Interference Pham Tan Thi, Ph.D Department of Biomedical Engineering Faculty of Applied Science Ho Chi Minh University of Technology A Single Oscillating Wave The formula y(x, t) = Acos(kx !t) describes a harmonic plane wave of amplitude A moving in the +x direction For a wave on a string, each point on the wave oscillates in the x direction with simple harmonic motion of angular frequency ω The wavelength 2⇡ = k ! The speed/velocity v = f = k The intensity is proportional to the square of the amplitude I / A2 Multiple Waves: Superposition The principle of superposition states that when two or more waves of the same type cross at a point, the resultant displacement at that point is equal to the sum of the displacements due to each individual wave For inequal intensities, the maximum and minimum intensities are: Imax = |A1 + A2|2 Imin = |A1 - A2|2 Multiple Waves: Superposition Constructive “Superposition” Destructive “Superposition” ou added the two sinusoidal waves shown, Superposing Sinusoidal Waves at would the result look like? If we added the two sinusoidal waves shown, what would the result look like? 0 900 800 700 600 500 400 300 200 1000 - 100 0 0 - 0 of two sines having the same frequency is another same frequency ude depends on their relative phases 0 0 ou added the two sinusoidal waves shown, Superposing Sinusoidal Waves at would the result look like? If we added the two sinusoidal waves shown, what would the result 0 1000 900 800 700 600 500 400 300 200 100 0 0 - - 0 look like? 0 0 0 - 1000 900 800 700 600 500 400 300 200 100 of two sines having the same frequency is another s same frequency The sum of two sines having the same frequency is another sine ude depends their→ relative with the sameon frequency Its amplitudephases depends on their relative - 0 phases of two sines having the same frequency is another same frequency ude depends on their relative phases 0 0 - 0 - 0 0 900 800 700 600 500 400 300 200 1000 - 0 100 - 0 0 Adding Sine Waves with Different Phases Suppose we have two sinusoidal waves with the same A1, ω and k: y1 = A1 cos(kx !t) y2 = A1 cos(kx and !t + ) One starts at phase 𝜙 after the other Spatial dependence of waves at t = Resulting wave: y = y1 + y ✓ ↵ ◆✓ +↵ A1 (cos↵ + cos ) = 2A1 cos y1 + y y = 2A1 cos(( /2) /2)cos(kx y = 2A1 cos( /2)cos(kx Amplitude !t + /2) Oscillation ◆ !t + /2) What happens when two waves are present at the same place? Interference of Waves Always add amplitudes (pressures or electric fields) WhatHowever, happens when two waves are present at the same place? we observe intensity (power) amplitude (e.g pressures or electric fields) ForAlways equal Aadd and ω: However, we observe intensity (i.e power) = ω: 2A1 cos(φ / 2) For equal A A and A = 2A1cos(𝜙/2) ⇒ I = 4I1 cos (φ / 2) I = 4I1cos2(𝜙/2) Example: Stereospeakers: speakers: Stereo Listener: Listener: Terminology: Constructive interference: Terminology: waves are “in phase” Constructive (φ = 0, interference: 2π, 4π, ) waves are “ininterference: phase” Destructive waves are “out of phase” (𝜙 = 0, 2π, 4π,…) (φ = π,interference: 3π, 5π, …) Destructive waves are “out of phase” (𝜙 = π, 3π, 5π,…) Of course, φ can take on an infinite number of values We won’t use terms like “mostly constructive” or “slightly destructive” Lec Quiz Each speaker alone produces an intensity of I1 = W/m2 at the Example: Changing phase of the Source listener: Each speaker alone produces an intensity of I = W/m2 at the listener: Example: Changing phase of the Source Each speaker alone produces an intensity of I1 = W/m2 at the listener: I = I = of A =the W/mSource Example: Changing phase I = I1 = A12 = W/m2 1 2 22 I =1 IW/m W/m Each speaker alone produces an intensity of I1 = listener: = A1 at=the Drive the speakers in phase What is the intensity I at the listener? is intensity I at the listener? DriveDrive thethespeakers in What phase IWhat = I1 = AI1at =the 1the W/m speakers in phase is the intensity listener? I= Drive the speakers in phase What is the intensity I =I at the listener? I =? Now shift phase of one speaker by 90o.What is the intensity I at the listener? I= Now shift phase of one speaker by 90o.What is the intensity I at the listener? Now shift phase of one speaker by 90° What is the intensity I at the I= listener? Now shift phase of one speaker by 90o.What is the intensity I at the listener? I= φ φ Lecture 2, p.7 I= I =? Lecture 2, p.7 φ Lecture 2, p.7 Quiz Each speaker alone produces an intensity of I1 = W/m2 at the Example: Changing phase of the Source listener: Each speaker alone produces an intensity of I = W/m2 at the listener: Example: Changing phase of the Source Each speaker alone produces an intensity of I1 = W/m2 at the listener: I = I = of A I=the W/m Example: Changing phase = I1Source = A12 = W/m2 1 2 22 I =1 IW/m W/m Each speaker alone produces an intensity of I1 = listener: = A1 at=the Drive the speakers in phase What is the intensity I at the listener? is intensity I at the listener? DriveDrive thethespeakers in What phase IWhat = I1 = AI1at =the 1the W/m speakers in phase is the intensity listener? I= Drive the speakers in phase What is the intensity I =I at the listener? I = (2A1)2 = 4I1 = W/m2 Now shift phase of one speaker by 90o.What is the intensity I at the listener? I= Now shift phase of one speaker by 90o.What is the intensity I at the listener? Now shift phase of one speaker by 90° What is the intensity I at the I= listener? Now shift phase of one speaker by 90o.What is the intensity I at the listener? I= φ φ Lecture 2, p.7 I= I =? Lecture 2, p.7 φ Lecture 2, p.7 ... Constructive interference: Terminology: waves are “in phase” Constructive (φ = 0, interference: 2π, 4π, ) waves are “ininterference: phase” Destructive waves are “out of phase” (