Physics
Properties of Waves
The properties of waves include amplitude, wavelength, frequency, and speed. Amplitude is the maximum displacement of a wave from its rest position, while wavelength is the distance between two consecutive points in a wave that are in phase. Frequency refers to the number of complete wave cycles that pass a given point per unit time, and speed is the rate at which a wave travels through a medium.
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10 Key excerpts on "Properties of Waves"
eBook - ePub- Alexander McPherson(Author)
- 2011(Publication Date)
- Wiley-Blackwell(Publisher)
CHAPTER 4 WAVES AND THEIR PROPERTIESIn a mathematical sense, a periodic wave is any function f (x ) whose value varies in a repetitive and perfectly predictable manner over discrete intervals of some variable x . A physical way of describing waves is that they are some property of the medium in which they exist that changes in a regular and periodic manner as a function of the distance from some point, or as a function of time if one stands at a fixed point in space and measures the unique property. For sound, the property may be pressure; for waves in the water, it may be height above or below the surface; for light or X rays, the properties are electromagnetic.THE Properties of WavesWaves have quantifiable properties. In our physical world of time and space, the maximum value that a wave can attain is its amplitude. The distance it travels before it repeats is its wavelength, the time required for it to travel one wavelength, or to complete one full oscillation is its period. The number of periods it completes per unit time is its frequency. Period and frequency are inverses of one another. We normally say that the period of a wave corresponds to 360° (or 2π ), and that any point within the period corresponds to some angle between zero and 360°.A simple wave with which most of us are familiar, seen inFigure 4.1a, is the sine wave. It is like the ripples on smooth water when you cast a stone. As the wave progresses from its point of origin it rises continuously to a maximum, decreases continuously through zero to its negative minimum, and then returns in the same way to its starting point at zero. We can have more complex waves like that inFigure 4.1b, that experience complicated variations over their period, and we can, as in Figures 3.22 and 3.23 from the previous chapter, have waves in two or three dimensions. The important point, however, is that any wave, no matter how complex, begins again and repeats its course after completing a period or traversing a wavelength. Aside from its complexity, it is otherwise no different than a sine wave. It follows that if we measure n (λ ) along the direction of propagation from any point on a periodic wave, we find ourselves with exactly the same value of f (x ), but n
eBook - PDFWaves and Oscillations in Nature
An Introduction
- A Satya Narayanan, Swapan K Saha(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
The incident idealized photon is monochromatic in nature. The corresponding classical wave has the same extent as well. For a wave traveling through a medium, a crest is seen moving along from particle to particle. This crest is followed by a trough which, in turn, is followed by the next crest. A distinct wave pattern in the form of a sine wave is observed traveling through the medium. This sine wave pattern continues to move in uninterrupted fashion until it encounters another wave along the medium or until it encounters a boundary with another medium. This type of wave pattern is referred to as a traveling wave; for instance, an ocean wave is falling under such category. The wave properties that are described by the following quantities are interrelated. 1. Amplitude: The amplitude of a wave is the maximum displacement of a particle from its equilibrium position as the wave passes through it (see Figure 1.3). It is measured in meters (m). amplitude y x λ FIGURE 1.3 : Amplitude pattern. 2. Frequency: The number of cycles per unit of time is called the frequency, ν , of oscillations caused by the wave. The unit of frequency is hertz (Hz; cycles per second). The quantity ν = ω 2 π = 1 T (1.1) Introduction to Waves and Oscillations 11 where ω is the angular frequency, which is 2 π times the frequency, ν , and T the period of the vibrations; one complete cycle of the wave is associated with an angular displacement of 2 π radians. The angular frequency, ω , of a wave is the number of radians per unit of time at a fixed position. 3. Path difference: The path length, l , is the distance through which a wavefront recedes when the phase increases by δ and is expressed as l = v ω δ = λ 2 π δ = λ 0 2 πn δ (1.2) where v is the velocity, λ the wavelength, λ 0 the wavelength in free space (vacuum), n = c v (1.3) the refractive index for refraction from vacuum into that medium, and c the speed of light in free space.
