Physics 1B03summer-Lecture 10 Superposition of Waves Identical waves in opposite directions: “standing waves” 2 waves at slightly different frequencies: “beats” 2 identical waves, but not in phase: “interference” Physics 1B03summer-Lecture 10

Principle of Superposition Two Waves In The Same Medium: The observed displacement y(x,t) is the sum of the individual displacements: y1(x,t) + y2(x,t) = y(x,t) (for a “linear medium”) Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 Quiz What do you get if you add two identical (but out-of-phase) square or triangular waves? + = ? + = ? Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 What’s Special about Sine Waves? Two waves, of the same frequency, arrive out of phase: Eg. From Trig: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2] Result: amplitude Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 Asin wt Asin (wt+f) + ARsin (wt+fR) = Resultant: Sine wave, AR depends on phase difference Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 “Constructive interference:” phase difference = 0, 2p, 4p, ... AR =A1 + A2 “Destructive interference:” phase difference = p, 3p, 5p,... AR =|A1 - A2| Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 Standing Waves A standing wave is an oscillation pattern with a stationary outline that results from the superposition of two identical waves traveling in opposite directions. Physics 1B03summer-Lecture 10

Sine Waves In Opposite Directions: y2 = Aosin(kx + ωt) y1 = Aosin(kx – ωt) Total displacement, y(x,t) = y1 + y2 Trigonometry : Then: Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 (where A = 2Ao ) The particle motions are simple harmonic oscillations which are all in phase (or ½ cycle out of phase) with each other, but with different amplitudes. y A t = 0 t = T/8 x t = 3T/8 -A t = T/2 node node node Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 node antinode Antinodes form where the waves always arrive in phase (“constructive interference”); nodes form at locations where the waves are 180o (½ cycle) out of phase (“destructive interference”). Physics 1B03summer-Lecture 10

Physics 1B03summer-Lecture 10 Nodes are positions where the amplitude is zero: at kx = 0 (x = 0) kx = π (x = λ/2) kx = 2π (x = λ), kx = 3π (x=3λ/2) etc. i.e., Nodes are ½ wavelength apart. Antinodes (maximum amplitude) are halfway between nodes. Physics 1B03summer-Lecture 10