Mathematical Tools for Physics
4—Differential Equations 113 4.24 Apply the Green’s function method for the forceF 0 ( 1 −e−βt ) on the harmonic oscillator with ...
4—Differential Equations 114 4.31 When you use the “dry friction” model Eq. ( 2 ) for the harmonic oscillator, you can solve the ...
4—Differential Equations 115 4.35 For the damped harmonic oscillator apply an extra oscillating force so that the equation to so ...
4—Differential Equations 116 twice with respect to time to verify that it really gives what it’s supposed to. This is a special ...
4—Differential Equations 117 4.45 In the equation of problem 17 , make the change of independent variablex= 1/z. Without actuall ...
Fourier Series Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. It h ...
5—Fourier Series 119 1 3 5 highest harmonic: 5 1 3 5 highest harmonic: 5 The same function can be written in terms of sines with ...
5—Fourier Series 120 And then use the orthonormality of the basis vectors,xˆ.ˆy= 0etc. Take the scalar product of the preceding ...
5—Fourier Series 121 Now all you have to do is to evaluate the integral on the left. ∫L 0 dxsin (mπx L ) 1 = L mπ [ −cos mπx L ] ...
5—Fourier Series 122 and Eq. ( 4 ) then becomes 〈 un,um 〉 = { 0 n 6 =m L/ 2 n=m where un(x) = sin (nπx L ) (9) The Fourier serie ...
5—Fourier Series 123 cosnπx/L (n= 0, 1 , 2 ,...), or you can choose a basis sin(n+^1 / 2 )πx/L (n= 0, 1 , 2 ,...), or you can ch ...
5—Fourier Series 124 This is the central identity from which all the orthogonality relations in Fourier series derive. It’s even ...
5—Fourier Series 125 No solutions there, so tryλ= 0 u(x) =A+Bx, then u(0) =A= 0 and so u(L) =BL= 0⇒B= 0 No solutions here either ...
5—Fourier Series 126 The periodic behavior of the exponential implies thatkL= 2nπ. The condition that the derivatives match at t ...
5—Fourier Series 127 am= 4 (2m+ 1)π Then the series is 4 π [ sin πx 2 L + 1 3 sin 3 πx 2 L + 1 5 sin 5 πx 2 L +··· ] (20) Parsev ...
5—Fourier Series 128 Rearrange this to get ∑∞ k=0 1 (2k+ 1)^2 = π^2 8 A bonus. You have the sum of this infinite series, a resul ...
5—Fourier Series 129 solve it, such as those found in section4.2. One part of the problem is to find a solution to the inhomogen ...
5—Fourier Series 130 Suppose that the viscous friction is small (bis small). If the forcing frequency,ωeis such that−mωe^2 +k= 0 ...
5—Fourier Series 131 All that I have to do now is to solve for an inhomogeneous solution one term at a time and add the results. ...
5—Fourier Series 132 The natural frequency of the system is (for small damping) still √ k/m. Look to see where a denominator in ...
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