Superposition and Standing Waves

Transcription

1 Physics 1051 Lecture 9 Superposition and Standing Waves

2 Lecture 09 - Contents 14.5 Standing Waves in Air Columns 14.6 Beats: Interference in Time 14.7 Non-sinusoidal Waves

3 Trivia Questions 1 How many wavelengths is this? If this is a standing wave, can you label the feature at the left and at the right? N A 4 3

4 14.5 Standing Waves in Air Columns We already looked at the idea of a standing wave for wave on a string which is transverse Now we can look at an example of a longitudinal wave in air or a sound wave. Two identical waves travelling in opposite directions in a enclosed column of air create a standing wave The details of this depend on the boundary conditions... 4

5 Boundary Conditions Two possible sets of boundary conditions Tube open at both ends Tube open at one end; closed at the other Closed End Is a displacement node Is a pressure antinode Open End Approximately a displacement antinode Approximately a pressure node 5

6 Finding Harmonic Frequencies We can derive the harmonic frequencies in a very similar way to what we did for string. Note boundary conditions Draw sketch Determine Wavelength from Length Use formula for frequency in terms of wave speed 6

7 Natural Frequencies of an Air Column Open at Both Ends First Harmonic Second Harmonic See Figure a L= 1/ 1 1 = /1 L L= / = / L f 1 = v = v L f = v = v L Third Harmonic L= 3/ 3 3 = /3 L f 3= v =3 v L 7

8 Harmonic Frequency Formula: Both Ends Open It is possible to write down the general formula for a tube closed at both ends. f n =n v L f n =n f 1 n=1,,3,... f 1 = v L 8

9 Natural Frequencies of an Air Column Open at One End First Harmonic See Figure b L= 1/4 1 1 = 4/1 L f 1 = v = v 4 L Third Harmonic L= 3/4 3 3 = 4/3 L f 3 = v =3 v 4 L Fifth Harmonic L= 5/4 5 5 = 4/5 L f 5= v =5 v 4 L 9

10 Harmonic Frequency Formula: One End Open It is possible to write down the general formula for a tube closed at both ends. f n =n v 4L f n =n f 1 n=1,3,5... f 1 = v 4L 10

11 GENERAL Summary Wavelength depends on L, length of given medium. Harmonic refers to multiple of the frequency or number of smallest segment of wavelengths that fit in to medium. Boundary Conditions Note: Need to understand and know how to derive these! Smallest Segment of λ Fundamental Frequency f 1 Frequency Formula Values for n Same at Both Ends 1/ 1 n/l f n =n f 1 1,,3... Different at Each End 1/4 1 n/4l f n =n f 1 1,3,5,... 11

12 Example Problem 14.30, page 455 The fundamental frequency of an open organ pipe corresponds to middle C (61.6 Hz on the chromatic musical scale). The third resonance of a closed organ pipe has the same frequency. What is the length of each of the two pipes? 1

13 14.6 Beats: Interference in Time Up to now, we have just only dealt with waves interfering with the same frequency. Spatial Interference Now let's investigate two waves with slightly different frequencies. Temporal Interference (in time) The result is what we call beats! 13

14 Interference Spatially We can consider the wave functions of two waves that are at a fixed point in space x=0 y 1 x,t =Asin 1 t 0 y x, t =Asin t 0 Interference; y x,t = y 1 x,t y x,t y x,t =Asin 1 Asin y x,t = Acos 1 sin 1 14

15 Simplifying Phase Terms 1 1 = 1t 01 t 0 = 1 t = f 1 f t 1 1 = 1t 01 t 0 = 1 t = f 1 f 1 t 15

16 Wave function y x,t =A cos f 1 f t sin f 1 f 1 t Observations: Effective wave has frequency equal to the average frequency f 1 f / The amplitude term depends on time With a frequency of Amplitude: f 1 f / A t =A cos f 1 f t 16

17 Beats Graph See Figure

18 Beat Frequency Let's look at amplitude to see how often we hear a beat Here we take a beat to be loud sound to no sound A t =A cos f 1 f This effective frequency is for one cycle; loud sound to loud sound f 1 f / Thus a beat is twice that: f b = f 1 f t 18

19 Example Problem 14.40, page 456 While attempting to tune the note C at 53 Hz, a piano tuner hears.00 beats/s between a reference oscillator and a string. a) What are the possible frequencies of the string? b) When the she tightens the string slightly, she hears 3.00 beats/s. What is the frequency of the string now? c) 19

20 14.7 Non-sinusoidal Waves Sound produce by most instruments is not sinusoidal. These waves are still periodic But it is composed of various sinusoidal waves with different frequencies that interfere. This has to do with Musical sounds Musical Sound: when composition is of frequencies that are multiples of a fundamental Noise: composition is of freququences that are NOT multiples of a fundamental 0

21 See Figure

22 Mathematics of Non-sinusoidal Waves Fourier's Theorem: Any periodic function (or any function over a finite interval) can be represented as a series of since and cosine terms using a mathematical technique known based on Fourier Series. Let y(t) be any periodic function: Where y t = n A n sin f n t B n sin f n t f 1 =1/T, f n =n f 1 An and Bn represent magnitudes of harmonics

23 Harmonic Analysis See Figure

24 Harmonic Synthesis See Figure