Vågrörelselära och optik

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1 Vågrörelselära och optik Kapitel 16 - Ljud 1 Vågrörelselära och optik Kurslitteratur: University Physics by Young & Friedman Harmonisk oscillator: Kapitel Mekaniska vågor: Kapitel Ljud och hörande: Kapitel Elektromagnetiska vågor: Kapitel 32.1 & 32.3 & 32.4 Ljusets natur: Kapitel & 33.7 Stråloptik: Kapitel Interferens: Kapitel Diffraktion: Kapitel &

Vågrörelselära och optik kap 14 kap 14+15 kap

33 kap 34 kap 34 kap 34+35 kap 35 kap 36 kap

2 Vågrörelselära och optik kap 14 kap kap 15 kap kap 16 kap kap kap 33 kap 34 kap 34 kap kap 35 kap 36 kap 36 3 Sound & Pressure Sound as pressure waves 4

3 Sound & Pressure Longitudinal sinusoidal wave ν = ω / k Amplitude y x k = 2π/λ ω = 2π/T 5 Sound & Pressure Piston moving in and out: Air molecule movement: y x Pressure: p x 6

soundwave is moving the area S to y 1 and the area S to y 2.

4 Sound & Pressure Bulk modulus The change in pressure after a change of volume: Δp = -B ΔV/V Pressure increase: Δp > 0 and ΔV < 0 7 Sound & Pressure A soundwave is moving the area S to y 1 and the area S to y 2. y 1 =y(x,t) V=Sy 1 V=Sy 2 y 2 =y(x+δx,t) V = S Δx Area = S S ΔV = Sy 2 Sy 1 ΔV = S[ y(x+δx,t) y(x,t)] 8

5 Sound & Pressure V= S Δx Δp = -B ΔV/V 9 Sound & Pressure The pressure amplitude The maximum pressure fluctuation 10

6 Sound & Pressure y x p x 11 Sound & Pressure Human hearing Audible range: khz the human frequency range. Loudness: Higher pressure amplitude (at constant frequency) Higher loudness Different frequency Different loudness (at constant pressure amplitude) Pitch: Higher frequency High pitch Higher pressure amplitude Usually higher pitch Timbre: Tone color or harmonic content. Instruments with the same fundamental frequency can have different harmonic content. 12

7 Sound & Problems Problem solving 13 Sound & Problems 14

8 Sound - velocity The velocity of sound in a liquid 15 Sound - velocity Kinematics Momentum: Impuls: The Momentum-Impuls theorem: The impulse is equal to the change of momentum! 16

Sound - velocity F 1 = F 2 = P 2 Pressure Momentum 17 Sound - velocity Sound in a liquid F 1 = Time = 0: P: pressure in the liquid A: area of the piston F 1 : force on the piston ρ: density of the

9 Sound - velocity F 1 = F 2 = P 2 Pressure Momentum 17 Sound - velocity Sound in a liquid F 1 = Time = 0: P: pressure in the liquid A: area of the piston F 1 : force on the piston ρ: density of the liquid F 2 = Time = t: ν y = velocity of the piston ν = velocity of the wave ν y t = distance the piston has moved νt = distance the wave has moved Δp = increase of pressure F 2 : force on the piston 18

10 Sound - velocity ΔV V Δp = -B ΔV / V Volume is decreasing 19 Sound - velocity F 1 = F 2 = 20

11 Sound - velocity General: String: Liquid: Solid: Gas: F: String tension μ: Mass per unit length B: Bulk modulus ρ: Density Y: Young s module ρ: Density γ: Adiabatic index P: Pressure = nrt / V ρ: Density = m/v R: Gas constant = 8.31 J/mol per K T: Absolute temperature in K M: Molar mass = m / n 21 Sound & Problems Problem solving 22

12 Sound & Problems 23 Sound & Problems 24

13 Sound power & intensity The power and intensity of sound 25 Sound power & intensity The power in general: Wave power (P): The instantaneous rate at which energy is transfered along the wave. Unit: W or J/s Wave intensity (I): Average power per unit area through a surface perpendicular to the wave direction. Unit: W/m 2 26

14 Sound power & intensity The wave function: The pressure function: Pressure is equal to force per unit area The wave power: The wave power per unit area: 27 Sound power & intensity The wave power per unit area: Intensity = Average wave power per unit area: ν = ω / k k = ω / 28

15 Sound power & intensity The pressure amplitude The maximum pressure fluctuation k = ω / p max = B A ω / A 2 ω 2 = p max2 / (ρb) I The intensity is proportional to the square of the pressure amplitude 29 Sound & Problems Problem solving 30

16 Sound & Problems I ν ρ = 31 Sound & Problems p max = 3.0 x 10-2 Pa, ρ = 1.20 kg/m 3, ν = 344 m/s, I = 1.1 x 10-6 W/m 2 ν ρ = I = ν ρ ω 2 A 2 / 2 A 2 = 2I / ( ν ρ ω 2 ) A 2 = 2x1.1x10-6 / (344x1.20x(40π) 2 ) A = 0.60 μm 32

17 Sound & Problems Intensity is average power per unit area The intensity through a sphere with radius r The intensity through a hemisphere with radius r 33 Sound - Decibel The decibel scale of the intensity 34

18 Sound - Decibel Intensity in the unit of decibel (db) I 0 = W/m 2 is a reference intensity It is roughfly the threshold of human hearing β = 0 db for I = I 0 β = 120 db for I = 1 W/m 2 35 Sound - Decibel 36

