Secions 1,,4,5, I. Oulook II. Wha is wave? III.Kinemaics & Eamples IV. Equaion of moion Wave equaions V. Eamples
Oulook Translaional and Roaional Moions wih Several phsics quaniies Energ (E) Momenum (p) Angular momenum (L) Wih Conservaion laws Conservaion of energ Conservaion of linear momenum Conservaion of angular momenum
Wha is Wave? Vibraion
Wha is Wave? Inerference
Wha is Wave? Various Tpes of Waves Movie Wave1 Mechanical vibraion Spring ssem Sring fied a boh ends Sound (vibraion of air densi) Waer wave Movie Wave Elecromagneic vibraion PHYS08 Ligh
Quick Look a Kinemaics Coninuum of SHOs Movie Wave3 [Q] How can ou describe he shape of he rope? [A] Well, I use period T (because he are SHOs)! [Q] Anhing else?
A Kinemaics π ( ) Asin λ Wave number is number of waves in uni lengh : k π/λ So, how man waves in his case? A 0 1 m [Q] How can ou describe he shape of he rope? [A] T, λ (or k), and A [Q] Anhing else?
[Q] When did I ake his snap sho? A A π (, 0) Asin λ
Pick one SHO and consider is moions. A A 1 4 3 π (, 0) Asin λ [Q] Can we find T? [A]
A A π (, 0) Asin λ [] Moion wih a consan veloci [] SHM angular frequenc
A A sin 0), ( A λ π sin ), ( v A 13 ω λ π λ π 4 λ π λ π v v + 4 λ π λ π λ π The shape of he rope (wave) moves o + direcion.
A T π π T (, ) A sin v 4 λ λ 4 13 ω π πvt A sin λ λ π π A sin λ π Acos λ A π (, 0) Asin λ S.H.M. Mah and Phsics Trig. Trig. funcions: funcions: S.H.O. S.H.O. sin(θ sin(θ + + π/) π/) cos(θ) cos(θ) 1) 1) Spring Spring plus plus block block sin(θ + π ) sin(θ) sin(θ + π ) sin(θ) Horizonal Horizonal cos(θ π/) sin(θ) cos(θ π/) sin(θ) Verical Verical Derivaive Derivaive and and inegral inegral Trig. funcions ) Pendulum Trig. funcions ) Pendulum
Eample 1 [19] A ransverse raveling wave (ampliude A, wave lengh λ, and frequenc f) on a cord a 0 is represened b A sin(π/λ + φ). Here φ is a consan phase facor. (a) Wha will be he equaion for a wave raveling o he lef along he ais as a funcion of and? [Hin] (,)? (b) Wha is is maimum acceleraion of paricles on he cord? [Hin] a (,)?
Eample [0] A ransverse raveling wave on a cord is represened b (,) 0.48 sin(0.56 + 84) where and are in meers and in seconds. For his wave deermine (a) he ampliude, (b) wavelengh, frequenc, veloci (magniude and direcion), (c) maimum and minimum speeds of paricles of he cord, and (d) maimum acceleraion (magniude) of he paricles. [A]
Eample 3 [3] A ransverse wave pulse ravels o he righ along a sring wih speed v.0 m/s. A 0, he shape of he pulse is given b he funcion 0.45 cos(3.0 + 1.) where and are in meers and. For his wave deermine (a) he wavelengh, frequenc, and ampliude, (b) maimum and minimum speeds of paricles of he sring, and (c) maimum and minimum acceleraions (magniudes) of he paricles. [A]
Secions 1,,4,5, I. Oulook II. Wha is wave? III.Kinemaics & Eamples IV. Equaion of moion Wave equaions V. More Eamples
Visualizaion π π (, ) Asin v λ λ 13 ω A A π (, 0) Asin λ [] Wave propagaion: moving wih a consan veloci [] SHM angular frequenc
Mah & Phsics Mah & Phsics Equaion of Moions Equaion of Moions? ) ( sin ) ( d d ) cos( ) ( ) ( 1 ) ( 0 0 + + v A, A m F / a a v λ π ω ω
Finding Wave Equaions - Transverse Wave on Rope - Consider a segmen (mass m) of he rope under F T.
Finding Wave Equaion 1 m F T Consider a segmen (mass m) of he rope under F T. F T Look a he verical () moion.
a F m Finding Wave Equaion 1 m F T F T
Finding Wave Equaion 1 m F T Consider a segmen (mass m) of he rope under F T. Look a he verical () moion. Wave F T Equaion: FT µ Wave veloci: v F T µ
Eample 4 [7] Deermine if he funcion A sin (k ω ) is a soluion of he wave equaion. [A]
Mah Differenial Equaion (,) where Aω cos [LH] [RH] k π Asin ( λ ω k π, λ [ k ω ] Aω sin Ak ω sin v ), π v λ [ k ω ] [ k ω ] v Soluion of D.Eq. Where is Phsics?
(,) where Aω cos [LH] [RH] Mah & Phsics Equaion of Moions k π Asin ( λ ω k π, λ [ k ω ] Aω sin Ak ω sin v ), π v λ [ k ω ] [ k ω ] v a ω (You have seen his!)
Eample 5 [71] The figure shows he wave shape of a sinusoidal wave raveling o he righ a wo insans of ime. Find he mahemaical represenaion of he wave? [A]
Eample 5 λ 6 cm 4