelength dependence of the frequency, that is, the dispersion of water waves. 3.3.3 ENERGY FLUX AND VELOCITY OF ENERGY PROPAGATION Next, let us consider the energy flux and energy propagation velocity associated with the linear sinusoidal wave (3.46). The energy flux, that is, the flow of energy F x in the positive x direc- tion per unit time, which crosses a vertical cross section perpendicular to the wave propagation directions, is composed of two parts: (1) by a portion of water with energy crossing the section; and (2) by the excess pressure generated by the wave doing work. If the wave amplitude is denoted as a, the energy density is O.a 2 / and the flow velocity is O.a/, so the part (1) is a quantity of O.a 3 /. On the other hand, in the part (2), the excess pressure by the wave is O.a/, and the displacement per unit time, that is, the flow velocity is O.a/, so the magnitude of its contribution is of O.a 2 /. Therefore, in linear theory where a is very small, the part (1) can be ignored compared to the part (2). Linearizing the Euler equation 12 which is an equation of motion for an inviscid fluid, and write its x component we get @u @t D 1 @p @x : (3.66) By using u D @ =@x here, we obtain p D @ @t D c @ @x : (3.67) Here, for the second equal sign, we have used the fact that @ @t D c @ @x holds because the sinu- soidal wave translates at a speed c .
eBook - PDF- Charles A. Bennett(Author)
- 2022(Publication Date)
- Wiley(Publisher)
1 1 The Physics of Waves The solution of the difficulty is that the two mental pictures which experiment lead us to form — the one of the particles, the other of the waves — are both incomplete and have only the validity of analogies which are accurate only in limiting cases. Heisenberg 1.1 Introduction The properties of waves are central to the study of optics. As we will see, light (or more prop- erly, electromagnetic radiation) has both particle and wave properties. These complementary aspects are a result of quantum mechanics, and prior to the early 1900s, there were two schools of thought. Newton postulated that light consists of particles, while contemporaries Huygens and Hooke promoted a wave theory of light. The matter seemed settled with Young’s important double-slit experiment offering clear experimental evidence that light is a wave. Maxwell’s sweeping theory of electromagnetism finally provided a deep and complete description of electromagnetic waves that we consider in detail in Chapter 2. Although current theories of optics include both wave and particle descriptions, the wave picture still forms the bedrock of most optical technology. In this chapter, we will outline some general properties that apply to traveling waves of all types. 1.2 One-Dimensional Wave Equation Mechanical waves travel within elastic media whose material properties provide restoring forces that result in oscillation. When a guitar string is plucked, it is displaced away from its equilibrium position, and the mechanical energy of this disturbance subsequently propagates along the string as traveling waves. In this case, the waves are transverse, meaning that the displacement of the medium (the string) is perpendicular to the direction of energy travel. Acoustic waves in a gas are longitudinal, meaning that the gas molecules are displaced back and forth along the direction of energy flow as regions of high and low pressure are created along the wave.
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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 3 Fundamental Physics Concepts 1. Wavelength Wavelength of a sine wave , λ, can be measured between any two points wi th the same phase, such as between crests, or troughs, or corresponding zero crossings as shown. In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats. It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to periodic waves of non -sinusoidal shape. The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. ________________________ WORLD TECHNOLOGIES ________________________ Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a periodic variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are periodic variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary periodically in both lattice position and time. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves.
eBook - PDF- Oliver Bühler(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
1.7 Notes on the literature 21 More specialized descriptions of geophysical fluid dynamics (GFD) appro- priate for atmosphere and ocean flows are available in the textbooks Salmon (1998) and Vallis (2006). 2 Linear Waves Linear Wave theory has a special place in applied mathematics. For exam- ple, the powerful concepts of Linear Wave theory, such as dispersion, group velocity or wave action conservation, are fundamental for describing the be- haviour of solutions to many commonly occurring partial differential equa- tions (PDEs). Also, whilst it is certainly not true that every Linear Wave problem has an explicit general solution, it is true that every linear problem can be approached by using linear thinking, i.e., by building up more com- plex solutions out of superpositions of simpler solutions. In some cases, this procedure can be carried to its logical conclusion and the complete general solution to a problem can be formulated as a sum over special solutions. For example, this works for PDEs with constant coefficients in a periodic domain, for which the general solution can be written as a sum of plane waves described mathematically by a Fourier series. But even in cases where there is no explicit general solution, the possibility to develop special solutions using asymptotic methods and the ability to combine several simple solutions to form a more complex solution always deepens our understanding of the underlying problem, and such an improved understanding could then be used to aid a numerical simulation for situations of particular interest, for example. Thus time spent studying Linear Wave theory is time well spent. We are particularly interested in the behaviour of small-scale waves prop- agating on an inhomogeneous basic state, because this is the natural setting for unresolved waves interacting with a resolved mean flow in a numerical model.
