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1 Chapter 21 Superposition and standing waves

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2 When waves combine Two waves propagate in the same medium Their amplitudes are not large Then the total perturbation is the sum of the ones for each wave On a string: Wave 1: y 1 Wave 2: y 2 Total wave: y 1 +y 2 To an excellent approximation: waves just add up

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3 The combination of waves can give very interesting effects On a string: Wave 1: y 1 = A sin(k 1 x - 1 t + 1 ) Wave 2: y 2 = A sin(k 2 x - ω 2 t + 2 ) Now suppose that k 1 = k + Δkk 2 = k - Δk ω 1 = ω + Δω ω 2 = ω - Δω 1 = + Δ 2 = - Δ note: 2 - 1 =2Δ And use sin(a+ b ) + sin(a- b ) = 2 sin(a) cos( b ) to get y = y1 + y2 = 2A cos(Δk x – Δω t + Δ ) sin(kx – ωt + )

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4 Assume Δk=0 Δω =0 y = [ 2A cos(Δ ) ] sin(kx – ωt + ) The amplitude can be between 2A and 0, depending on Δ When Δ =0 y = [ 2A ] sin(kx – ωt + ) The waves add up The phase difference in WAVELENTGHS is 0 When Δ = /2 or 2 - 1 =2Δ =π y = 0 The waves cancel out The phase difference in WAVELENTGHS is λ/2

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5 More Generally y = [ 2A cos(Δk x – Δω t + Δ ) ] sin(kx – ωt + ) Behaves as a slowly changing amplitude

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6 Two waves are observed to interfere constructively. Their phase difference, in radians, could be a. /2. b. . c. 3 . d. 7 . e. 6 .

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7 Standing waves A pulse on a string with both ends attached travels back and forth between the ends If we have two waves moving in opposite directions: Wave 1: y 1 = A sin(kx - ωt) Wave 2: y 2 = A sin(kx + ωt) y = y 1 + y 2 = 2A sin(k x) cos(ωt) If this is to survive for an attached string of length L y(x=0)=y(x=L)=0 L sin(k L)=0 If λ does not have these values the back and forth motion will produce destructive interference. For example, if λ=L/2 λ /2 applet

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8 A pulse on a string with both ends attached travels back and forth between the ends Standing wave: y = 2A sin(k x) cos(ω t) λ = 2L/n, n=1,2,3,… Smallest f is the fundamental frequency The others are multiples of it or, harmonics

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Two pulses are traveling on the same string are described by a)In which direction does each pulse travel? b) At what time do the two cancel everywhere? c)At what point do the two waves always cancel?

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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

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Two 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 intensity b) How far is he from the pole at these moments? Let v represent the speed for sound and assume that the ground is carpeted and will not reflect the sound L h d h

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12 Exercise 14.9 This is a case of interference I will need Pythagoras theorem Cancellations when kx=… -5p,-3p,-p,p, 3p, 5p … L h d h

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13 Allowed n: all such that the right hand side is positive λ=v/f

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14 Destructive interference occurs when the path difference is A. /2 B. C.2 D.3 E.4

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Two sinusoidal waves combining in a medium are described by the wave functions y 1 =(3cm)sin( x+0.6t) y 2 =(3cm)sin( x-0.6t) where x is in cm an t in seconds. Describe the maximum displacement of the motion at a)x=0.25 cm b) x=0.5 cm c) x=1.5 cm d) Fine the three smallest (most negative) values of x corresponding to antinodes nodes when x=0 at all times

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16 Exercise Superposition problem sin(a+b) + sin(a-b)=2 sin(a) cos(b)

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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 & Superposition Dimensions restrict allowed wavelengths-Resonant Frequencies Initial disturbance excites various resonant frequencies

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18 Standing Wave Patterns for Air Columns Pipe Closed at Both Ends Displacement Nodes Pressure Anti-nodes Same as String Pipe Open at Both Ends Displacement Anti-nodes Pressure Nodes (Atmospheric) Same as String 3. Open One end /2211/main/demos/standing_wwaves/stlwaves.htm beats demo v/λ=f

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19 Organ Pipe-Open Both Ends A N A Turbulent Air flow excites large number of harmonics “Timbre” n = 1 λ = 2L

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20 Problem 9 v sound = 331 m/s m/s T C T C is temp in celcius

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