tailieunhanh - Ebook The physics of vibrations and waves (6th edition): Part 2

(BQ) Part 2 book "The physics of vibrations and waves" has contents: Waves in more than one dimension, fourier methods, waves in optical systems, interference and diffraction, wave mechanics, non linear oscillations and chaos, non linear waves, shocks and solitons. | 9 Waves in More than One Dimension Plane Wave Representation in Two and Three Dimensions Figure shows that in two dimensions waves of velocity c may be represented by lines of constant phase propagating in a direction k which is normal to each line, where the magnitude of k is the wave number k ¼ 2 = . The direction cosines of k are given by l¼ k1 ; k m¼ k2 k 2 2 where k 2 ¼ k 1 þ k 2 and any point rðx; yÞ on the line of constant phase satisfies the equation lx þ my ¼ p ¼ ct where p is the perpendicular distance from the line to the origin. The displacements at all points rðx; yÞ on a given line are in phase and the phase difference between the origin and a given line is ¼ 2 2 (path difference) ¼ p ¼ k Á r ¼ k 1x þ k 2y ¼ kp Hence, the bracket ð!t À Þ ¼ ð!t À kxÞ used in a one dimensional wave is replaced by ð!t À k Á rÞ in waves of more than one dimension, . we shall use the exponential expression e ið!tÀkÁrÞ In three dimensions all points rðx; y; zÞ in a given wavefront will lie on planes of constant phase satisfying the equation lx þ my þ nz ¼ p ¼ ct The Physics of Vibrations and Waves, 6th Edition H. J. Pain # 2005 John Wiley & Sons, Ltd 239 240 Waves in More than One Dimension k2 y l= k k1 m= k1 Crest Trough k k2 k lx + my = p = ct p λ 2 r(x ⋅y) k ⋅ r = k1x + k2y = kp x Figure Crests and troughs of a two-dimensional plane wave propagating in a general direction k (direction cosines l and m). The wave is specified by lx þ my ¼ p ¼ ct, where p is its perpendicular distance from the origin, travelled in a time t at a velocity c where the vector k which is normal to the plane and in the direction of propagation has direction cosines l¼ k1 ; k m¼ k2 ; k n¼ k3 k 2 2 2 (so that k 2 ¼ k 1 þ k 2 þ k 3 Þ and the perpendicular distance p is given by kp ¼ k Á r ¼ k 1 x þ k 2 y þ k 3 z Wave Equation in Two Dimensions We shall consider waves propagating on a stretched plane membrane of negligible thickness having a mass .