12. Superposition

Superposition Principle of Superposition – when two or more waves of the same nature meet at a point, the resultant displacement is the vector sum of the individual displacements due to each waves. Interference – two or more waves of the same type superpose to produce a resultant wave. Constructive Interference  Two waves meet in phase, resultant displacement is greater than individual displacements by each wave Destructive Interference  Two waves meet in antiphase, resultant displacement is minimum. Whether Constructive or Destructive Interference at a point, P? 1. Determine Path Difference, ΔL 21L L L  2. Determine ΔL as a full integer multiple or half-integer multiple of the wavelength Ln or (2n 1)L 2  3. Look for whether sources are in phase or antiphase. (Note: S1 and S2 are coherent: constant phase difference) Sources start in Phase Sources start in antiphase Path Difference n λ (2n+1)/2 λ n λ (2n+1)/2 λ Will Meet In Phase Antiphase Antiphase In Phase Conclusion Constructive Destructive Destructive Constructive Stationary Waves Application of Stationary Waves String Open-Closed Tube Open-Open Tube  Closed at both ends, thus  Node at both ends.  Antinode at open end  Node at closed end  Antinode at both open ends S1 S2 P L1 L2

Single Source Diffraction Equation: sin b   Also, xtan D  Small Angle Approximation, sin tan , therefore, sin tan x bD   (Note: in some reference, 2x ≈ x since x is very small) Young’s Double Slit Experiment Assumption:  d << D (~103)  Infinite numbers of bright and dark fringes which are equally spaced Equation: Dx d  Small Angle Approximation, Consider 0th order to 1st order maxima, xtan D  Diffraction Grating Equation: d sin n 1d N Small Angle Approximation, Consider nth order maxima, sin tan xn Dd  There exist a maximum order of maxima, sin 1  n 1d   , dn  , round down to nearest integer. First minima screen b D x x θ θ λ First order maxima screen d D x x Bright Dark Dark Bright Bright Dark Dark Bright Bright First order maxima Second order maxima Second order maxima First order minima First order minima Second order minima Second order minima x λ screen D λ θ nth order maxima θ nth order maxima Grating of N number of lines per length x

Rayleigh Criterion Equation: minsin b   Two objects are resolved when θ ≥ θmin. Achievable by  Increasing x  Decreasing D At angle of resolution θmin, 1. As θmin is very small, min minsin tan  x bD   2. As light from each source diffract through aperture, the central maxima of one object falls exactly at the first minima of the other object. Further Note:  Each equation above applies to only one wavelength. If there are multiple wavelengths (e.g. magenta) entering each slit / grating / aperture at the same time, treat each wavelength of l