长为[tex=0.357x1.0]5vVfAZliYwqMw8JaLE+iEA==[/tex]的弦，两端固定，弦中张力为[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]，在距一段为[tex=0.929x1.0]XQ8c0totc8uufRPOvpPxwQ==[/tex]的一点以力[tex=1.0x1.214]kLwCQn0d73TWOUUlnBVGBA==[/tex]把弦拉开，然后突然撤除这力，求解弦的振动[img=291x147]178efbe78e53a93.png[/img]
解：由题意得方程[img=396x160]178f211209b6e2a.png[/img]令u(x,t)=X(x)T(t)，代入泛定方程得[tex=11.0x2.429]Q03B3rKHsohITpeV7soeSpeNCOlk/q0ppnHpON/BThvWjdpL3YmOZUE/3bRwp7MGTUozfJQxpv26ES09Jj7FNKTEs76/3e5m+9IElvXPX+HC5FYmBfZAx/yuTWLkGFaB[/tex].进一步可得[tex=11.714x3.214]a0s3MH7cLIdmiBRR0YN069DZiBF0qjveYZ8YvfEOQDgjJEFyOqWsk4msdOOx5NfL2CRZ1frt32VqH/P2Y5CmrKIMnVIgVVXCz0CqGT8Y7uBtSSqBhd4v4kkbjhtdLik8YCEWfXyftOPNKr9osADR8FDVMcl4OXrSFQdwXp4X0iU=[/tex].对时间变数t，解得[tex=11.0x1.357]NXlNY7vc/t6RZ3u5O41BdNRVarrbXkWh69jNAOIYMlwiCg/bLmZvARGslc+feqk0[/tex].对空间变数x，解得[tex=10.857x1.357]dH0PzSOUjmE63PcYANW+++vm+FJyo+hrN4HS/FaKYMLNTiekFil8h8NGPmi4A3nV[/tex].由X(0)=0可知C=0，由X(l)=0可知，[tex=13.429x1.357]S8h0US4Xd82meSsyjOQr0d7+HBSeE+E2RubLXiAyg08H6xsZqCVtcE5Z96yT91/PkG39SiB41Kce5xWHPTsjGw==[/tex]，[tex=22.214x5.786]c8gX0O6CKBpyqTBZ2fB4DgmFxt7hCZBSvzKPEfIWOEySHwLpKTA0CFrJVEpTKaSzvBDEu2da8XK04iAzx2HZ3dHJ1/BhvCcw0AIBVdRBT9EWTWQau2SikptXIQ+7enpnTtDl9F+apmXDGHnBUV1LrQfMnsDVrl3k68ddZx2R4I3ONNwpNEe67y2Cqxz181U6m4BcncsI2HgNSyWaoCPdrY14FEeZ9ifPkE+jDEDYvfmOe9aLSaSJV8MQgVRIUs8z0AokjMGNu0ogLF9+HLUxbw==[/tex].由初始条件[tex=3.714x1.357]BFPW9eltEHdVdC7BRHXNyHeGEgSPIbL4f02rRn6Srok=[/tex]知[tex=28.929x8.214]a0s3MH7cLIdmiBRR0YN06x5UlLgdntIVjl/BrMh2qUsS+6a9S0StrzfZ9vvYFxm6y8aybL4z70gseZKlPzfYsDaMiTUnfYdc/5MFJF5wYqtUzihu4nymDPpcb/3j17GPTH9Q9ja6R3DRLzN4yVnevbX62Wv9RstHLDo6CpyVli+diBD9qN3Zb753Trc85A18hUREa250vCTacaLVitmu3mhpVwAjaeKtb0EYO3RkOo8MlTWygJuTICExgUASDdLLp/9d9dQNLBlFgwmB7hBnjIMES9sq4NId1zmrKY+Hj7zJ37q+wlyEhGjFyMR+Rum3LG56Goh/WprFJ3gOSB7aYxD1gbGMw4VGBwoUMjQ/M4/MTL4SlGeXpYFOkzPUf18XXVoqui2mUFcTLe8WTI/WK84UlVJuqq060PHMUl1xd98=[/tex][tex=2.5x1.214]Dw3fAPVvFgPgUP5YGHMPbQ==[/tex]，则[tex=13.857x3.214]bmpWSUUSUiF+8E3qoAGsPxIMoKvqRppvok/zC/lccVAR6J/2roJnjP72CSmFJ8kblq1f2LLxSa42++SLrI0N6kmBuIUzba82EfIhui6JioOqkM8DmztfiE6eqFUEkYhm[/tex].