19 Sound & Problems Problem solving 37 Sound & Problems I 0 = W/m 2 38

20 Sound & Problems r 1 β 1 I 1 r 2 =2r 1 β 2 I 2 39 Sound Standing waves Sound and standing waves 40

21 Sound Standing waves Kundt s tube 41 Sound Standing waves Displacement antinode Maximum movement Minimum pressure changes Displacement node Minimum movement Maximum pressure changes λ = 95 cm ν = λ f = 0.95 x 357 = 339 m/s 42

22 Sound Standing waves Antinode Antinode 43 Sound Standing waves Here the pressure is atmospheric giving displacement antinode (pressure node) 44

time = 0 time = T/2 Standing wave: If the airspeed and pipelengths are choosen correctly.

23 Sound Standing waves Open-open pipe Open-closed pipe 45 Sound Standing waves Organpipe: Airflow from below. time = 0 time = T/2 Standing wave: If the airspeed and pipelengths are choosen correctly. Mouth: Pipe is open at the bottom and gives a pressure node (displacement antinode). Airflow: Depending on time the air flow will either go into the pipe or out through the mouth. 46

24 Sound Standing waves The pipe can be open-open or open-closed 47 Remember: The distance between two nodes is λ/2 47 Sound & Problems Problem solving 48

25 Sound & Problems Displacement nodes Pressure nodes A N A N A N A 49 Sound & Problems 50

26 Sound & Problems Fundamental frequency First overtone Second overtone Fundamental Second harmonic Third harmonic 51 Sound Resonance Sound and resonance 52

Sound & Problems Resonance Many mechanical systems have normal mode frequencies of oscillation. In these modes every particle in the system oscillates with simple harmonic oscillation.

27 Sound & Problems Resonance Many mechanical systems have normal mode frequencies of oscillation. In these modes every particle in the system oscillates with simple harmonic oscillation. If an outside drivingforce is applied that varies with a normal mode frequency then the system is in resonance and the amplitude of the oscillations can increase. In this case the drivingforce is continuously adding energy to the system. 53 Sound & Problems Problem solving 54

28 Sound & Problems 55 Sound Interference Sound and interference 56

Sound Interference Interference Two waves that arrives at a

..) undergo contructive interference and have a doubled

29 Sound Interference Interference Two waves that arrives at a point where the distance is different with nλ (n=0,1,2,3...) undergo contructive interference and have a doubled amplitude. Two waves that arrives at a point where the distance is different with nλ/2 (n=1,3,5...) undergo destructive interference and have a zero amplitude. 57 Sound Interference 58

Sound Interference BEAT: If two sound waves with slighty

pulsating sound is only heard if the difference in frequency is <

30 Sound Interference BEAT: If two sound waves with slighty different frequencies are added up they give a sound that is going up and down in intensity. Two waves with different frequency Their superposition This pulsating sound is only heard if the difference in frequency is < 7 Hz 59 Sound Interference What is the frequency of the beat? T beat = 9T red = 8T blue T beat = nt a = (n-1)t b 60

31 Sound Doppler effect The Doppler effect 61 Sound Doppler effect Doppler effect 62

Sound Doppler effect The time for a sound wave to reach a

L f ν ν s L The time for a sound wave to reach a listener (L)

λ behind longer λ in front shorter 63 Sound Doppler effect What

32 Sound Doppler effect The time for a sound wave to reach a listener (L) gets longer if the source (S) is moving away. L f ν ν s L The time for a sound wave to reach a listener (L) gets shorter if the source is moving closer. λ behind longer λ in front shorter 63 Sound Doppler effect What if the listener is also moving? The wave speed relative to L is ν + ν L change in frequency 64

33 Sound Doppler effect always works if the positive direction is defined as going from the listener to the source. positive direction positive direction L S S L L S S L L S S L L S S L 65 Sound Doppler effect Electromagnetic waves such as light also have a Doppler shift. It can be calculated using the theory of relativity: f S = the frequency of the source f O = the frequency detected by an observer c = the speed of light v = the relative velocity of the source with respect to the observer v is positive if the observer and the source is moving apart v is negative if the observer and the source is moving towards each other 66

34 Sound & Problems Problem solving 67 Sound & Problems f = 300 Hz speed of sound = 340 m/s What frequency does the listener hear? 68

35 Sound shockwave Shockwave 69 Sound shockwave Shock waves ν: Speed of sound ν s : Speed of the plane ν s > ν Shockwave is created (not only when ν s = ν) ν s > ν No sound in front of the plane 70

Sound A conical shock wave is produced if a plane flies

36 Sound A conical shock wave is produced if a plane flies faster than the speed of sound. A series of circular wave crests from the plane interfere constructively along a line that is given by an angle α. ν: Speed of sound ν s : Speed of the plane Speed of the plane in Mach number: Ν Μ = ν s /ν 71 Sound & Problems Problem solving 72

37 Sound & Problems Ν Μ = ν s /ν = 1.75 sin α = ν / ν s = 1 / N M = 1 /

Vågrörelselära och optik

Vågrörelselära och optik Harmonic oscillation: Experiment Experiment to find a mathematical description of harmonic oscillation Kapitel 14 Harmonisk oscillator 1 2 Harmonic oscillation: Experiment Harmonic