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equation. To demonstrate that the spherical wave (1/r)cos(kr Ét) is a solution of (12.9), we must transform the Laplacian operator from Cartesian to polar coordinates, "2(x, y, z) ’! "2(r, ¸, Æ). The transformation is "2/"x2 + "2/"y2 + "2/"z2 ’! (1/r2)[("/"r)(r2("/"r)) + (1/sin¸)("/"¸)(sin¸("/"¸)) + (1/sin2¸)("2/"Æ2)]. (12.17) This transformation is set as a problem. If there is spherical symmetry, there is no angular-dependence, in which case, "2(r) = (1/r2)("/"r)(r2("/"r)) = "2/"r2 + (2/r)("/"r). (12.18) We can check that È = È0(1/r)cos(kr Ét) is a solution of the radial form of (12.9), Differentiating twice, we find "2È/"r2 = È0[( k2/r)cosu + (2k/r2)sinu + (2/r3)cosu], where u = kr Ét, 176 W A V E M O T I O N and "2 È/"t2 = È0(É2/r)cosu, É = kV, from which we obtain (1/V2)"2È/"t2 ["2È/"r2 + (2/r)"È/"r] = 0. (12.19) 12.6 The superposition of harmonic waves Consider two harmonic waves with the same amplitudes, È0, travelling in the same direction, the x-axis. Let their angular frequencies be slightly different É ± ´É with corresponding wavenumbers k ± ´k. Their resultant, ¨, is given by ¨ = È0ei{(k + ´k)x (É + ´É)t} + È0ei{(k ´k)x (É ´É)t} = È0ei(kx Ét)[ei(´kx ´Ét) + e i(´kx ´Ét)] = È0ei(kx Ét)[2cos(´kx ´Ét)] = AcosÆ, (12.20) where A = 2È0ei(kx Ét), the resultant amplitude, and Æ = ´kx ´Ét, the phase of the modulation envelope . The individual waves travel at a speed É/k = vÆ, the phase velocity, (12.21) and the modulation envelope travels at a speed ´É/´k = vG, the group velocity. (12.22) W A V E M O T I O N 177 In the limit of a very large number of waves, each differing slightly in frequency from that of a neighbor, dk ’! 0, in which case dÉ/dk = vG. For electromagnetic waves travelling through a vacuum, vG = vÆ = c, the speed of light. We shall not, at this stage, deal with the problem of the superposition of an arbitrary number of harmonic waves. 12.7 Standing waves The superposition of two waves of the same amplitudes and frequencies but travelling in opposite directions has the form ¨ = È1 + È2 = Acos(kx Ét) + Acos(kx + Ét) = 2Acos(kx)cos(Ét). (12.23) This form describes a standing wave that pulsates with angular frequency É, associated with the time-dependent term cosÉt. In a travelling wave, the amplitudes of the waves of all particles in the medium are the same and their phases depend on position. In a standing wave, the amplitudes depend on position and the phases are the same. For standing waves, the amplitudes are a maximum when kx = 0, À, 2À, 3À, ... and they are a minimum when kx = À/2, 3À/2, 5À/2, ...(the nodes). PROBLEMS The main treatment of wave motion, including interference and diffraction effects, takes place in the second semester (Part 2) in discussing Electromagnetism and Optics. 178 W A V E M O T I O N 12-1 Ripples on the surface of water with wavelengths of about one centimeter are found to have a phase velocity vÆ = "(±k) where k is the wave number and ± is a constant characteristic of water. Show that their group velocity is vG = (3/2)vÆ. 12-2 Show that y(x, t) = exp{x vt} represents a travelling wave but not a periodic wave. 12-3 Two plane waves have the same frequency and they oscillate in the z-direction; they have the forms È(x, t) = 4sin{20t + (Àx/3) + À}, and È(y, t) = 2sin{20t + (Ày/4) + À}. Show that their superposition at x = 5 and y = 2 is given by È(t) = 2.48sin{20t (À/5)}. 12-4 Express the standing wave y = Asin(ax)sin(bt), where a and b are constants as a combination of travelling waves. 12-5 Perhaps the most important application of the relativistic Doppler shift has been, and continues to be, the measurement of the velocities of recession of distant galaxies relative to the Earth. The electromagnetic radiation associated with ionized calcium atoms that escape from a galaxy in Hydra has a measured wavlength of 4750 × 10 10m, and this is to be compared with a wavelength of 3940 × 10 10m for the same process measured for a stationary source on Earth. Show that the measured 13 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 13.1 Definitions Two n-vectors An = [a1, a2, ...an] and Bn = [b1, b2, ...bn] are said to be orthogonal if "[i = 1, n] aibi = 0. (13.1) (Their scalar product is zero). Two functions A(x) and B(x) are orthogonal in the range x = a to x = b if +"[a, b] A(x)B(x)dx = 0. (13.2) The limits must be given in order to specify the range in which the functions A(x) and B(x) are defined. The set of real, continuous functions {Æ1(x), Æ2(x), ...} is orthogonal in [a, b] if +"[a, b] Æm(x)Æn(x)dx = 0 for m `" n. (13.3) If, in addition, +"[a, b] Æn2(x)dx = 1 for all n, (13.4) the set is normal, and therefore it is said to be orthonormal. The infinite set {cos0x, cos1x, cos2x, ... sin0x, sin1x, sin2x, ...} (13.5) in the range [ À, À] of x is an example of an orthogonal set. For example, +"[ À, À] cosxÅ"cos2xdx = 0 etc., (13.6) 180 O R T H O G O N A L F U N C T I O N S and +"[ À, À] cos2xdx `" 0 = À, etc. This set, which is orthogonal in any interval of x of length 2À, is of interest in Mathematics because a large class of functions of x can be expressed as linear combinations of the members of the set in the interval 2À. For example we can often write Æ(x) = c1Æ1 + c2Æ2 + where the c s are constants = a0cos0x + a1cos1x + a2cos2x + ... + b0sin0x + b1sin1x + b2sin2x + ... (13.7) A large class of periodic functions ,of period 2À, can be expressed in this way. When a function can be expressed as a linear combination of the orthogonal set {1, cos1x, cos2x, ...0, sin1x, sin2x, ...} , it is said to be expanded in its Fourier series. 13.2 Some trigonometric identities and their Fourier series Some of the familiar trigonometric identities involve Fourier series. For example, cos2x = 1 2sin2x (13.8) can be written sin2x = (1/2) (1/2)cos2x and this can be written sin2x = {(1/2)cos0x + 0cos1x (1/2)cos2x + 0cos3x + ... + 0{sin0x + sin1x + sin2x + ...} (13.9) ’! the Fourier series of sin2x. O R T H O G O N A L F U N C T I O N S 181 The Fourier series of cos2x is cos2x = (1/2) + (1/2)cos2x. (13.10) More complicated trigonometric identies also can be expanded in their Fourier series. For example, the identity [ Pobierz caÅ‚ość w formacie PDF ] |