2. Waves in Elastic Media, Mechanical Waves


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1 2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in air, are called mechanical waves. If a vibraor disurbance occurs a an poin in an elasic medium, his disurbance will be ransmied from one laer o he ne hrough he medium, because of he elasic forces on adjacen laers. The medium iself does no move as a whole. In a mechanical wave he energ is also ransmied form one poin o he ne, b he moion or propagaion of a disurbance, wihou an corresponding bulk moion of he maer iself. A maerial medium is necessar for he ransmission of mechanical waves. Such a medium is no needed o ransmi elecromagneic waves. I propagaes in vacuum. 2.1 Tpes of Waves We can disinguish differen kinds of mechanical waves b considering how he moions of he paricles of maer are relaed o he direcion of propagaion of he waves hemselves. The waves are called longiudinal if he moion of each individual paricle of he medium is in he same direcion as he wave propagaion hrough ha poin. Sound waves in air are longiudinal waves. If he moions of he maer paicles are perpendicular o he direcion of propagaion of he wave iself, we hen have a ransverse wave. If we have a sreched sring and we oscillae he endpoin perpendicular o he direcion of he sring, he wave is ransverse wave. If our moion is periodic, we produce a periodic rain of waves in which each paricle of he sring has a periodic moion. The simples special case of a periodic wave is a simple harmonic wave, which gives each paricle a simple harmonic moion. A wave fron is a surface connecing medium paricles all of which are moving in he same manner a an given momen. A wave called spherical wave or plane wave, if he wave fron has he corresponding shape. 2.2 Traveling waves Le us consider a long sring sreched in he direcion along which a ransverse wave is raveling. A = 0 he shape of he sring can be represened b = f(), where is he ransverse displacemen of he sring a he posiion. Eperimen shows ha as ime goes on such a wave ravels along he sring wihou changing is form, if he inernal fricional losses are small enough. A some ime laer he wave has ravelled a disance c o he posiive  direcion, where c is he magniude of he wave veloci. The equaion of he curve a he ime : = f ( c) B. Palásh 1
2 =0 c The above equaion is he mahemaical form of a pulse raveling along he posiive  direcion. In more general, his (, ) funcion ma represen several differen phsical quaniies, such as he deformaion in a solid, he pressure in a gas, and so on. An especiall ineresing case is ha in which (, ) a sinusoidal or harmonic funcion. Suppose a wave ravels from lef o righ, he direcion of increasing. In a sinusoidal wave, all poins in he medium move wih he same frequenc bu wih phase difference. Suppose, ha a ime = 0, we have a wave rain along he sring given b: = Asin The wave shape is a sine curve, he maimum displacemen is A. The value of he ransverse displacemen is he same a, +, , and so on. The smbol is called wavelengh of he wave rain and represens he disance beween wo adjacen poins in he wave having he same phase. As he ime goes on he wave ravels o he righ wih a phase speed c. Hence he equaion of he wave a ime is: 2 (, ) = Asin c π ( ) A =0 0 c A The period T is he ime required for he wave o ravel a disance of one wavelengh, so ha Puing his relaion o he above equaion: = ct. B. Palásh 2
3 (, ) = Asin ( c) = Asin Asinω ct T = c c Where he angular frequenc ω is defined: ω =, T so he equaion of a sine wave raveling o he righ: (, ) = Asinω c I is one wa o describe a simple harmonic raveling wave. Le us now consider he oher wa o obain he mahemaical form of a sinusoidal raveling wave. Suppose he displacemen of a paricle a he lef end of he sring (a = 0 ), where he moion originaes, is given b = Asinω The cause of he negaive sign is shown on he previous figure. The oscillaing poin a = 0 posiion sars o move ino he negaive direcion. The ime required for he wave disurbance o ravel from = 0, o some poin o he righ is given b, where c is he c phase speed. The moion of poin a ime is he same as he moion of poin = 0 a he earlier ime c. Thus he displacemen of poin a ime is obained b replacing in he above equaion b c, we find (, ) = Asinω c sin α = sinα is known, ha is The relaion ( ) The same formula as we have alread. (, ) = Asinω c. To reduce he equaion o a more compac form we define he wave number k as: k =. B. Palásh 3
