1、利用旋轉液體特性測量液體折射率（全面版）資料第27卷第7期2007年7月物 理實驗 P H YSICS EXPERIM EN TA TIONVol.27 No.7J ul.,2007圓柱容器XYJ旋轉液體物件儀轉盤電動機ft細唱收稿日期：2006212215修改日期：2007203219 作者簡介:高 嚴（1986-,男山東日照人，中國石油大學（北京資源與信息地質工程專業2004級本科生.指導教師:王愛軍（1968-，男，河北沽源人,中國石油大學（北 京數理系副教授，碩士,從事凝聚態物理研究.利用旋轉液體特性測量液體折射率高 嚴 1, 范 凱 1, 王愛軍 2, 唐軍杰 2(1.中國石油大學

2、(北京資源與信息學院 ,北京 102249;2.中國石油大學 (北京數理系 ,北京 102249摘 要:在XY 21型旋轉液體實驗儀的圓柱形容器底部加裝了圓形平面鏡，利用旋轉液體的幾何特性和折射定律 ,即可測量液體的折射率 .關鍵詞:旋轉液體;旋轉拋物面;折射率中圖分類號 :O351;O552.423 文獻標識碼 :A 文章編 號:100524642(200707200422031引言盛有液體的圓柱形容器繞其圓柱面的對稱軸勻速轉動時 ,旋轉液體的表面將成為拋物面 .拋物面的參數與重力加速度有關 ,利 用此性質可以測重力加速度 ;旋轉液體的上凹面可作為光學系統加以研究,還可測定液體折射率等 ;因

3、此旋轉液體實驗是一個內容十分豐富的綜合性實驗12.但是,在旋轉液體特性研究實驗中測量測液體折射率采用的是光柵衍射的方法,比較繁瑣 ,沒有充分運用旋轉液體的特性 .本文對旋轉液體實驗裝置的圓柱形容器稍作改進,根據旋轉液體的幾何特性和折射定律 ,提出一種利用旋轉液體特性測量液體折射率的方法 .2 實驗原理及儀器改進2.1 勻速旋轉液體的幾何性質半徑為R盛有液體的圓柱形容器，當圓柱體繞對稱軸以角速度 勻速穩定轉動 時，液體的表面將成為拋物面.設x軸為水平方向,y軸為垂直水平面向上方向，拋物面 的方程為 1,3y = 3 2x 22g +h-Qo 2R24gJ(1其中|x | < R ,1是圓柱

4、形容器內液體靜止時液面高度,g是重力加速度.當x =x0=R2時,由(1式得y (x 0=h 0,即液面在x 0處的高度是恒定的，它不隨旋轉圓柱體的轉動角速 度改變而改變 ,圖1為旋轉液體特性實驗裝置示意圖 .透明屏幕上有毫米坐標 ,用于實驗中讀取 入射光點與反射光點的距離 ,屏幕可在豎直方向上下移動 .圓筒側壁有毫米刻度線 ,用 于讀取液面高度 ,也可以用直尺測量 .圓筒底部正中央有小標識 ,用以確定光軸 .圓形轉 盤由直流電動機驅動，可通過調節直流電源的電壓改變液體轉動的角速度.用XY 21旋轉液體物性儀測量液體旋轉周期 .儀器底座有氣泡式水平儀 ,圓柱形容器的內徑用游標卡尺測量 .圖 1

5、 旋轉液體特性實驗裝置示意圖為了便于測量液體折射率 ,在原儀器的圓形容器底面加裝圓形平面鏡 ,用以加強容器底面反射光線的強度.2.3利用旋轉液體特性測液體折射率實驗原理如圖2所示，設圓柱形容器內液體靜止時液面高度為h 0當液體旋轉起來后，根據旋轉液體性質，在距圓柱形容器中心軸 R/2 處旋轉液體的高度仍為h 0,不隨旋轉液體的轉動角速度改變而改變.調節激光筆使讓 激光束豎直入射到旋轉液體液面的不動點B處.設入射光線為A B ,經拋物液面反射的光線為B C ,其中A和C分別為入射光線和反射光線與水平半透明屏幕的交點，經過拋物面折射后的光線為B D. 9為入射光線A B的入射角，B為折射光線B D

