高中物理 钢琴振动曲线研究_乐音由若干个振幅和频率不同的简谐振动合成教学论文 新人教版选修3-4

1、钢琴振动曲线研究高中物理教科书中关于“乐音由若干个振幅和频率不同的简谐振动合成的”知识，多年来由于没有先进的教学工具，以致于不容易说清楚，目前，可以借助于电脑的Excel进行研究，可以说明白了。人民教育出版社物理室编的全日制普通高级中学教科书（试验修订本 必修）物理第一册第166页写道“钢琴的振动是由16个频率不同的简谐振动合成的，乙图（本文图1）表示出了这16个简谐振动的频率和强度的关系”相对声强0 400 800 1200 1600 2000 频率 图1为了研究钢琴的振动曲线是怎样由这16个频率不同的简谐振动合成的，先看一看比较简单的情况。1 两个频率不同的简谐振动合成怎样的曲线？设y1=

2、sinx, y2=sin2x, y=sinx+sin2x,令x从-6.3到6.3（单位：弧度）即从-2到2两个周期，每次梯增0.1。作出数据表及图象如下。xsinxsin2xsinx+sin2x-6.3-0.01681-0.03362-0.05044-6.20.0830890.1656040.248694-6.10.1821630.3582290.540392-60.2794150.5365730.815988-5.90.3738770.6935251.067402-5.80.4646020.8228291.287431-5.70.5506860.9193291.470014-5.60.631

3、2670.9791781.610444-5.50.705540.999991.705531-5.40.7727640.9809361.753701-5.30.8322670.9227751.755043-5.20.8834550.8278261.711281-5.10.9258150.6998751.625689-50.9589240.5440211.502945-4.90.9824530.3664791.348932-4.80.9961650.1743271.170491-4.70.999923-0.024780.975148-4.60.993691-0.222890.770801-4.50

4、.97753-0.412120.565412-4.40.951602-0.584920.366685-4.30.916166-0.73440.181769-4.20.871576-0.85460.016977-4.10.818277-0.94073-0.12245-40.756802-0.98936-0.23256-3.90.687766-0.99854-0.31078-3.80.611858-0.96792-0.35606-3.70.529836-0.89871-0.36887-3.60.44252-0.79367-0.35115-3.50.350783-0.65699-0.3062-3.4

5、0.255541-0.49411-0.23857-3.30.157746-0.31154-0.1538-3.20.058374-0.11655-0.05818-3.1-0.041580.0830890.041509-3-0.141120.2794150.138295-2.9-0.239250.4646020.225353-2.8-0.334990.6312670.296278-2.7-0.427380.7727640.345385-2.6-0.51550.8834550.367953-2.5-0.598470.9589240.360452-2.4-0.675460.9961650.320701

6、-2.3-0.745710.9936910.247986-2.2-0.80850.9516020.143106-2.1-0.863210.8715760.008366-2-0.90930.756802-0.15249-1.9-0.94630.611858-0.33444-1.8-0.973850.44252-0.53133-1.7-0.991660.255541-0.73612-1.6-0.999570.058374-0.9412-1.5-0.99749-0.14112-1.13861-1.4-0.98545-0.33499-1.32044-1.3-0.96356-0.5155-1.47906

7、-1.2-0.93204-0.67546-1.6075-1.1-0.89121-0.8085-1.6997-1-0.84147-0.9093-1.75077-0.9-0.78333-0.97385-1.75717-0.8-0.71736-0.99957-1.71693-0.7-0.64422-0.98545-1.62967-0.6-0.56464-0.93204-1.49668-0.5-0.47943-0.84147-1.3209-0.4-0.38942-0.71736-1.10677-0.3-0.29552-0.56464-0.86016-0.2-0.19867-0.38942-0.5880

8、9-0.1-0.09983-0.19867-0.2985-2E-14-2E-14-3.9E-14-5.9E-140.10.0998330.1986690.2985030.20.1986690.3894180.5880880.30.295520.5646420.8601630.40.3894180.7173561.1067740.50.4794260.8414711.3208970.60.5646420.9320391.4966820.70.6442180.985451.6296670.80.7173560.9995741.716930.90.7833270.9738481.75717510.8

