Interference and Diffraction 14.1 Superposition of Waves Consider a region in space where two or more waves pass through at the same time. According to the superposition principle,the net displacement is simply given by the vector or the algebraic sum of the individual displacements.Interference is the combination of two or more waves to form a composite wave,based on such principle. The idea of the superposition principle is illustrated in Figure 14.1.1. constructive interference b） Figure 14.1.1 Superposition of waves.(a)Traveling wave pulses approach each other, (b)constructive interference,(c)destructive interference,(d)waves move apart. Suppose we are given two waves, y(x,t)=Ψ。sin(kx±0t+9) Ψ2(x,1)=Ψ20sin(kx±0,1+，).(14.1.1) The resulting sum of the two waves is y(x,)=Ψosin(kx±ω，t+9)+Ψ2osin(k3x±ω，1+92) (14.1.2) The interference is constructive if the amplitude of y(x,t)is greater than the individual ones(Figure 14.1.1b),and destructive if smaller (Figure 14.1.1c). As an example,consider the superposition of the following two waves at t=0: y,(x)=sinx,Ψ，(x)=2sin(x+π/4) (14.1.3) The resultant wave is given by 14-214-2 Interference and Diffraction 14.1 Superposition of Waves Consider a region in space where two or more waves pass through at the same time. According to the superposition principle, the net displacement is simply given by the vector or the algebraic sum of the individual displacements. Interference is the combination of two or more waves to form a composite wave, based on such principle. The idea of the superposition principle is illustrated in Figure 14.1.1. (a) (b) (c) (d) Figure 14.1.1 Superposition of waves. (a) Traveling wave pulses approach each other, (b) constructive interference, (c) destructive interference, (d) waves move apart. Suppose we are given two waves, ψ 1 (x,t) = ψ 10 sin(k1 x ± ω1 t + φ1 ), ψ 2 (x,t) = ψ 20 sin(k2 x ±ω2 t + φ2 ). (14.1.1) The resulting sum of the two waves is ψ (x,t) = ψ10 sin(k1 x ±ω1 t + φ1) +ψ 20 sin(k2 x ±ω2 t + φ2 ). (14.1.2) The interference is constructive if the amplitude of ψ (x,t) is greater than the individual ones (Figure 14.1.1b), and destructive if smaller (Figure 14.1.1c). As an example, consider the superposition of the following two waves at t = 0 : ψ1(x) = sin x, ψ 2 (x) = 2sin(x + π / 4) (14.1.3) The resultant wave is given by