VIEWS: 115 PAGES: 18 POSTED ON: 11/11/2009
Wave propagation Progressive (traveling) wave (前進波) • Definition: a progressive or traveling wave is a disturbance 干擾 that moves from a source to surrounding places, and as a result of which energy 能量 is transferred from one point to another. Wave equation 波動方程 y(x,t) = A sin(ωt – kx) represents the displacement of a particle at time t and at position x 粒子在某時間t，位置x的位移是y y(x,t) = A sin(ωt – kx) • A progressive wave propagating in the positive x-axis direction. • y: displacement 位移 • x: particle position 位置 • t: time • A: amplitude 振幅 • ω: angular speed 角速度, ω= 2πf • k: wave number 波數, 2π/λ • The wave equation y(x,t) =A sin(ωt – kx) represents the displacement of a particle at time t and at position x. • Wave speed v = ω/k • ω is related to f • k is related to wavelength λ • sin(-θ) = - sinθ [t=0] • sin(π/2 – θ) = cos θ [t=π/4] • sin (θ- π/2) = - sin (π/2 – θ)[x=λ/4] = - cos θ • At t = 0, y O 0 y(x,0) = A sin( kx) y(x,0) = A sin(kx) t=0 P Q x • At point O, x = 0, y 0 y(0,t) = A sin(ωt) x=0 T t y 0 O P Q t=0 x t = T/4 y 0 O P Q x O, x = 0 y 0 y 0 T T t Q, x = /4 t Rotating vector 旋轉矢量 diagram y The variation of displacements at the points O, P and Q with time can be represented by the projections 投影 on the y axis of the vectors rotating anti-clockwise 逆時針 with angular velocity , starting at the time as shown. O P Q • y-x graph y x 0 y • y-t graph P t 0 At the point P Huygen’s principle 惠更斯原理 • 1. any point on a wavefront 波陣面上的每 一個點 can be considered as a source of secondary wavelets 次級球面子波的波源 which spread out with the wave velocity. • 2. the new wavefront is the envelope of these secondary wavelets 子波的包絡. A B ray wavefronts By Huygen’s principle, explain rectilinear propagation of light waves 光波是行直線 A B ray • Explanation of law of reflection 反射定律 i=r • Explanation of law of refraction 折射定律 1n2 = v1/v2 = λ1/λ2 Section 12.3 of further physics. Reflection and phase changes 反射和相位變化 • When a wave is reflected at a denser medium「較 密」介質 , an inversion of wave shape or a phase change of 180o takes place 180o的相位變化 . (air -> water or rope fixed on a wall) Fixed end C R C R Applications of reflected waves • Sonar & radar 聲納及雷達 • Long distance propagation of radio wave (low frequency EM wave) by reflection from the ionosphere電離層. ionosphere ground Speed of mechanical waves 機械波 • Speed of transverse wave in a stretched string: • T: tension 張力 T • μ: mass per unit length v • Speed of longitudinal waves in a solid rod: • E: Young modulus 楊氐模量 E • ρ: density 密度 of the rod v • Speed of wave depends on the elasticity 彈性 and density 密度 of the medium. Phase difference (相差) between two particles having the same period of oscillation. From y-t graph, y Q D t t P ΔΦ y P Q ω ΔΦ is the phase difference ΔΦ/ 2π=Δt / T • From y-x graph, we can compare their distance apart with ΔΦ. M’ y yM M ΔΦ Dx N ω N