9、414710.9092971.7507681.10.8912070.8084961.6997041.20.9320390.6754631.6075021.30.9635580.5155011.479061.40.985450.3349881.3204381.50.9974950.141121.1386151.60.999574-0.058370.9411991.70.991665-0.255540.7361241.80.973848-0.442520.5313271.90.9463-0.611860.33444220.909297-0.75680.1524952.10.863209-0.871

10、58-0.008372.20.808496-0.9516-0.143112.30.745705-0.99369-0.247992.40.675463-0.99616-0.32072.50.598472-0.95892-0.360452.60.515501-0.88345-0.367952.70.42738-0.77276-0.345382.80.334988-0.63127-0.296282.90.239249-0.4646-0.2253530.14112-0.27942-0.13833.10.041581-0.08309-0.041513.2-0.058370.1165490.0581753

11、.3-0.157750.3115410.1537963.4-0.255540.4941130.2385723.5-0.350780.6569870.3062033.6-0.442520.7936680.3511473.7-0.529840.8987080.3688723.8-0.611860.967920.3560623.9-0.687770.9985430.3107774-0.75680.9893580.2325564.1-0.818280.9407310.1224534.2-0.871580.854599-0.016984.3-0.916170.734397-0.181774.4-0.95

12、160.584917-0.366684.5-0.977530.412118-0.565414.6-0.993690.22289-0.77084.7-0.999920.024775-0.975154.8-0.99616-0.17433-1.170494.9-0.98245-0.36648-1.348935-0.95892-0.54402-1.502955.1-0.92581-0.69987-1.625695.2-0.88345-0.82783-1.711285.3-0.83227-0.92278-1.755045.4-0.77276-0.98094-1.75375.5-0.70554-0.999

13、99-1.705535.6-0.63127-0.97918-1.610445.7-0.55069-0.91933-1.470015.8-0.4646-0.82283-1.287435.9-0.37388-0.69353-1.06746-0.27942-0.53657-0.815996.1-0.18216-0.35823-0.540396.2-0.08309-0.1656-0.248696.30.0168140.0336230.050437 图2 两个频率不同的简谐振动的合成由上图可以看出，两个频率不同的简谐振动的合振动，虽然不是简谐振动，但也是周期性振动，可以成为乐音。2 图中16个频率不同的

14、简谐振动怎样合成钢琴的振动曲线？先把图1中表达的16个简谐振动表达为三角函数的正弦函数。为方便起见，设基音为，根据图1中的表示，其余的15个应该是： ，那么，钢琴的振动就是。0000000000000000000.10.0998330.1192020.073880.1168260.0599280.2823210.0805270.2869420.2349980.2524410.2673620.3728160.2890670.2956350.2974990.194773.3240480.20.1986690.2336510.1411610.2152070.1051840.466020.123181

15、0.3998290.2921540.2727890.2425490.2701850.154650.100496-0.07666-0.08853.050560.30.295520.3387850.1958320.2796120.1246870.4869240.1079010.2701850.1282140.042336-0.04732-0.17701-0.20633-0.26147-0.27774-0.154551.1455640.40.3894180.4304140.233010.2998720.1136620.3377320.041874-0.02335-0.13276-0.22704-0.

16、28548-0.39847-0.26504-0.189380.1482340.1587340.6314390.50.4794260.5048830.2493740.2727890.0748090.07056-0.04385-0.30272-0.29326-0.28768-0.21166-0.111770.0645360.1970960.2395460.0824240.9845080.60.5646420.5592230.2434620.2026390.01764-0.22126-0.10895-0.39847-0.23183-0.083820.0934620.3174670.2995630.2

17、5638-0.20996-0.196191.1040020.70.6442180.591270.2158020.100496-0.04385-0.43579-0.12281-0.252510.0050440.1970960.296450.341840.09573-0.10994-0.185440.0067251.3443370.80.7173560.5997440.168866-0.01751-0.0946-0.49808-0.078910.046620.23810.2968070.175475-0.06973-0.24835-0.293750.2577490.1931321.3929140.