eBook - PDF- John Kimball(Author)
- 2015(Publication Date)
- CRC Press(Publisher)
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. ( Wave speed ) = ( Wavelength ) × ( Frequency ) For example, if the frequency is 6 hertz (cycles per second), then six waves pass a fixed point each second. If the wavelength is 8 meters, then the six wave peaks will have moved 6 × 8 meters in 1 second, and the speed is 48 meters/second. The speed of a light wave (in vacuum) is a constant regardless of its frequency or its amplitude. Audible sound wave speed is almost constant. The constant speed means plane wave shapes do not change in time. Water wave speeds do depend on the wavelength, which Amplitude Distance along wave Figure 5.2 The sine wave shape. For water waves, the amplitude is the height. For sound, it is atomic displacement or pressure. For electromagnetic waves, it is the electric field. Figure 5.1 Water waves spreading from a central point. 149 WAVES means the wave shapes evolve. This is one reason water waves are more complicated. Waves contain energy and transmit energy at the wave speed (assuming a constant speed). The energy in each cubic meter of wave is proportional to the square of its amplitude. Multiplying the energy in each cubic meter times the wave speed gives the “energy flux,” or the number of joules hitting 1 square meter of surface each second. The most important energy flux is from sunlight. The sun’s 1370 watts/ square meter keep us from freezing in the dark. 5.2.2 Sound in Solids The basic mechanism of sound propagation in solids is shown in Figure 5.3. This is a greatly magnified segment of a crystal lattice showing only three of the atoms (spheres) and the atomic forces (springs) that make up the periodic crystal structure.
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- Charles A. Bennett(Author)
- 2015(Publication Date)
- Wiley(Publisher)
CHAPTER 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 Contents 1.1 Introduction 1 1.2 One-Dimensional Wave Equation 2 1.3 General Solutions to the 1-D Wave Equation 5 1.4 Harmonic Traveling Waves 8 1.5 The Principle of Superposition 11 1.6 Complex Numbers and the Complex Representation 12 1.7 The Three-Dimensional Wave Equation 17 1.1 INTRODUCTION The Properties of Waves are central to the study of optics. As we will see, light (or more properly, electromagnetic radiation) has both particle and wave properties. These comple- mentary aspects are a result of quantum mechanics, and prior to the early 1900s there were 1 2 THE PHYSICS OF WAVES two schools of thought. Newton postulated that light consists of particles, while contem- poraries Huygens and Hooke promoted a wave theory of light. The matter seemed settled with Young’s important double-slit experiment that offered 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.
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
2 The waves move with a velocity depending upon the properties of the medium. The waves remain stationary and do not move. 3 Each particle of the medium executes periodic motion about their mean position with the same amplitude. Except the node, all the particles of the medium execute SHO with varying amplitude. 4 There is a continuous change of phase from particle to particle. All the particles between two consecutive nodes are at the same phase, but differ in phase by p from those in the preceding as well as succeeding similar segments. 5 At any instant all the particles do not come together in the mean position, they pass their mean position in succession but with the same velocity. All the particles pass their mean position at a time, but with different velocities. Oscillations and Waves 57 6 Each particle of the medium undergoes similar change of pressure and density There is no change of pressure and densities at the antinodes while there is maximum change of pressure and densities at the nodes. 7 There is transmission of energy across every plane in the direction of propagation of waves. There is no flow of energy across any plane. 8 A complete wavelength contains a compression and rarefaction in the case of longitudinal waves and crest and trough in the case of transverse waves. The wavelength is the distance between two alternate nodes and anti nodes. 9 Compression and rarefaction move from point to point throughout the medium. The compression and rarefaction do not move from point to point; they simply appear at and disappear at certain equidistance fixed points. 10 No particle of the medium is permanently at rest.