6、的 折射角.設液體折射率為n ,根據折射定律有：n =si n9 1sin 9 2圖2旋轉液體中的光路下面來計算9 和 9 2設B點在圓柱形容器底面的投影點為F.設經過圓柱形容器底面反射鏡反射的光線為D E,E為反射光線D E和透明圓柱形容器側壁交點,E在圓柱 形容器底面的投影為G點.由圖2有：12/ A B C =12arctan A C A B,(3 9 2二(E1FB D = 9 -arctanFD.（4另有幾何關系 B D F sED G,則有FD GD =B FEG.（5又因GD =FG -FD ,將GD和（5式代入（4式有:0 2= BdrctanFGEG +B F（6設透明屏幕上

7、部到圓柱形容器底面的距離為H ,EG =h 1,A C =d.已知 B F =h 0,A B =H -h 0,FG =R -R2將上述A B ,A C ,FG,EG和B F的結果代入（3式和（6式得:1=12arctan dH -h 0,(70 2= BdrctanR -R2h 1+h 0.(8調節轉速使點E恰好打在拋物液面與容器壁的交線上，則E點距圓柱形容器底面的距離h 1滿足拋物線方程 (1式,則有h 1= 3 2R 22g +h 0L 2R 24g = 3 2R24g +h 0.(9又由3 =2n T ,為液體旋轉周期，將3 =2nT和(9 式代入 (8 式有2=0-a1rctan1-1

8、2R 2h 0+ n 2R 2g T2.（10實驗中只要測出R ,h 0,d ,H和T ,代入（7和（10式算得9 1和 9 2再根據式（2即可 求得液體折射率 n.3 實驗內容利用氣泡水平儀和平臺下的 3個可調螺絲 ,調節平臺水平用游標卡尺（精度為0.02mm測量出圓柱形容器的直徑2R.利用 自準法調節激光筆使其發出的激光豎直照射于液面,然后保持激光筆的豎直狀態 ,將豎直光線平移到距圓柱容器底面中心R/2處.用直尺測量液體靜止時液面到容器底部的距離 h 0及透明屏幕到容器底面的距 離H.打開電機并調節電機轉速 ,使經圓柱形容器底面平面鏡反射的光線剛好打在拋 物液面與圓柱容器側壁的交線上 .待

9、穩定后記下 XY 21 旋轉液體物性儀計時器顯示 的容器旋轉10周所用時間10T 用直尺測量激光束入射光線和經拋物液面反射的光線與透明屏幕的交點之間的距離 d.改變透明屏幕到圓柱形容器底面的距離 H 及液面高度 h 0,重復上述過程 ,記錄試驗數據 .4 實驗結果待測液體為甘油 ,實驗測量的數據如表 1 所34第 7期高 嚴,等:利用旋轉液體特性測量液體折射率示.表 1 中 R=5.564cm.表 1 實驗測量數據5 結束語本文對旋轉液體實驗裝置的圓柱形容器稍作改進,根據旋轉液體的幾何特性和折射定律 ,提出一種測量液體折射率的新方法.新方法原理簡單 ,儀器改進非常容易 ,只需在原有實驗裝置中添

10、加平面反射鏡即可.用這種新方法測液體折射率是對旋轉液體特性研究實驗實驗內容的一個很好的擴充 .參考文獻 :1 包奕靚黃吉,陸申龍新型旋轉液體實驗J.大學物理 ,2003,22(2:2730.2 袁野,晏湖根,陸申龍,等.旋轉液體實驗裝置的設計J.物理實驗,2004,24(2:4346.3 賈起民，鄭永令.力學（上冊M.上海:復旦大學出版社 ,1989.116117.R efractive index of liquids determined by rotating liquid methodGAO Yan1,FAN Kai1,WAN G Ai2jun2,TAN G J un2jie2(1.D

11、epartment of Resource and Information,China University of Petroleum2Beijing,Beijing102249,China;2.Department of Mathematics and Physics,China University of Petroleum2Beijing,Beijing102249,ChinaAbstract:The XY12type rotating liquid apparat us is modified by adding circular plane mirror un2 der t he c

