高中物理教学论文 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

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=sinx, y2=sin2x, y=sinx+sin2x,令x从-6.3到6.3（单位：弧度）即从-2π到2π两个周期，每次梯增

0.1。作出数据表及图象如下。

x sinx sin2x sinx+sin2x

-6.3-0.01681-0.03362-0.05044

-6.20.0830890.1656040.248694

-6.10.1821630.3582290.540392

-60.2794150.5365730.815988

-5.90.3738770.693525 1.067402

-5.80.4646020.822829 1.287431

-5.70.5506860.919329 1.470014

-5.60.6312670.979178 1.610444

-5.50.705540.99999 1.705531

-5.40.7727640.980936 1.753701

-5.30.8322670.922775 1.755043

-5.20.8834550.827826 1.711281

-5.10.9258150.699875 1.625689

-50.9589240.544021 1.502945

-4.90.9824530.366479 1.348932

-4.80.9961650.174327 1.170491

-4.70.999923-0.024780.975148

-4.60.993691-0.222890.770801

-4.50.97753-0.412120.565412

-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

-3.30.157746-0.31154-0.1538 -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 -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 -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 -0.2-0.19867-0.38942-0.58809 -0.1-0.09983-0.19867-0.2985 -2E-14-2E-14-3.9E-14-5.9E-14

0.10.0998330.1986690.298503

0.20.1986690.3894180.588088

0.30.295520.5646420.860163

0.50.4794260.841471 1.320897 0.60.5646420.932039 1.496682 0.70.6442180.98545 1.629667 0.80.7173560.999574 1.71693 0.90.7833270.973848 1.757175 10.8414710.909297 1.750768 1.10.8912070.808496 1.699704 1.20.9320390.675463 1.607502 1.30.9635580.515501 1.47906 1.40.985450.334988 1.320438 1.50.9974950.14112 1.138615 1.60.999574-0.058370.941199 1.70.991665-0.255540.736124 1.80.973848-0.442520.531327 1.90.9463-0.611860.334442 20.909297-0.75680.152495 2.10.863209-0.87158-0.00837 2.20.808496-0.9516-0.14311 2.30.745705-0.99369-0.24799 2.40.675463-0.99616-0.3207 2.50.598472-0.95892-0.36045 2.60.515501-0.88345-0.36795 2.70.42738-0.77276-0.34538 2.80.334988-0.63127-0.29628 2.90.239249-0.4646-0.22535 30.14112-0.27942-0.1383 3.10.041581-0.08309-0.04151 3.2-0.058370.1165490.058175 3.3-0.157750.3115410.153796 3.4-0.255540.4941130.238572 3.5-0.350780.6569870.306203 3.6-0.442520.7936680.351147 3.7-0.529840.8987080.368872 3.8-0.611860.967920.356062 3.9-0.687770.9985430.310777 4-0.75680.9893580.232556 4.1-0.818280.9407310.122453