2 The principle of superposition When waves combine Two waves propagate in the same mediumTheir amplitudes are not largeThen the total perturbation is the sum of the ones for each waveOn a string:Wave 1: y1Wave 2: y2Total wave: y1+y2To an excellent approximation: waves just add upThe principleof superposition
3 The combination of waves can give very interesting effects On a string:Wave 1: y1=A sin(k1x-ω1t+1)Wave 2: y2=A sin(k2x- ω2t+ 2)Now suppose thatk1=k+Δk k2=k- Δkω 1= ω + Δω ω 2= ω - Δω 1= + Δ 2= - Δ note: 2- 1=2ΔAnd usesin(a+b) + sin(a-b)=2 sin(a) cos(b)to gety = y1 + y2 = 2A cos(Δk x – Δω t + Δ) sin(kx – ωt + )
4 constructive interference destructive interference Assume Δk=0 Δω =0 y = [2A cos(Δ ) ] sin(kx – ωt + )The amplitude can be between 2A and 0, depending on ΔconstructiveinterferenceWhen Δ =0y = [2A ] sin(kx – ωt + )The waves add upThe phase difference in WAVELENTGHS is 0When Δ =π/2 or 2- 1=2Δ =πy = 0The waves cancel outThe phase difference in WAVELENTGHS is λ/2destructiveinterference
5 y = [2A cos(Δk x – Δω t + Δ) ] sin(kx – ωt + ) More Generallyy = [2A cos(Δk x – Δω t + Δ) ] sin(kx – ωt + )Behaves as a slowly changing amplitudebeatsLOUDLOUD
6 Two waves are observed to interfere constructively Two waves are observed to interfere constructively. Their phase difference, in radians, could bea. /2.b. .c. 3.d. 7.e. 6.
7 y = y1 + y2 = 2A sin(k x) cos(ωt) Standing wavesA pulse on a string with both ends attached travels back and forth between the endsIf we have two waves moving in opposite directions:Wave 1: y1=A sin(kx- ωt)Wave 2: y2=A sin(kx+ ωt)y = y1 + y2 = 2A sin(k x) cos(ωt)sin(k L)=0If this is to survive for an attached string of length Ly(x=0)=y(x=L)=0For example, if λ=L/2λ/2If λ does not have these values the back and forth motion will produce destructive interference.Lapplet
8 Smallest f is the fundamental frequency A pulse on a string with both ends attached travels back and forth between the endsStanding wave:y = 2A sin(k x) cos(ω t)λ=2L/n, n=1,2,3,…Smallest f is the fundamental frequencyThe others are multiples of it or, harmonics
9 14.3Two pulses are traveling on the same string are described byIn which direction does each pulse travel?At what time do the two cancel everywhere?At what point do the two waves always cancel?
10 Exercise 14.3 A wave moving right looks like f(x-vt) A wave moving left looks like f(x+vt)I need the superposition principle
11 14.9Two speakers are driven in phase by the same oscillator of frequency f. They are located a distance d from each other on a vertical pole as shown in the figure. A man walks towards the speakers in a direction perpendicular to the pole.a) How many times will he hear a minimum in the sound intensityb) How far is he from the pole at these moments?Let v represent the speed for sound and assumethat the ground is carpeted and will not reflectthe soundLhd
12 Exercise 14.9 This is a case of interference I will need Pythagoras theoremLhdCancellations whenkx=…-5p,-3p,-p,p, 3p, 5p …
13 all such that the right hand side is positive Allowed n:all such that the right hand side is positiveλ=v/f
14 Destructive interference occurs when the path difference is /2234
15 14.13Two sinusoidal waves combining in a medium are described by the wave functionsy1=(3cm)sin(x+0.6t)y2=(3cm)sin(x-0.6t)where x is in cm an t in seconds. Describe the maximum displacement of the motion atx=0.25 cm b) x=0.5 cm c) x=1.5 cmd) Fine the three smallest (most negative) values of x corresponding to antinodesnodes when x=0 at all times
16 Exercise 14.13 Superposition problem sin(a+b) + sin(a-b)=2 sin(a) cos(b)
17 Sources of Musical Sound Oscillating strings (guitar, piano, violin)Oscillating membranes (drums)Oscillating air columns (flute, oboe, organ pipe)Oscillating wooden/steel blocks (xylophone,marimba)Standing Waves-Reflections& SuperpositionDimensions restrict allowedwavelengths-ResonantFrequenciesInitial disturbance excitesvarious resonantfrequencies
18 Standing Wave Patterns for Air Columns v/λ=fPipe Closed at Both EndsDisplacement NodesPressure Anti-nodesSame as String/2211/main/demos/standing_wwaves/stlwaves.htmPipe Open at Both EndsDisplacement Anti-nodesPressure Nodes (Atmospheric)Same as String3. Open One endbeats demo
19 Organ Pipe-Open Both Ends λ = 2LANATurbulent Air flow excites large number of harmonics“Timbre”
20 Problem 9vsound = 331 m/s m/s TCTC is temp in celcius
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