机械波的种类

機械波MechanicalWaves•Mechanicalwaveisadisturbancethattravelsthroughsomematerialorsubstancecalledthemediumforwave.•Transversewaveisthewavethatthedisplacementofthemediumareperpendicularortransversetothedirectionoftravelofthewavealongthemedium•Longitudinalwaveisthewavethatthemotionsofparticlesisthesameasthedirectionofwave.機械波的種類LM,TMLTL),(2v單位因次分析（DimensionalAnalysis）和波速有關的物理量有繩子單位長度的質量m及繩上之張力τ←單位因次唯一可能Cv其中C為一無單位之常數。繩波的速度力學分析(,)(,)yyxtFxtFxxxxyyyxyxyF21FFF22222),(1),(ttxyvxtxyWaveEquation22FtyxxyxyFxxxy22tyFxxyxyxxxFvMathematicalDescriptionofaWavetvxπfAtvxωAtxy2coscos),(sinusoidalwavemovingin+x-directionTtxAtxyλ2cos),()(cos),(ωtkxAtxyPhaseEnergyinWaveMotionttxyxtxyFtxvtxFtxPy),(),(),(),(),(TheinstantaneousrateofdoingworkatthepointxandtimetisForsinusoidalwavewithwavefunctiongivenas)cos(),(tkxAtxy)(sin)(sin),(22222tkxAFtkxAFktxP22maxPFAMaximumPoweraveragepower,sinusoidalwaveonastring22av21AωμFP212221rrIIinverse-squarelawforintensity1214PIrAverageintensityWaveInterferenceBoundaryConditionsandSuperpositionOverlapoftwowavepulsesstandingwaveonastring,fixedendatx=0ωtkxAtxysin)sin(),(SWNormalModesofaString12nfLvnfnμFLf211100)cos(2)(nnntnAAtfT20)cos(101tA)cos(0nntnA傅立葉分析(FourierAnalysis)f(t)是週期性函數注意事項：稱為基頻(fundamentalfrequency)T為f(t)之週期。稱為基頻項或基頻分量，其他項稱為諧波(harmonics)n次諧波項：10)sin()(nntnAtf100)cos(2)(nntnAAtf若f(t)為t之奇函數若f(t)為t之偶函數作業:以下列函數近似鋸齒波(sawtoothwave)ntnntysin1)(利用電腦繪出(1)n=3(2)n=7之圖形SoundWavesSoundisalongitudinalwaveinamediumAudiblerangeofhumanearis20to20000HzGaugepressureGaugepressure210()[(,)(,)][()(,)](,)lim(,)/(/)(,)(,)sin()xVSyySyxxtyxtdVSyxtyxtyxtVSxxAndBpxtdVVyxtpxtBBkAkxtxBkApmaxInasinusoidalsoundwavethemaxpressurevariationsare3.0*10-2Paabovetheatmosphericpressure.Findthemaxdisplacementiffrequencyis1000Hz.Givenspeedofsoundis344m/sandbulkmodulusis1.42*105Pa2max58218.3/3.010(1.4210)(1.83/)1.210fkradmvvpPaABkParadmmTheAnatomyofHumanEarAeardrum=43mm2Astirrup=3.2mm2Vfluid=1500m/sPaBfluid91018.2PaPearinner4.0)max(mAinner11104.42221AωρBIintensityofasinusoidalsoundwaveBpvpI222max2max0logdB10IIβdefinitionofsoundintensitylevelBvBkApmaxSoundVelocityandGaugePressurepBVpdVdpVdppdVisothermal,0MRTpvpBVpdVdpdVpVVdpconstpVa,0.1空氣的bulkmodulus若為等溫過程，pV=nRT，用此結果代入所得之聲速較實際值小。在聲音的傳遞，速度很快，空氣和周圍來不及做熱交換，要考慮絕熱過程，即。m/s972kg/kmole00.4K273KJ/kmole10314.8335vHe在STP，M0=4.00kg/kmole,=5/3＊＊吸一口He氣體，講話音調會變高，為什麼？求在0℃的聲速BasicPrincipleofDopplerFlowmeter16.8TheDopplereffectMovinglistenerandstationarysource1LLsvffvSSLLfvvvvfMovingsourceandmovinglistenerttssttsttsssstsstssmmmmmcoscos2)(),()(cos)(cos2coscoscos,cos21212121212121212121221121beat21beat2fff2smcos't拍音可利用來做樂器的調音。人類可感覺之拍音約7HzBeats拍音PeopleusedDopplerflowmetertomeasurethebloodflowrates.Thespeedofsoundinbloodisknowntobe1570m/sandatypicalredbloodcellhasaradiusof5μm.Pleaseestimatewhatwouldbethefrequencyfortheflowmeter?IfthefrequencyshiftmeasuredbytheDopplerflowmeteris100Hz(themeterwasoperatedat107Hz),pleasefindtheaveragevelocityofbloodflow.1.5cm/scscsvvvvffShackWave(SonicBoom)Gv橫波(transversewave)彈性體所產生之形變為剪應變(shearstrain)Yv縱波(longitudinalwave)彈性體所產生之形變為拉伸應變(tensilestrain)恢復力為拉伸應力(tensilestress)m/s3000kg/m1083.7Pa103.83310Gv(a)鋼之密度為7.83×103kg/m3，G=8.3×1010Pa，橫波在鋼中傳遞之速率為(b)鑽石的橫波聲速高達12800m/s，密度為3.51×103kg/m3，他的shearmodulus為2vG=5.75×1011Pa