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Unformatted text preview: Waves on a String Newtons law of motion This is one dimensional scalar wave equation propagating with velocity, v Tension force density A simplified case for the wave equation is the plane wave propagating in the xdirection. In this case, the wave equation can be written as &#2; One solution for a plane wave propagating in an unbounded, uniform medium can be expressed as &#2; In this equation, u0 is the amplitude, is the angular frequency; k is called the wave number. &#2; We will show the relationship of k with respect to angular frequency . &#2; Taking the secondary derivative of u with respect to space, here the x coordinate, is 2 2 2 2 ) , ( 1 ) , ( t t x u v x t x u = ) cos( kx t u u o + = and putting the second derivative of u with respect to time: &#2; Thus by combining these two equations we will get ) cos( 2 2 2 kx t u k x u + = ) cos( 1 2 2 2 2 2 kx t u v t u v + = v k = Some Relation to Remember Reflection and Transmission A C B Reflection and Transmission BC1 : Displacement must be continuous across the boundary Reflection and Transmission BC2 : Y component of the tensional force on two sides of the string must be equal Reflection and Transmission Reflection and Transmission Kinetic and Potential energy Kinetic and Potential energy Kinetic and Potential Energy Pulse incident on junction between strings of different properties gives transmitted and reflected pulses. Reflected wave inverted because impedance is greater in the right string. Transmitted pulse has a smaller length because velocity is lower in the right string. In Seismic wave propagation both the displacement and the force vector vary in space and time. Our Goal : Use Newton second law to characterize a continuous media and measure its response to applied force . Steps followed 1.To describe forces acting on a deformable continuous medium > Stress Tensor 1.Relate stress to Displacement of a continuous medium > Equation of motion 1.Measure the variation of displacement within the medium that give rise the internal deformation  > Strain Continuum mechanics Stress Let us consider forces acted on a small volume, V with surface S Where F is the surface force on element d s General description of a propagating wave: f t x v If: &#2; Then: &#2; Where: is the wavelength,...
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 Spring '08
 Duta

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