eBook - PDFOcean Waves and Oscillating Systems: Volume 8
Linear Interactions Including Wave-Energy Extraction
- Johannes Falnes, Adi Kurniawan(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
CHAPTER FOUR Gravity Waves on Water The subject of this chapter is the study of waves on an ideal fluid, namely a fluid which is incompressible and in which wave motion takes place without loss of mechanical energy. It is also assumed that the fluid motion is irrotational and that the wave amplitude is so small that linear theory is applicable. Starting from basic hydrodynamics, we shall derive the dispersion relationship for waves on water which is deep or otherwise has a constant depth. Plane and circular waves are discussed, and the transport of energy and momentum associated with wave propagation is considered. The final parts of the chapter introduce some concepts and derive some mathematical relations, which turn out to be very useful when, in subsequent chapters, interactions between waves and oscillating systems are discussed. 4.1 Basic Equations: Linearisation Let us start with two basic hydrodynamic equations which express conservation of mass and momentum, namely the continuity equation ∂ρ ∂ t + ∇ · ( ρ v) = 0 (4.1) and the Navier–Stokes equation D v Dt ≡ ∂ v ∂ t + ( v · ∇) v = − 1 ρ ∇p tot + ν∇ 2 v + 1 ρ f. (4.2) Here ρ is the mass density of the fluid, v is the velocity of the flowing fluid element, p tot is the pressure of the fluid and ν = η/ρ is the kinematic viscosity coefficient, which we shall neglect by assuming the fluid to be ideal. Hence, we set ν = 0. Finally, f is external force per unit volume. Here we consider only gravitational force—that is, f = ρ g, (4.3) 62 4.1 BASIC EQUATIONS: LINEARISATION 63 where g is the acceleration due to gravity. With the introduced assumptions, Eq. (4.2) becomes ∂ v ∂ t + ( v · ∇) v = − 1 ρ ∇p tot + g. (4.4) For an incompressible fluid, ρ is constant, and Eq. (4.1) gives ∇ · v = 0. (4.5) Further, we assume that the fluid is irrotational, which mathematically means that ∇ × v = 0.
- Lokenath Debnath(Author)
- 2008(Publication Date)
- ICP(Publisher)
Its absence in the solution for fully-dispersed waves can only be due to the heavy attenua-tion associated with the frequency range where dispersion is present, which causes the wave energy propagated upstream to be supercritically damped. As a follow-up of his work in 1956 that was dedicated to Sir G.I. Tay-lor, Lighthill developed mathematical theory of gas dynamics interacting with gas physics. He then published two major papers “Dynamics of a Dis-sociating Gas, Part 1” and “Equilibrium Flow, Part 2, Quasi-equilibrium Transfer Theory” in the Journal of Fluid Mechanics in 1957. He dealt with predictions on the velocity field originated from acoustic noise and a gener-alized turbulence in a layer overlaying a convectively unstable atmospheric region. 8.2 Kinematic Waves Classical wave motions are described by Newton’s second law of motion together with some reasonable assumptions relating a stress to a displace-ment (as in gravity waves), to a strain (as in nondispersive longitudinal and transverse waves), or to a curvature (as in capillary waves and flexural waves). In contrast with the case of dynamic waves, a class of waves is called kinematic waves when an appropriate functional relation exists between the density and the flux of some physically observed quantity. Kinematic waves are not at all waves in the classical sense, and they are physically Linear and NonLinear Waves in Fluids 189 quite different from the classical wave motions involved in dynamical sys-tems. They describe, approximately, many important real-world problems including traffic flows on long highways, flood waves in rivers, roll waves in an inclined channel, and chromatographic models in chemistry. Lighthill and Whitham (1955a,b) first gave a general and systematic treatment of kinematic waves and applications.
eBook - PDF- David G. Andrews, Conway B. Leovy, James R. Holton(Authors)
- 1987(Publication Date)
- Academic Press(Publisher)
Waves can also be separated into stationary waves, whose surfaces of constant phase are fixed with respect to the earth, and traveling waves, whose phase surfaces move. Since information propagates with the group velocity (Section 4.5) and not with the phase speed, propagation can still occur in stationary waves. (We here use the adjective steady to denote waves whose amplitudes are independent of time, and transient for waves whose amplitudes are time-varying; see Section 3.6. Some authors use these terms as synonyms for our stationary and traveling, while another definition of transient is mentioned in Section 5.1. What we have called stationary waves are sometimes also known as standing waves; however, the latter name is best reserved for waves with fixed nodal surfaces as typified, say, by a velocity disturbance u' oc cos kx cos ωί.) The final general form of classification that we shall mention distinguishes waves that do not lead to any mean-flow acceleration from those that do. The former category includes waves that are linear, steady, frictionless, and adiabatic (see Section 3.6), while the latter usually includes any wave that is transient or nonconservative; however, nonLinear Waves can sometimes satisfy nonacceleration conditions if they are steady and conservative. 4.2 Wave Disturbances to a Resting Spherical Atmosphere When a stratified spherical atmosphere, at rest with respect to the rotating planet, undergoes small disturbances, an important class of wave motions 152 4 Linear Wave Theory results. The study of such a system originated with Laplace in the early nineteenth century, and has led to many insights into atmospheric behavior; in particular it underlies the theory of tides (Section 4.3) and global normal modes (Section 4.4). We start with the linearized equations [Eqs. (3.4.2)], and set the basic flow ü to zero; thus θ φ also vanishes, by Eq. (3.4.1c). Moreover, we use Eq. (3.4.2c) to substitute for Θ' in Eq.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