18、90.7833270.5843090.106845-0.13276-0.12219-0.386380.0021020.3174670.2909670.123636-0.13726-0.39237-0.22860.0100870.119022-0.094480.84371710.8414710.5455780.03528-0.22704-0.11987-0.139710.0821230.3957430.123636-0.16321-0.3-0.214630.126050.297182-0.28842-0.15020.8440011.10.8912070.485098-0.03944-0.2854

19、8-0.088190.1557710.1235210.233967-0.13726-0.3-0.134890.2368290.2960320.090936-0.04470.1627351.6461341.20.9320390.405278-0.11063-0.29885-0.034930.3968340.106825-0.06973-0.29428-0.160970.1776220.3862630.032326-0.266270.2999380.076251.5777151.30.9635580.309301-0.17194-0.265040.026890.4992720.039887-0.3

20、3113-0.22860.126050.2960320.043101-0.27874-0.18145-0.03259-0.197380.6172261.40.985450.200993-0.21789-0.189380.0821230.427299-0.04581-0.391670.0100870.2971820.090936-0.35503-0.181450.204589-0.291540.0134420.639331.50.9974950.084672-0.24438-0.083820.117250.206059-0.10996-0.214630.2411350.195086-0.2135

21、4-0.300390.1816620.2509970.1077180.1912751.406621.60.999574-0.03502-0.249040.0349650.12367-0.08716-0.12240.0926040.289697-0.08637-0.284650.1373260.278639-0.119270.263782-0.100361.1359811.70.991665-0.15332-0.231450.1482340.099811-0.34994-0.077270.3436650.119022-0.28842-0.04470.399917-0.03259-0.29154-

22、0.17569-0.145670.3117181.80.973848-0.26551-0.193190.23810.051515-0.490470.0042030.386263-0.14173-0.22530.2441020.1525-0.296070.020162-0.218510.1665520.4064681.90.9463-0.36711-0.137670.290376-0.00939-0.459660.0836960.194559-0.295220.0449630.266147-0.2894-0.125810.2983930.2319980.069990.74215420.90929

23、7-0.45408-0.069850.296807-0.068-0.268290.123826-0.11516-0.22530.273884-0.00266-0.362230.2287680.0812720.158725-0.198360.3086542.10.863209-0.522950.0042030.25638-0.109960.0168120.105718-0.355030.0151270.250997-0.268560.0268830.248198-0.27077-0.27290.0201430.0075142.20.808496-0.570960.0778850.175475-0

24、.1250.2960370.03789-0.379540.244102-0.00266-0.240980.381714-0.09598-0.17331-0.08840.1892030.5339742.30.745705-0.596210.144610.066867-0.109430.471848-0.04776-0.173830.288346-0.253870.0499440.249751-0.299550.2118510.29568-0.106120.9378382.40.675463-0.59770.198417-0.0523-0.067070.482829-0.110950.137326

25、0.114375-0.271670.286286-0.20072-0.064280.245330.012208-0.140980.6465722.50.598472-0.575350.2345-0.16321-0.008290.325144-0.121950.365178-0.14615-0.039710.209772-0.395210.265161-0.12845-0.298830.1701810.2912522.60.515501-0.530070.249636-0.248350.0525210.053877-0.07560.371518-0.296070.228768-0.09598-0

26、.08570.206136-0.2890.0647960.0636510.1856262.70.42738-0.463660.242472-0.294280.100473-0.236210.0063030.1525-0.221930.286913-0.296850.333104-0.154880.0302150.282129-0.1991-0.005422.80.334988-0.378760.21365-0.293750.123826-0.443780.085245-0.159020.0201620.081272-0.173310.327107-0.2890.299267-0.13750.0

27、26822-0.362792.90.239249-0.278760.165742-0.246850.116862-0.496330.124096-0.374080.247-0.199090.139617-0.096040.0002660.071516-0.24670.186916-0.6465930.14112-0.167650.10303-0.160970.081286-0.375490.104582-0.362230.286913-0.296410.299974-0.396710.289139-0.274960.201069-0.11176-0.639073.10.041581-0.049

28、850.031114-0.049680.025808-0.123490.035882-0.130650.109696-0.121210.132517-0.191460.154422-0.164980.194884-0.13613-0.241563.2-0.058370.06993-0.043580.069453-0.035990.171657-0.049690.180176-0.150540.165428-0.179760.257959-0.206520.218873-0.251290.1736170.3313523.3-0.157750.186925-0.114380.177622-0.08

