Superposition & Standing Waves Superposition Principle - PDF document

Superposition & Standing Waves • Superposition Principle • Interference of Waves • Standing Waves • Homework 1

Superposition Principle • Overlapping waves algebraically add to pro- duce a resultant wave (or net wave). • Overlapping waves do not in any way alter the travel of each other. 2

Superposition Principle (cont’d) 3

Interference of Waves Consider two sinusoidal waves that are travel- ing in the positive x-direction and have the same frequency, wavelength and amplitude but differ in phase y 1 = A sin ( kx − ωt ) y 2 = A sin ( kx − ωt + φ ) y = y 1 + y 2 = A [sin ( kx − ωt ) + sin ( kx − ωt + φ )] a + b a − b     sin a + sin b = 2 sin  cos             2 2     2 A cos φ  kx − ωt + φ     y =  sin             2 2  When φ = 0 , 2 π, 4 π, . . . then 2 A cos φ 2 = ± 2 A ⇒ constructive interference When φ = π, 3 π, . . . then 2 A cos φ 2 = 0 ⇒ destructive interference 4

Interference of Waves (cont’d) 5

Example Two identical sinusoidal waves, moving in the same direction along a stretched string, inter- fere with each other. The amplitude of each wave is 9.8 mm, and the phase difference be- tween them is 100 ◦ . (a) What is the amplitude of the resultant wave due to the interference of these two waves, and what type of interference occurs? (b) What phase difference, in radians and wavelengths, will give the resultant wave an amplitude of 4.9 mm? 6

Example Solution (a) 2 φ = 2 (9 . 8 mm ) cos 100 ◦ A ′ = 2 A cos 1 = 13 mm 2 (b) A ′ = 2 A cos 1 2 φ φ = 2 cos − 1 A ′ 2 A 4 . 9 mm φ = 2 cos − 1 2 (9 . 8 mm ) = ± 2 . 6 rad 1 wavelength   φ = ± 2 . 6 rad       2 π rad   φ = ± 0 . 41 wavelengths 7

Standing Waves Consider two waves with the same amplitude, wavelength, and frequency traveling in oppo- site directions y 1 = A sin ( kx − ωt ) y 2 = A sin ( kx + ωt ) y = y 1 + y 2 = A [sin ( kx − ωt ) + sin ( kx + ωt )] a + b a − b     sin a + sin b = 2 sin  cos             2 2    y = (2 A sin kx ) cos ωt y = 0 when x = nλ 2 where n = 0 , 1 , 2 , . . . ⇒ nodes  n + 1 λ   y = 2 A cos ωt when x =       2 2  where n = 0 , 1 , 2 , . . . ⇒ antinodes 8

Standing Waves (cont’d) 9

Homework Set 25 - Due Fri. Nov. 12 • Read Sections 14.1-14.4 • Answer Questions 14.1 & 14.2 • Do Problems 14.1, 14.5, 14.8 & 14.10 10