Wave Motion Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition 1
Waves: propagation of energy, not particles 2
Longitudinal Waves: disturbance is along the direction of wave propagation 3
Transverse Waves: disturbance is perpendicular to the direction of wave propagation 4
Waves with Longitudinal Transverse Motions 5
Amplitude of a Wave pressure height longitudinal displacement transverse displacement 6
Waveforms a pulse a wave train a continuous wave 7
PhET 8
Wavelength in a continuous wave 9
Wave Speed vv = λλ TT 10
PhET 11
Two Snapshots of a Wave Pulse tt = 0 yy = ff xx tt 0 yy = ff xx vv tt 12
PhET 13
Fingerprint of a Wave: ψψ(xx, tt) = ff xx vv tt 14
A Harmonic Wave tt = 0 ψψ xx, tt = 0 = AA cccccc 2 ππ xx λλ tt 0 ψψ xx, tt = AA cccccc 2 ππ xx vv tt λλ 15
PhET 16
A Couple of Definitions ψψ xx, tt = AA cccccc 2 ππ xx vv tt λλ Wave Number Angular Frequency kk 2 ππ λλ ωω 2 ππ TT = 2 ππ ff ψψ xx, tt = AA cccccc kk xx ωω tt vv = ωω kk 17
Propagation towards Positive x-direction ψψ xx, tt = AA cccccc kk xx ωω tt Propagation towards Negative x-direction ψψ xx, tt = AA cccccc kk xx + ωω tt 18
Got It? 14.1 19
The Wave Equation ψψ(xx, tt) = AA cccccc kk xx ωω tt ψψ = kk AA ssssss kk xx ωω tt ψψ = ωω AA ssssss kk xx ωω tt 2 ψψ xx 2 = kk2 AA cccccc kk xx ωω tt 2 ψψ tt 2 = ωω2 AA cccccc kk xx ωω tt 1 2 ψψ kk 2 xx 2 = 1 2 ψψ ωω 2 tt 2 2 ψψ xx 2 = 1 2 ψψ vv 2 tt 2 20
Waves on a String 21
An example on how the properties of the carrying medium determines the wave speed: 22
Wave Speed FF nnnnnn 2 FF θθ FF = mm aa mm 2 θθ RR μμ aa = vv2 RR 2 FF θθ 2 θθ RR μμ vv2 RR vv = FF μμ 23
Example 14.2 mm = 5.0 kkkk FF =?? xx = 43 mm tt = 1.4 ss vv = FF μμ 24
Got It? 14.2 25
Wave Power yy xx, tt = AA cccccc kk xx ωω tt PP = FF. vv = FF vv ssssss θθ vv = tttttt θθ = xx = ωω AA ssssss kk xx ωω tt = kk AA ssssss kk xx ωω tt FF vv tttttt θθ = FF ωω kk AA 2 ssssss 2 kk xx ωω tt = μμ vv ωω 2 AA 2 ssssss 2 kk xx ωω tt PP = 1 2 μμ vv ωω2 AA 2 PP = μμ vv ωω 2 AA 2 ssssss 2 kk xx ωω tt 26
Wave Intensity II PPPPPPPPPP AAAAAAAA II = PPPPPPPPPP 4 ππ rr 2 27
Example of Wave Intensities 28
Example 14.3 PP 1 = 9.2 WW xx 1 = 1.9 mm xx 2 =?? PP 2 = 4.9 WW II 1 = II 2 II = PPPPPPPPPP 4 ππ rr 2 II 1 = PP 1 4 ππ xx 1 2 II 2 = PP 2 4 ππ xx 2 2 PP 1 4 ππ xx 1 2 = PP 2 4 ππ xx 2 2 xx 2 = xx 1 PP 2 PP 1 29
Got It? 14.3 30
Sound Waves vv = γγ PP ρρ γγ is a constant characteristic of the gas 31
Audible Frequencies for Human Ears ββ dddd 10 log 10 II II oo II oo 1 10 12 WW mm 2 32
Example 14.4 ββ 1 = 75 dddd ββ 2 = 60 dddd PP 2 PP 1 =?? PP 2 PP 1 = II 2 II 1 II = II oo 10 ββ 10 ββ 2 10 II 2 = II oo 10 II 1 II oo 10 ββ 1 10 = 10 ββ 2 ββ 1 10 33
Interference or what happened when two waves are present in the same region of space at a particular time? Just add them up!! When wave crests coincide with crests, the interference is constructive. When crests coincide with troughs, the interference is destructive. 34
2016 Pearson Education, Inc. Co-Propagating Waves Constructive Destructive
An application of destructive interference: getting waves to cancel each other: 36
2016 Pearson Education, Inc. Adding Multiple Harmonic Waves: Fourier Analysis
Sound 2016 Pearson Education, Inc.
Dispersion: vv ωω No dispersion With dispersion 39
2016 Pearson Education, Inc. Beats: two co-propagating waves of slightly different frequencies tt yy 1 tt = AA cccccc ωω 1 tt yy 2 tt = AA cccccc ωω 2 tt yy 1 tt + yy 2 tt = 2 AA cccccc 1 2 ωω 1 ωω 2 tt cccccc 1 2 ωω 1 + ωω 2 tt
41
Interference in 2D yy 1 rr 1 = AA cccccc kk rr 1 yy 2 rr 2 = AA cccccc kk rr 2 Constructive Interference: kk rr 1 rr 2 = ππ 2 mm Destructive Interference: kk rr 1 rr 2 = ππ 2 mm + 1 42
Reflection and Refraction PhET 43
Partial Reflection and Transmission 44
2016 Pearson Education, Inc. Standing Waves: two counter-propagating waves yy 1 xx, tt = AA cccccc kk xx ωω tt yy 2 xx, tt = AA cccccc kk xx + ωω tt yy 1 tt + yy 2 tt = 2 AA cccc ss ωω tt cccccc kk xx
46
Doppler Effect λλ = λλ uu TT = λλ uu λλ vv 47