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1、第五讲 机 械 波,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,声波、水波、电磁波都是常见的波。,各种类型的波有其特殊性，但也有普遍的共性，可以用相同的数学形式来描述。,波动的分类：机械波、电磁波、物质波。,第五讲 机 械 波,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright
2、2004-2011 Aspose Pty Ltd.,一、机械波产生的条件（源和路）,1.波源,2.连续介质, 1 机械波的形成和传播,二、机械波的类型,1.横波：介质中质点振动的方向与波的传播方向垂直,2.纵波：介质中质点振动的方向与波的传播方向平行,1)横波只能在固体中。,2)纵波在所有物质中都可以传播。,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,波速,注意,1、有些波既不是横波也不是纵波。如：水表面的
3、波既非横波又非纵波。水波中的质元是做圆（或椭圆）运动的。,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,flash,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,三、波长、波的周期和频率 波速波动描述,1、波长l 同一时刻，
4、两个相邻的相位差 为2的振动质点间的距离。,3、频率n 单位时间内波向前传播的完整波 的数目. (1s内向前传播了几个波长）,2、波的周期T 波传过一个波长的时间。,周期或频率只决定于波源的振动,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,在波动过程中，某一振动状态在单位时间内传播的距离。,4、波速：,Evaluation only. Created with Aspose.Slides for .NET
5、3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,平面波,球面波,各向同性均匀介质中，波线恒与波面垂直。,沿波线方向各质点的振动相位依次落后。,四、波线和波面,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,五、简谐波,简谐波是最基本的波动，波源和介质 的各质点均作简谐振动。数学上的傅里叶 分解法告诉我们，任何非简谐振动都可以 分解为
6、许多简谐振动的组合。因此，研究 简谐波特别重要。,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,一、平面简谐波的波函数,1、右行波的波动方程：,（1）已知O点振动表达式：,P点在t时刻的振动状态, 与O点在(t -x/u)时刻的振动状态相同。, 2 平面简谐波的波动方程,P点的相位比O点相位落后 。,参考点O,（滞后效应）,Evaluation only. Created with Aspose.Slides
7、 for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,（2）如图，已知 P 点的振动方程：,思考,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,（1）已知O点振动表达式：,思考,2、左行波的波动方程：,（超前效应）,Evaluation only. Created with Aspose.Slides for .NE
8、T 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,（2）如图，已知 P 点的振动方程：,思考,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,注意：,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright
9、2004-2011 Aspose Pty Ltd.,1、t 一定时的波形图 （快照）,讨论各质点在给定时刻的振动方向,二、波函数的物理意义,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,例1：如图所示，为一简谐波在t=0时刻的波 形图。试写出O、1、2、3点的初位相,解：画出下一时刻的波形图,（1）,（2）,Evaluation only. Created with Aspose.Slides for .NE
10、T 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,讨论质点在某一时刻的振动方向,质点的振动速度,2、x一定，t 变化。,表示 点处质点的振动方程（ 的关系）,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,波速即波形平移的速度！,3、t、x都变化时：,对应跑动的波形（行波）,Evaluation only. Created wit
11、h Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,通常：,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,例2：一平面简谐波某时刻的波形图如下，则OP之间的距离为多少厘米。,解题思路：,设波向右传播（P点落后于O点),思考：若设波朝左传，则,Evaluation only. Creat
12、ed with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,例3：一平面波在介质中以速度 u=20m/s 沿x 轴正方向传播，已知在传播路径上某点a点的振动方程为 ya=3cos4t (m) （1）以a为坐标原点写出波动方程。 （2）以距a点5m处的o点为坐标原点， 写出 波动方程。 (3) 求出oc 两点之间的位相差,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.
13、2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,解： ya=3cos4t,（1） a为坐标原点：,(2) 0点为坐标原点：,0 点振动方程,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,(3) 求出oc 两点之间的位相差,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.
14、0.0. Copyright 2004-2011 Aspose Pty Ltd.,例4：沿X轴正方向传播的平面简谐波、在 t=2s 时的波形如图，已知A、u及 ，问：1）原点O的初相及P点的初相各为多大？2）写出波动方程。,解题思路：,思考:求O、P两点之间的位相差。,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,设有一行波：,质量为 的媒质元其动能为：,3 波的能量,质元的速度：,一、媒质中振动动能,Eva
15、luation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,二、媒质中振动势能,可以证明：,势能：,总能：,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd., 3 波的能量（补充）,胡克定律： 在弹性限度内，应力和应变成正比。,式中E为关于线变的比
16、例系数，它随材料的不同而不同，叫杨氏模量。,补充：,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,振动动能,以固体棒中传播的纵波为例分析波的能量.,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,弹性势能,Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.,能量极小,能量极大,体积元在平衡位置时，动能、
