It oscillates at a characteristic frequency \u03c0 c LAnd the higher values of

It oscillates at a characteristic frequency π c land

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It oscillates at a characteristic frequency π c / L And the higher values of n represent the harmonics. We call these the normal modes of the string We see that the full motion of the string is a superposition of harmonics set by the initial conditions. D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics May 25, 2012 35 / 36
Normal modes The initial configuration of the string (i.e. its position and velocity) are Fourier analyzed into harmonics. And then each harmonic vibrates at its particular frequency independently of the others. Note too that the vibration according to this equation never stops and the amplitude of vibration repeats forever. The wave equation has no dissipation and so energy in the string stays there forever. In reality, of course, real strings have dissipation and the vibration will eventually die out. D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics May 25, 2012 36 / 36
ACM 100c The Wave Equation - part 2 Dan Meiron Caltech June 1, 2012 D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics June 1, 2012 1 / 38
Recap In our previous lecture we introduced the wave equation 2 u t 2 = c 2 u 2 x 2 We developed a general solution over the interval -∞ < x < We discussed the basic characteristics of the solutions This included the concept of domain of dependence and region of influence D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics June 1, 2012 2 / 38
Recap We then turned our attention to the wave equation in a finite domain u tt = c 2 u xx 0 < x < L t > 0 , We applied clamped end boundary conditions: u ( 0 , t ) = 0 u ( L , t ) = 0 t > 0 , And initial conditions u ( x , 0 ) = h ( x ) 0 < x < L u t ( x , 0 ) = p ( x ) 0 < x < L . D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics June 1, 2012 3 / 38
Normal modes We solved this using the method of finite transforms and obtained the following solution u ( x , t ) = X n = 1 h n cos ( n π ct / L ) + p n L n π c sin ( n π ct / L ) sin ( n π x / L ) . Note that the solution is a linear superposition of the functions sin ( n π ct / L ) sin ( n π x / L ) and cos ( n π ct / L ) sin ( n π x / L ) D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics June 1, 2012 4 / 38
Normal modes Looking at these functions we see that they are the oscillatory modes of the string. For example sin ( π x / L ) represents the fundamental mode of vibration of the string. It oscillates at a characteristic frequency π c / L And the higher values of n represent the harmonics. Note that the spacing between these frequencies is constant This is what allows us to hear pure tones from vibrating strings We call these the normal modes of the string We see that the full motion of the string is a superposition of harmonics set by the initial conditions. D. Meiron (Caltech) ACM 100c - Methods of Applied Mathematics June 1, 2012 5 / 38
Normal modes The initial configuration of the string (i.e. its position and velocity) are Fourier analyzed into harmonics. And then each harmonic vibrates at its particular frequency independently of the others.

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