29、8970.406837-0.11190.381714-0.296850.299974-0.295590.378405-0.264910.239386-0.130130.057240.6676243.4-0.255540.296468-0.174970.257749-0.120170.499896-0.121470.351709-0.218510.158725-0.08840.0162770.064796-0.13750.284821-0.199630.6142483.5-0.350780.394192-0.219920.297182-0.121950.418328-0.073920.10836

30、20.025192-0.128450.215392-0.366610.299577-0.286130.0567340.0334710.3006623.6-0.442520.476201-0.245230.289697-0.093870.1906250.008401-0.200720.249828-0.297530.283804-0.281960.0954770.040234-0.299440.184418-0.04263.7-0.529840.539225-0.248640.236476-0.04281-0.103670.086771-0.388040.285399-0.193060.0420

31、720.162265-0.24850.2998030.020428-0.11727-0.199393.8-0.611860.580752-0.229830.145920.018735-0.361750.124331-0.339990.1049850.088911-0.245640.39956-0.228420.061680.294177-0.13113-0.329563.9-0.687770.599126-0.19050.0323260.075692-0.493460.103416-0.0857-0.154880.289139-0.264910.1273030.126292-0.27884-0

32、.096230.176857-0.722134-0.75680.593615-0.13414-0.086370.114118-0.452790.0338630.220571-0.297530.2235340.005311-0.30730.295988-0.15647-0.269380.050765-0.923024.1-0.818280.564438-0.06581-0.191430.124604-0.25395-0.051620.393047-0.21502-0.047590.269729-0.350010.0320610.2256480.16565-0.19992-0.418454.2-0

33、.871580.5127590.008406-0.266270.1045820.033604-0.112820.3271070.030215-0.274960.2393860.053645-0.278840.2331710.2266920.0400820.0051914.3-0.916170.4406380.081869-0.299070.0589550.309418-0.120960.0627470.252585-0.24953-0.052560.388886-0.18124-0.14638-0.224070.181711-0.413174.4-0.95160.350950.148018-0

34、.28465-0.001110.477143-0.07221-0.239670.2838040.005311-0.287070.2281870.181874-0.28293-0.16895-0.12265-0.735574.5-0.977530.2472710.200946-0.2253-0.06090.4781880.010497-0.396710.1002450.255271-0.20787-0.223520.278540.0502070.267604-0.12598-0.329034.6-0.993690.1337340.235924-0.13037-0.105780.3121890.0

35、88271-0.31311-0.159180.2705370.098494-0.39017-0.032860.2999990.0999940.179897-0.406114.7-0.999920.0148650.249827-0.01486-0.124760.0371330.12453-0.03958-0.298140.0370720.297219-0.05925-0.296120.051773-0.293370.044232-1.269354.8-0.99616-0.10460.2414140.102994-0.1132-0.250890.1022210.257959-0.21147-0.2

36、30480.171140.347234-0.12557-0.2824-0.0244-0.2-1.31624.9-0.98245-0.219890.2114370.204589-0.07392-0.451280.0318350.3990230.035229-0.28613-0.141960.3108940.22894-0.147770.2996570.046648-0.535145-0.95892-0.326410.1625720.273884-0.01654-0.49402-0.053520.2980450.255271-0.07871-0.29993-0.121920.2480490.232

37、167-0.052820.178799-0.754025.1-0.92581-0.419920.0991850.2999380.044882-0.36418-0.113710.0162770.2821290.201069-0.13013-0.39925-0.096230.226692-0.28605-0.12789-1.693025.2-0.88345-0.49670.0269380.2786390.09532-0.10713-0.12042-0.275360.0954770.2959880.181874-0.16742-0.29953-0.155110.126533-0.12068-1.525035.3-0.83227-0.55367-0.047710.2133480.122420.18735-0.07049-0.39997-0.163430.1187780.2951240.277921-0.06402-0.279420.2534390.182734-0.759865.4-0.77276-0.58856-0.118110.1143750.1195470.416380.01259-0.28196-0.29866-0.167640.0858610.3688360.2652860.060123-0.19