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2.1.1 Wave Motion
ОглавлениеThe wave motion could be treated as the transfer of oscillation from one location to another location. A harmonic oscillation is described either by a sine or by a cosine function. A periodic oscillation has a fixed period. An oscillation repeats itself after the periodic time (T). In general, an oscillation, i.e. an oscillatory motion can have any shape such as square, triangular, and so on. However, with the help of the Fourier series, such periodic oscillations can be decomposed to the harmonic functions. Likewise, wave motion can also acquire an arbitrary shape. The arbitrary periodic shape of a wave can be decomposed into the harmonic waves.
Figure (2.1) shows the instantaneous amplitude v(t) of the wave generated by an oscillating quantity, such as an oscillating ball (particle) at the location A. It has an angular frequency of oscillation ω radian/sec and maximum amplitude Vmax. The ball is attached to a spring and immersed in a water tank. It generates a wave motion at the surface of the water. The water wave travels with velocity vp and causes another delayed oscillation, at the location B, in a similar ball attached to a spring. Through the mechanism of wave motion, the oscillation is transferred from the location A to the location B separated by distance x. Likewise, the oscillating quantity could be a charge, electric field, magnetic field, or voltage obtained from an oscillator connected to a cable.
The equation of the harmonic oscillation at location A is described by the cosine function,
(2.1.1)
The equation of harmonic oscillation that appears at location B after a delayed time t′ is
(2.1.2)
The wave motion, created by the oscillation at the location A, travels with velocity vp. The time delay in setting up the oscillation at the location B is t′ = x/vp. So the equation of oscillation at the location B is
Figure 2.1 Delayed oscillation as a wave motion‐initial oscillation v(t) at location A, and delayed oscillation v(t − x/vp) at location B. The waves propagate from the locations A to B.
Equation (2.1.3) describes the propagating wave through the medium between locations A and B. It is called the wave equation. The phase constant β of the propagating wave is defined as β = ω/vp. On dropping the subscript “b,” equation (2.1.3) is rewritten in the usual form,
In equation (2.1.4), the lagging phase ϕ = βx is caused by the delay in oscillation at the receiving end. The medium supporting the wave motion is assumed to be lossless. The wave variable v(t, x) is a doubly periodic function of both time and space coordinates as shown in Fig (2.2a and b), respectively.
The temporal period, i.e. time‐period T is related to the angular frequency ω as follows:
Figure 2.2 Double periodic variations of wave motion.
(2.1.5)
where f is the frequency in Hz (Hertz), i.e. the number of cycles per sec. The wavelength (λ) describes the spatial period, i.e. space periodicity. It is related to the phase‐shift constant (or phase constant), i.e. the propagation constant β, as follows:
(2.1.6)
The wave motion or velocity of a monochromatic wave (the wave of a single frequency) is the motion of its wavefront. For the 3D wave motion in an isotropic space, the wavefront is a surface of the constant phase. In the case of the 1D wave motion, the wavefront is a line, whereas, for the 2D wave motion on an isotropic surface, the wavefront is a circle.
Figure (2.3) shows the peak point P at the wavefront of the ID wave. The motion of a constant phase surface is known as its phase velocity vp. Thus, the peak point P at the wavefront has moved to a new location P′ in such a way that phase of the propagating wave remains constant, i.e. ωt − βx = constant. On differentiating the constant phase with respect to time t, the expression for the phase velocity is obtained:
Figure 2.3 Wave motion as a motion of constant phase surface. It is shown as the moving point P.
(2.1.7)
In a nondispersive medium, the phase velocity of a wave is independent of frequency, i.e. the waves of all frequencies travel at the same velocity. Figure (2.4) shows the nondispersive wave motion on the (ω − β) diagram. It is a linear graph. The slope of the point Q on the dispersion diagram subtends an angle θ at the origin that corresponds to the phase velocity of a propagating wave,
(2.1.8)
Thus, for a frequency‐independent nondispersive wave, the phase constant is a linear function of angular frequency.
(2.1.9)
The dispersive nature of the 1D wave motion is further discussed in sections (3.3) and (3.4) of the chapter 3. The phase velocity in a dispersive medium is usually frequency‐dependent. It is known as the temporal dispersion. In some cases, the phase velocity could also depend on wavevector (β, or k). It is known as spatial dispersion. The spatial dispersion is discussed in the subsection (21.1.1) of the chapter 21. The dispersion diagrams of the wave propagation in the isotropic and anisotropic media are further discussed in the section (4.7) of the chapter 4, and also in the section (5.2) of chapter 5.
Figure 2.4 The ω‐β dispersion diagram of nondispersive wave.
The one‐dimensional (1D) wave travels both in the forward and in the backward directions. It can have any arbitrary shape. In general, the 1D wave propagation can be described by the following wave function, known as the general wave equation:
(2.1.10)
On taking the second‐order partial derivative of the wave function ψ(t, x) with respect to space‐time coordinate x and t, the 1D second‐order partial differential equation (PDE) of the wave equation is obtained. Similarly, wave functions ψ(t, x, y) and ψ(t, x, y, z) are solutions of two‐dimensional (2D) and three‐dimensional (3D) wave equations supported by the surface and free space medium, respectively. These PDEs are summarized below:
The dispersion diagram of the 2D wave propagation over the (x, y) surface is obtained by revolving the slant line of Fig (2.4) around the ω‐axis with propagation constants βx, βy in the x‐ and y‐direction. It is discussed in subsection (4.7.4) of chapter 4.