12、ylindrical glass bottom.The refractive index of liquids can be measured using t he geometric characteristic of rotating liquid and t he Fresnell reflective law.K ey w ords:rotating liquid;rotating parabolic;ref ractive index責任編輯 :郭 偉利用透鏡組及相關成像設備，求出已知濃度鹽水的折射率，進 而求出凸透鏡的曲率半徑。並討論鹽水的折射率與濃度大小的關係 為何？原理1. 將

13、一物體置於凸透鏡焦點上，經平面鏡反射，再經此凸透鏡 聚光成像後，在原焦點上得到大小不變的倒立影像。用此成 像原理，將可經由實驗求出任何凸透鏡組的焦距。光線成像RiR22.由造鏡者公式f ( n-1)11可以得到以液體為材料的平凹透鏡焦距f液滿足(1)( n 液-1)- 可以得到配合組合透鏡焦距公式F丄nw 1 Fw f 凸廠丄丄Fxf凸其中，nw :水的折射率 nx :待求鹽水的折射率Fw :的透鏡及水組合透鏡'勺焦距 Fx:的透鏡及未知液體組合透鏡 '勺焦距 f凸：凸透鏡的焦距Fx及f凸，代入式，得鹽水的折射率nx，(2)式的推導見附錄。 經由實驗可測得Fw、再將nx代入(1

14、)式，得凸透鏡的曲率半徑。 探實驗步驟：1製作箭頭光柵後，架設實驗裝置，如下頁裝置圖:2. 移動光柵位置，找到與箭頭光柵同大小但倒立之清晰像，此光柵與透鏡的距離即為透鏡之焦距f凸。3. 在水平放置的平面鏡上滴上幾滴水，再將雙面對稱的薄凸透 鏡置於水滴上，兩鏡之間不能有任何氣泡，水填滿兩鏡間的 空隙。則此系統可視為雙凸透鏡（以玻璃為折射材料）及平凹 透鏡（以水為折射材料）的薄透鏡組。4. 移動光柵位置找出 Fw （凸透鏡及水之組合透鏡的焦距）。5. 將水改成5%鹽水，找出Fs （凸透鏡及5%鹽水之組合透鏡 的焦距）。6. 改變鹽水的濃度為10%、15%重複步驟5,找出Fi。、F15。 探結果：測

15、量出f凸、Fw、F5、F10、F15平均值見附表（一）（五），將其值 代入（2）式計算鹽水的折射率，並由（1）式計算出凸透鏡的曲率半徑 R 列於下表。水5%鹽水10%鹽水15 %鹽水平均值標準差折射率1.33001.34271.34751.3644計算的R（cm）35.300035.298135.306035.303135.30180.0030此實驗求出凸透鏡之曲率半徑 R，都非常接近，標準差亦非常小, 可見相當精確。從上表亦可得到，鹽水的濃度愈大折射率也愈大。附錄1利用£(n液-1)1及1 1f液RFf凸1 “111(nw1)()f水RF wf凸1111(眼1)()f鹽水RFxf凸

16、(2)式的推導如下:丄2得兩式可得表(一)凸透鏡的焦距f凸第一次第二次第三次第四次第五次平均值焦距(cm)33.633.333.233.333.433.36表(二)加水透鏡組的焦距第一次第二次第三次第四次第五次平均值焦距(cm)48.848.748.848.748.548.7表(三)加5 %鹽水透鏡組的焦距第一次第二次第三次第四次第五次平均值焦距(cm)49.549.249.349.249.549.34表(四)加10%鹽水透鏡組的焦距第一次第二次第三次第四次第五次第六次第七次平均值焦距(cm)49.749.550.249.749.750.049.949.67表(五 )加15%鹽水透鏡組的焦距第

17、一次第二次第三次第四次第五次第六次平均值1焦距(cm)51.051.050.850.950.850.850.88利用星載 SAR 差分干涉測量改進變形含水層系統的繪圖、監測和分析美國 研究委員會地面沉降座談會（NRC; 1991 ）就3種信息需求達成了共 識：“第一，有關地面沉降大小和分布的基本的地球科學數據和信息要得到認 可并用來評價未來的問題。這些數據不僅能夠幫助研究局部地區的沉降問 題，也能識別 范圍內的問題。第二，針對地面沉降開展沉降治理和工程方 法的研究為了有效阻止或控制破壞第三，盡管美國現行的地面沉降減輕 方法有很多種，但是對這些方法的成本效益進行研究將有助于決策者做出更好 的選擇

