Physics
Amplitude of Wave
The amplitude of a wave refers to the maximum displacement of a particle from its equilibrium position as the wave passes through it. In simpler terms, it is the height of the wave from its resting position to its highest point. The amplitude of a wave is directly proportional to the energy it carries.
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4 Key excerpts on "Amplitude of Wave"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Such a graph is equivalent to a photograph of the wave taken at one instant in time and shows the disturbance that exists at each point along the Slinky’s length. As marked on this graph, the amplitude A is the maximum excursion of a particle of the medium (i.e., the Slinky) in which the wave exists from the particle’s undisturbed position. The amplitude is the distance between a crest, or highest point on the wave pattern, and the undisturbed position; it is also the distance between a trough, or lowest point on the wave pattern, and the undisturbed position. The wavelength λ is the horizontal length of one cycle of the wave, as shown in Figure 16.5a. The wave- length is also the horizontal distance between two successive crests, two successive troughs, or any two successive equivalent points on the wave. Part b of Figure 16.5 shows a graph in which time, rather than distance, is plotted on the horizontal axis. This graph is obtained by observing a single point on the Slinky. As the wave passes, the point under observation oscillates up and down in simple har- monic motion. As indicated on the graph, the period T is the time required for one complete up/down cycle, just as it is for an object vibrating on a spring. The period T is related to the frequency f, just as it is for any example of simple harmonic motion: f = 1 _ T (10.5) Vertical position of the Slinky Undisturbed position (a) At a particular time Distance Wavelength = λ A A Vertical position of one point on the Slinky (b) At a particular location Time Period = T A A FIGURE 16.5 One cycle of the wave is shaded in color, and the amplitude of the wave is denoted as A. 16.3 The Speed of a Wave on a String 481 The period is commonly measured in seconds, and frequency is measured in cycles per second, or hertz (Hz).
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Many mechanical quantities can occur as excitation or disturbance in a wave, e.g., pressure p or displacement r. The corresponding wave equation with velocity v then follows from the fundamental equations of mechanics. We will look more closely at these basic equations later, but in essence they will have the same form as the equations discussed earlier. If we find such a wave equation in the treatment of a system, we know that waves can occur and we know their speed of propagation. 4.4 Waves Are Transporting Energy With a continuous wave of any type, energy is transported. The energy flux of a wave depends on the kinetic energy of the oscillating elements, as well as the propagation velocity. As an example, consider again the wave traveling along a rope transversally. The energy of the moving rope element dx with the mass dm = Aρ dx (ρ = density, A = cross-section) is as follows: dE = dm 2 v 2 max = Aρ dx 2 u 2 0 ω 2 Here, we have used the fact that the maximum speed of the rope element, v max , is given by the maximum of the time derivative of the disturbance u = u 0 sin(kx − ωt): v max = ∂ u ∂ t max = u 0 ω Thus, the energy dE that traverses the rope cross-section during the time dt is given by the following: dE dt = Aρ dx 2dt u 2 0 ω 2 = ρ A 2 u 2 0 ω 2 v This energy that flows through the cross-section A per unit of time is called the intensity of the wave I: I = dE dt · 1 A = ρ v 2 u 2 0 ω 2 This definition of the intensity is valid for all types of waves. Therefore, we find that intensity ∝ u 2 0 amplitude of the wave squared ∝ ω 2 frequency of the wave squared ∝ v propagation speed of the wave Because v = λ · ν , we also find that the smaller the wavelength, the larger the transported energy. The unit of intensity is [I] = W/m 2 . We will see later that in the case of sound, another unit is often used to indicate intensities based on its logarithm. This is the decibel (dB).
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- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Thus, the resultant is a wave with a new amplitude and a phase angle whose values can be calculated from Eqs (1.116) and (1.117) respectively. From Eqs (1.116) and (1.117), it is clear that the phase angle and the amplitude R of the resultant wave changes with the initial phase difference d of the superposing waves because tan q and R is a function of d . Case 2: Two waves having the same frequency travelling in opposite direction (Production of stationary waves) Let two harmonic waves Y 1 and Y 2 having the same frequency and amplitude travelling, in opposite directions in a line in a medium be superimposed on each other. This is the superimposition of the incident wave and the reflected wave moving along a string, which Oscillations and Waves 79 gives rise to stationary waves. The incident wave and the corresponding reflected wave may be represented mathematically by ( ) sin I r t kx ω Ψ = − and ( ) sin R r t kx ω Ψ = + respectively. By the principle of superposition, the resultant displacement will be given by ( ) ( ) 1 sin sin R r t kx r t kx ω ω Ψ = Ψ + Ψ = − + + or ( ) ( ) sin sin 2 cos sin r t kx t kx r kx t ω ω ω Ψ = − + + = (1.118) Putting R = 2 r cos kx in this equation, we have sin R t ω Ψ = (1.119) Equation (1.119) is also a wave equation representing stationary wave or standing wave with amplitude R = 2 r cos kx. Thus, amplitude is a function of the position ‘ x ’ and varies from 0 to 2 r . The amplitude is 2 r when 2 cos 1 r kx = or , kx n π = n is a whole number, i.e., n = 0, 1, 2, 3, … or 2 2 2 n n n x n k π π λ λ π π π λ = = = × = The amplitude is 0 when 2 cos 0 r kx = or ( ) 2 1 2 kx n π = + or ( ) ( ) 2 1 2 1 2 4 n x n k π λ + = = + Thus amplitude is maximum (i.e., displacements of the particles of the medium are at maximum distance from the mean position) at 2 3 4 0, , , , 2 2 2 2 x λ λ λ λ = … , etc and amplitude
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