由初始条件[tex=2.0x1.357]WfzXYNdOb3O1ndzxlvYEHgBnqd9Ex3qZn/5sHEuRl7U=[/tex]知[tex=24.286x5.214]WfzXYNdOb3O1ndzxlvYEHv6XV51DspYIpU5N3Ccyre8hIfUEwhLTd3fQVvavFryhYI1Pr7Ylc4GmdY9SkSCgxtqtEE2GMfjeW8Wt1jfWJ0NnebrJV1Rg8pNEL7p6Hww5E2YEc5IarEVU1vhhm99G2GVQnUZX+i30fO3ivw69X1PXMjJS4vH93WGhrYaB7npcEcy4M5cJDIBy/sDLgQuRa22WOkyAVEpXnh7F6IYOvZsN2M87ZZBUK7+VssRyZhsbZ4VlAcou7nvKWmIRSwmDM1pg8kfqFFm2CSupL3YFW5lhSuR+7Vk45FIbAdMQtgFCu5wKuuiO8VVPW+Py6WegUA==[/tex].[tex=1.571x1.214]n8gAIpzbZFpSUMrEFivKuA==[/tex]为等号右边傅里叶展开系数[tex=27.857x9.643]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[/tex][tex=32.357x6.071]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[/tex][tex=30.714x6.071]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[/tex][tex=10.0x5.929]Ck4j1YFlvVH5wCAykOEMi5f9AJS71n26D//YdPjvVPxAaeruGThpPTDzU7QWYQlyVn4toFm+FmjaAk1Gn3qKsHV9GglfedOz+/qEE93eiebMQrFB/A266rbwnh7ooog+2BJRr1FGpw2cZ4qHoGxogmQmGeUlmCkHuu3ktycSy9GrOcjBDSr20gWKRoEGYpqbc32ozcOajGz2fXiNoDpQqbAny3y42kqFzajaYY2iXV5XBItNDMtTgr9DYpzp3veK[/tex]所以，[tex=20.214x3.5]bmpWSUUSUiF+8E3qoAGsP4XYbWLrgWX0uGBrA3Wk0NQ+4RJEr/3yzZDYhl9iP11f/bxlz7qndBLIRebpig2gkHDILUZ3AroXo10xz+kX6A3PW0zDZ+gFwgxHEFzgwMIXUnY1Z1cSOvOXbNT18udjHL11etfxHSHZfE0O7z6gmViU0rrwP6pqOd9pW+6mDjOqRoCHPsUltVYTWCm1dzB9qA==[/tex].
举一反三
- 长为[tex=0.357x1.0]5vVfAZliYwqMw8JaLE+iEA==[/tex]的均匀弦,两端[tex=1.857x1.0]sQ8UKBTHa4u9aJQTaFsBAg==[/tex]和[tex=1.714x1.0]z+3PraJ7SDoHa3jz672t+w==[/tex]固定,弦中张力为[tex=1.0x1.214]EsoJUyZj/6yD36MxA5tpQA==[/tex],在[tex=1.929x1.0]CE85dl2/W/1pUEtOAO/mFQ==[/tex]点,以横向力[tex=1.0x1.214]hzi2MsSHFnOjpdFGEAvoCA==[/tex]拉弦，达到稳定后放手任其自由振动.写出初始条件，问是否需要衔接条件？
- 设一条长度为I的均.匀弦,两端[tex=1.857x1.0]sQ8UKBTHa4u9aJQTaFsBAg==[/tex]和[tex=1.714x1.0]z+3PraJ7SDoHa3jz672t+w==[/tex]固定,弦中的张力为T.在[tex=2.571x1.0]fkJwwtH+18Pyc8GAZLQXq/XnkjMPe6rbDKfMvE6lelw=[/tex]点,以横向力Fo拉弦,达到平衡后放开手而使弦自由振动.试写出这个问题的定解条件.