18、。”有各種基于地面和衛星的方法可用來測量含水層系統的壓縮和地面沉降（表 1）。 SAR 干涉測量理論上適合測量與含水層系統壓縮相關的地面變形的 空間范圍和大小。 InSAR 可以提供一個區域內覆蓋整個含水層系統的數百萬個 數據點，與使用大量人力而只能獲得有限個點測量數據的水準測量，和GPS測量相比，通常而言，花費要更低一些。通過識別研究區內某一變形的特定區 域， SAR 干涉測量也可以用于定點測量并同時監測局部和區域尺度上的地面沉 降（如鉆孔伸長計、GPS監測網絡、水準路線；Bawden等，2003）。SAR干 涉測量的這些優勢，尤其是InSAR，能夠滿足NRC提出的每一種信息需求。 SAR干

19、涉測量的另一個重要優勢就是 SAR歷史數據的存檔文件越來越多。在很 多地區，從上世紀 90年代初開始，就已經有了大量的數據集，因而這一時期的 地面形變歷史測量數據即可應用。此外，為滿足新需求可以定制新數據。詳細 的過程和費用要依賴于使用的傳感器Space-based Tect onic Modeli ng in Subducti onAreas Usi ng PSI nSARR. M. W. Muss onBritish Geological SurveyM. HaynesNPA GroupA. FerrettiTeleRilevame nto EuropaINTRODUCTIONWhile

20、the applicati on of In SAR (INterferometric Syn thetic ApertureRadar) tech niq ues to seismology has bee n well known since themid-1990s (Massonnet et al. ,1993 ; Massonnet et al. , 1996 ), PSInSAR is gen erally un familiar to the Earth scie nee com mun ity.The PS sta ndsfor "perma nent scatter

21、er", and it is the use ofthese (al ong with thevolume of sce nes employed) that dist in guishesthe method from morefamiliar In SAR tech niq ues. A perma nent scattereris any persiste ntlysuch as build ing roofs, boulders. The use of thesemeasureme nts of groundreflective pre-existing ground fea

22、ture, metallic structures, and eve n large features offers the possibility ofand over periods of time, in terferometry. Furthermore, of displaceme nts over the full (started in 1991) for anydisplaceme nts to a degree of accuracy, previously un obta in able from conven ti onal it is possible to con s

23、truct histories temporal exte nt of the SAR data archivetherefore represe ntsextremely dense twelve years.part of the globe with data coverage. PSIn SAR the equivale nt of a n ewly discovered, superaccurate, GPS network that has been in existence for the lastThe high resolution of PSInSAR data, coup

24、led with its being particularly suited to urba ni zed areas (nu merous build in gs,thereforemany PS poin ts), makes it an excelle nt tool for study ingthings suchas urban subsidence ( Ferretti et al., 2000; Mizuno and Kuzuoka, 2003 Dehls and Nordgule n, 2004). It also has applicati ons in seismology

25、:as a substitute for GPS data where these do not exist, and asanenhan ceme nt where they do .In this paper we report on a pilotprojectin Japa n, the prin cipal aim of which was to calibrateand test thePSI nSAR measureme nts in an area where ground truth is very well established from GPS and leveli n

26、g data. This workresults from aEuropea n Space Age ncy (ESA) "Earth Observati onMarket Developme nt"project en titled "Developi ng markets forEO-derived land motio nmeasureme nt products", i nvolving NPA (lead), the British Geological Survey (UK), Imperial College (UK), TeleRilev

27、ame nto Europa (Italy), Image One (Japa n), the Geographic Survey In stitute (Japa n), Oyo Corporatio n (J apan),Fugro (Netherla nds), and SARCOM (the ESA datadistribut ing en tity).TECHNICAL BASIS OF PSI nSARConven ti onal satellite radar in terferometry invo Ives the phase comparis on of syn theti

28、c aperture radar (SAR) images gatheredatdifferent times (Massonnet et al. ,1993 ; Massonnet et al. ,1994 ;Zebker et al. , 1994 ; Gens and van Genderen, 1996 ; Massonnet and Feigl, 1998). This technique has the potential to detect millimeter-level target displaceme nts along the lin e-of-sight (LOS)