eBook - ePub- Nils O. Petersen(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
.Figure 17.1: Illustration of the spatial and temporal properties of a wave.The parameter ω is the angular frequency which is inversely proportional to the period, τ , which describes the temporal separation of the peaks of the wave so that ω = 2π /τ = 2πν . The frequency, ν , is the inverse of the period: ν = 1/τ .The parameter v in Equation 17.1 is the phase velocity, which is related to the wavelength and the period as, and describes the speed with which the peak of the wave moves either along the position axis or the time axis.v =ω k=λ τSome of these parameters are illustrated in Figure 17.1 .Note that the arguments used in Chapter 4 to define the concept of a wave packet (see Figures 4.3 and 4.4 ) apply equally to the general wave equations above, so that if there are uncertainties in the wavelength or period, there will be a corresponding wave packet, which in the context of the electromagnetic radiation, constitutes a photon.17.2 ELECTROMAGNETIC RADIATION
17.2.1 The Maxwell equations
The properties of electromagnetic radiation is defined by the Maxwell Equations, which interrelate the electric field vector,(17.3), and the magnetic field vector,E →. They areB →∇ →·E →=ρ ε∇ →×E →= -∂B →∂ t∇ →·B →= 0∇ →×B →= μ ε.∂E →∂ tThe Maxwell Equations depend on the properties of the medium that the electric and magnetic fields are progressing through. Thus, ρ is the charge density of the medium —in most cases there are no free charges, and hence this may be zero; ɛ is the permittivity of the medium, which affects both the electric and the magnetic field; and μ is the permeability of the medium, which affects only the magnetic field.The divergence (the dot-product) is a scalar while the curl (the cross-product) is a vector perpendicular to the plane of the two vectors. This places the electric field in one plane and the magnetic field in a plane perpendicular to the electric field.
eBook - PDF- Kenneth S. Krane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
Chapter 4 THE WAVELIKE PROPERTIES OF PARTICLES Just as we produce images from light waves that scatter from objects, we can also form images from “particle waves.” The electron microscope produces images from electron waves that enable us to visualize objects on a scale that is much smaller than the wavelength of light. The ability to observe individual human cells and even subcellular objects such as chromosomes has revolutionized our understanding of biological processes. It is even possible to form images of a single atom, such as this cobalt atom on a gold surface. The ripples on the surface are caused by electrons from gold atoms reacting to the presence of the intruder. Drs. Ali Yazdani & Daniel J. Hornbaker / Science Source 106 Chapter 4 The Wavelike Properties of Particles In classical physics, the laws describing the behavior of waves and particles are fundamentally different. Projectiles obey particle-type laws, such as Newto- nian mechanics. Waves undergo interference and diffraction, which cannot be explained by the Newtonian mechanics associated with particles. The energy carried by a particle is confined to a small region of space; a wave, on the other hand, distributes its energy throughout space in its wavefronts. In describing the behavior of a particle, we often want to specify its location, but this is not so easy to do for a wave. How would you describe the exact location of a sound wave or a water wave? In contrast to this clear distinction found in classical physics, quantum physics requires that particles sometimes obey the rules that we have previ- ously established for waves, and we shall use some of the language associated with waves to describe particles. The system of mechanics associated with quantum systems is sometimes called “wave mechanics” because it deals with the wavelike behavior of particles. In this chapter, we discuss the experimental evidence in support of this wavelike behavior for particles such as electrons.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
Wavelength is commonly designated by the ________________________ WORLD TECHNOLOGIES ________________________ Greek letter lambda (λ). The concept can also be applied to periodic waves of non -sinu-soidal 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. 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. For example, in waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength. Sinusoidal waves In linear media, any wave pattern can be described in terms of the independent propa-gation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by: ________________________ WORLD TECHNOLOGIES ________________________ Refraction: when a plane wave encounters a medium in which it has a slower speed, the wavelength decreases, and the direction adjusts accordingly. where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×10 8 m/s.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
Wavelength is commonly designated by the ________________________ WORLD TECHNOLOGIES ________________________ 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. 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. For example, in waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength. Sinusoidal waves In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by: ________________________ WORLD TECHNOLOGIES ________________________ Refraction: when a plane wave encounters a medium in which it has a slower speed, the wavelength decreases, and the direction adjusts accordingly. where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×10 8 m/s.
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