- 长为[tex=0.357x1.0]5vVfAZliYwqMw8JaLE+iEA==[/tex]的杆,一端固定,另一端受力[tex=1.0x1.214]kLwCQn0d73TWOUUlnBVGBA==[/tex]而伸长.求解杆在放手后的振动
- 求解一无限长弦的自由振动,设弦的初始位移为 [tex=2.071x1.357]eAvaTAXWWX5VwHAZCgurVQ==[/tex],初速度为 [tex=3.571x1.429]soZP84EF2c2mAbXzgnyfBhREeOSdm1bQ6TvYRrDkKo0=[/tex],并解释物理意义.
- 一长度为I的弦两端固定,在[tex=1.643x1.0]e6RhHIicI4xKNcYb53RxjQ==[/tex]时刻弦上[tex=2.286x1.0]DSJKaWfJALImFxxTg/8qhA==[/tex]处受到外力的作用,其冲量为I.试求弦振动的情况.
内容
- 0
写出弦的横振动问题在下列情况下的初始条件：弦的两端点[tex=1.857x1.0]bOlCq/PHWhsSVMaVf7Obdg==[/tex]和[tex=1.714x1.0]OFSQaAQTidbnVE7HphlqPw==[/tex]固定，用手将弦上的点[tex=7.643x1.357]GU2la7DF1Ucu2jrntZ7CiOitnmMoWmBbkrcYdYHDI8Q=[/tex]拉开使之与平衡位置偏离[tex=0.643x1.0]uPu/UBwxTDghY6MHYDLmcA==[/tex]（设[tex=2.5x1.071]g26D1stzaus10xjhmccClA==[/tex]），然后放手.
- 1
一长为l的弦两端固定，在开始的时刻弦在平衡位置,用宽为[tex=1.0x1.0]n6gFQY8INNupwCoF50b1Vw==[/tex]的平面锤敲击弦的中点，使弦的长为[tex=1.0x1.0]n6gFQY8INNupwCoF50b1Vw==[/tex]的小段得到初始速度0。试求弦自由振动的情况。
- 2
求解无限长弦的自由振动,设弦的初始位移为[tex=2.071x1.357]Wcz3USjhGhKpAyRt+u5BdA==[/tex],初始速度为[tex=3.571x1.429]vS5XtdtZPo9P8Dp/XjpbYzzQaUxoPsGs3s+pAY4CfFQ=[/tex].
- 3
一无穷长弦,[tex=1.857x1.143]SxqOD6r7KmOtF7uEqCCPtQ==[/tex]时在[tex=2.286x1.0]ii77lCTXExv3mnaX1dHV/A==[/tex]处受到瞬时的打击,冲量为[tex=0.5x1.0]5jdh+DZyTF+GNqVGQs9NCQ==[/tex].试求解弦的横振动,设初位移和初速度均为 0 .
- 4
一根均匀弦两端分别在 [tex=1.857x1.0]bOlCq/PHWhsSVMaVf7Obdg==[/tex] 及 [tex=2.071x1.0]jA1m6CKNNVX9j/JBOhtCnw==[/tex] 处固定，设初始速度为零，初始时刻弦的形状为一抛物线，抛物线的顶点为 [tex=3.857x1.357]yAvWLLgXgIYOSlJSlTP+RJrxdUk9PiuA2CXu1oTMdEI=[/tex] 求弦振动的位移。