29、directio n. Theaimof the in terferometric tech niq ues is to highlight possiblerangevariati ons of the target by means of a simple phase differe nee betwee n two images gathered at differe nt times. If the local reflectivity remains unchanged in time, its phase contribution disappears in the differe

30、 ntiatio n and possible range variatio nscanthen be detected.Si nee the wavele ngth of the illu min at ing radiati on is usuallya fewcen timeters (satellite SAR operates in the microwave doma in),eve n amillimetric range variatio n tran slates to a phase cha ngethat can bedetected. Due to low sig na

31、l-to-no ise ratio valuestypically prese ntin SAR phase values (n ever greater tha n 12dB), however, themon itori ng of subside nee rates of more tha n6-7 cm/year is notfeasible.Apart from cycle ambiguity problems, other limitati ons are duetotemporal and geometrical decorrelati on and to atmospheric

32、artifacts.Temporal decorrelatio n makes in terferometric measureme ntsso con sta nt be ane.g. , seismicun achievable where the electromag netic profiles an d/or the positi onsisi.e.of the scatterers change with time within the resolution cell, that the reflectivity phase contribution cannot be assum

33、ed with time. The use of short revisiting times proves to unsuitable solution, since very slow terrain motion ( creep) cannot be detected.Reflectivity variations as a function of the incidence angle ( geometrical decorrelatio n) further limit the nu mber of image pairs suitable for in terferometric

34、applicati ons, uni ess the cha ngeconfined to a pointwise character of the target (e.g. , a cornerreflector). In areas affected by either kind of decorrelati on, gen erati on of the in terferogram no Ion ger compe nsates the reflectivity phase con tributi on, and possible phase variati ons due to ta

35、rget motion cannot be highlighted.Fin ally, atmospheric heteroge neity creates an atmospheric phase scree n superimposed on each SAR image that can seriously compromise accurate deformati on mon itori ng. In deed, eve n con sideri ng areas slightly affected by decorrelati on, it may prove extremely

36、difficult to discrimi nate the sig nal of in terest from the atmosphericsig nature,at least using in dividual in terferograms.The PSInSAR method, developed by TeleRilevame nto Europa of the Politecnico di Mila no in Italy, provides a way to overcome these limitati ons. Although temporal decorrelatio

37、 n and atmospheric disturba nces still stro ngly affect in terferogram quality, reliable deformati on measureme nts can be obta ined in a multi-image framework on a small subset of image pixels corresp onding to stable areas.These poin ts, the perma nent scatterers (PS), can be used asa"n atura

38、l GPS n etwork" to mon itor terrain moti on, by an alyz ingthephase history of each one.every in time.in time and depe nding on the water pump ing, fault buildi ngs, etc.). removed by such as those gatheri ng data sinceAtmospheric artifacts show a stro ng spatial correlati on within single SAR

39、acquisition (Hanssen, 1998 ) but are uncorrelatedimages, and improve the that are not greatly are selected.Conv ersely, target moti on is usually stro ngly correlated can exhibit differe nt degrees of spatial correlatio n phe nomenon at hand ( e.g. , subside nee due to displaceme nts, localized slid

40、 ing areas, collaps ing Atmospheric effects can therefore be estimated and combi ning data from Ion g-time series of SAR images, available in the ESA ERS archive, which has bee n late 1991. To exploit all the available accuracy of the estimatio n, only scatterers affected by temporal and geometrical

41、 decorrelati onPossible stable and point targets, known as perma nent scatterers(PS),are detected on the grounds of the stability of theiramplitudereturns ( Ferretti et al. , 2001 ): i.e. , how constant their brightness or in te nsity remai ns from one SAR image tothe n ext. This allowspixel-by-pixe

42、l select ion with no spatialaverag ing. Due to the highspatial correlati on of the atmosphericcon tributi on, proper sampli ngof the atmospheric comp onents can be achieved with a sparse grid of measureme nts, provided that the PS den sity is high eno ugh (greater than 4-5 PS/km2; Ferretti et al. ,

43、2000, 2001). A sufficient number ofimages is needed (usually more than 30) to identify PS and separate the differe nt phase con tributi ons.areEve n though precise satellite positi on and velocity state vectorsimpact on atmospheric atmospheric corresp ond to cha nge the low wave- the sparse PS grid.