4 We have alread used ω = T, ha is For a sine wave raveling o he lef: ω = = = k c ct ω (, ) = Asin ω c (, ) sin( ω ) = A k. (, ) sin( ω ) = A k+. In he raveling waves we have assumed ha he displacemen is zero a he posiion = 0. This, of course, need no o be he case. The general epression for a sinusoidal wave raveling in he + direcion: (, ) = sin( ω ϕ) A k Where he quani in parenheses is called he phase and ϕ is called phase consan. 2.3 Superposiion and inerference I is epeimenal fac, ha for man kinds of waves wo or more waves can raverse he same space independenl of one anoher. I means ha he displacemen of an paricle a a given ime is simpl he sum of he displacemens ha he individual waves alone would give i. This pe of addiion is called superposiion. The wave rains involved are said o inerfere. Inerference refers o he phsical effecs of superimposing wo or more wave rains. Le us consider wo waves of equal frequenc and ampliude raveling wih he same speed in he same direcion ( + ) bu wih a phase difference ϕ beween hem. The equaion of he wo waves are: 1 = Asin( k ω) 2 = Asin( k ω ϕ). Now le us find he resulan wave, if superposiion occurs, ha is he sum: ( ω ) ( ω ϕ) = 1+ 2 = A sin k + sin k From he rigonomeric equaion for he sum of he sine of wo angles: B. Palásh 4
5 we obain α + β α β sinα+ sin β = 2sin cos 2 2 ϕ ϕ = 2Asin k ω cos 2, or 2 2Acos ϕ ϕ = sin k ω 2 2. This resulan wave corresponds o a new wave having he same frequenc bu wih am ϕ ampliude 2Acos The ver special cases: ϕ When ϕ = 0, cos = 1, he wo waves have he same phase. The waves inerfere 2 o ϕ consrucivel, and he resulan ampliude is 2A. If ϕ = 180, cos = 0, he wo waves are 2 in opposie phase, he inerfere desrucivel, and he resulan ampliude is zero. The principle of superposiion is of cenral imporance in all pes of wave moion. I applies no onl o waves on a sring, bu also o sound waves, elecromagneic waves (such as ligh) and all oher wave phenomena in which he wave equaion is linear. 2.4 Sanding Waves Le us ie one end of an elasic sring o a suppor and oscillae on he oher end. We can see a wave rain raveling owards he suppor. The raveling wave in he sring is refleced from he boundar, which is he suppor. Such reflecion gives rise o a wave raveling in he sring in he opposie direcion. The refleced wave adds o he inciden wave according o he principle of superposiion. Consider now wo wave rains of he same frequenc, speed and ampliude which are raveling in opposie direcion along a sring. Two such waves ma be represened b he ne equaions: B. Palásh 5
6 1 = Asin( k ω), 2 = Asin( k+ ω). We can wrie he resulan as: ( ω ) ( ω ) = 1+ 2 = A sin k + sin k. Using he same rigonomerical equaion as before we obain: = 2Asink cosω. This is he equaion of a sanding wave. Noice ha a paricle a an paricular poin eecues simple harmonic moion as ime goes on, and ha all paricles vibrae wih he same frequenc. In a raveling wave each paricle of he sring vibraes wih he same ampliude. Characerisic of a sanding wave, however, is he fac ha he ampliude is no he same for differen paricles bu varies wih he locaion of he paricle. The ampliude is 2Asin k, and i has maimum value of 2A a posiions where: or π 3π 5π k =,,,... ec =,,,... ec These poins are called aninodes and are separaed or spaced onehalf wavelengh apar. The ampliude has a minimum value of zero a posiions where: or k = π, 2 π, 3 π,... ec 3 =,,,... ec. 2 2 These poins are called nodes and are spaced onehalf wave lengh apar. The separaion beween a node and he adjacen aninodes is onequarer wave lengh. The superposiion of B. Palásh 6
7 an inciden wave and a refleced wave, being he sum of wo waves raveling in opposie direcions, will give rise o a sanding wave. B. Palásh 7
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