44、available for ERS satellites, orbit ambiguities and their the in terferograms cannot be n eglected. The estimated phase screen is actually the sum of two contributions: effects and frin ges due to orbital errors. The latter low-order phase polyno mials, however, and do not nu mber character of the s

45、ig nal to be estimated onAt the PS point, submeter accuracy elevati on and millimetricterrainmoti on detect ion (due to the high phase cohere nee ofthese scatterers)can be achieved once atmospheric con tributi onsare estimated andremoved. Relative target LOS velocity canbe estimated withun precede n

46、ted accuracy, sometimes eve n better tha n 0.1 mm/year, due to the long time spa n of the data used.The higher the accuracy ofthe measureme nts, the more reliable the differe ntiati ons betwee n models of the deformatio n processun der study.PILOT PROJECTThe area selected for the pilot project was t

47、he Tokai area inJapa n,in itially around Hamamatsu and the n exte nded to coverthe rest of thewest side of Suruga Bay and the n orther n partof the Izu Penin sula(Figure 1 ). This area was attractive forthe project for severalreas on s. It is one of the most inten sivelystudied areas in Japa n,becau

48、se it was ide ntified as the likely locati onof the next majorearthquake as long ago as the 1970s. It isan area of activetectonics in a complex structural setti ng.The prin cipal comp onent of the tecto nic structure in the Tokaidistrict is the collisio n of the n orthward-movi ng Philippi neSeaPlat

49、e (PSP) with Japa n ( Figure 1 ). This collisi onal processstartedabout 6-7 million years ago ( Niitsuma, 1982 ) and has been responsibleas the PSP subductsHon shu the situatio n subduct ion model. district is very muchThe subducti on tren ch, which,for most of the seismicity of souther n Japa n und

50、er the overlying Eurasian Plate. In southern is relatively simple and follows the conventional The plate boun dary geometry around the Tokai more complex, however ( Takahashi, 1994 ).as the Nan kai Trough, is orie nted n ortheast-southwestto the south ofHon shu, bends to an almost n orthsouth orie n

51、tatio nas the SurugaTrough in Suruga Bay. The most n ortherly pointof the PSP is occupiedby the Izu Penin sula, which is collidi ngwith Hon shu rather tha nbeing subducted ben eath it. The reas onfor this is believed to be therelative light ness of the volca nicrocks of the Izu Penin sula, thebuoya

52、ncy of which preve nts subduct ion (Takahashi, 1994 ). Theno rthward moveme nt of the PSP in the Izu area is thereforea processof collisi on tect onics akin to continen tal collisi onrather tha nnormal subduction ( Niitsuma and Matsuda, 1985; Koyama, 1991). Priorto the collisio n of the Izu Penin su

53、la, the Tan awa Blockcollided withthe Hon shu mainland duri ng the Mioce ne, and the process by which the Tan awa Block accreted to the mai nland is now being repeated with the Izu Penin sula ( Ama no, 1991). The process is described in detail by Takahashi ( 1994).The possibility, or eve n the proba

54、bility, of a large and disastrous earthquake in the Tokai district has been of concern since the area was ide ntified as a dan ger area and seismic gap by Mogi (1970) andIshibashi (1976). The subduct ion front from Shikoku to Hamamatsu hasbee n ide ntified as being partiti oned into several prin cip

55、al fault pla nes that appear to rupture in characteristic earthquakes. These segme nts were labeled A to D (from west to east) by Ando (1975), andthis system was expa nded by Sugiyama (1994) to in clude segme ntZ(Bungo Chann el) in the west and segme nt E (Suruga Bay) in theeast(Figure 1 ). Any larg

56、e earthquake may rupture one of thesesegme ntsen tirely, or in the worst case all six at on ce, asappare ntlyoccurred in the 1707 Hoei earthquake ( Sugiyama, 1994 ). For historical eve nts, the exte nt of the rupture can be deducedfrom in formatio n oninten sity distributi on, tsun ami distributi on,and ground deformati on(Utsu, 1974 ; An do, 1975)