sol3 - PHYS 218 SOLUTION TO ASSIGNMENT 3 16 Feb 2007 By...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
PHYS 218 - SOLUTION TO ASSIGNMENT 3 16 Feb 2007 By Chung Koo Kim e-mail : [email protected] 1. Fourier Transform Examples 1 (a) −τ τ h h (b) The integral is the same for complex integrand as long as the integral variable (t here) is real. g 1 ( ω ) = -∞ f 1 ( t ) e - iωt dt = 0 - τ he - iωt dt + τ 0 ( - h ) e - iωt dt = ih ω e - iωt 0 - τ - ih ω e - iωt τ 0 = ih ω [2 - ( e iωτ + e - iωτ )] = 2 ih ω [1 - cos ωτ ] * Subscript of g is for later use in problem 3. (c) We see that g 1 ( ω ) is pure imaginary, so the real part is zero. Refer to the follow- ing graph. Blue and red curves represent real and imaginary parts, respectively. 4 π τ 2 π τ 2 π τ 4 π τ 4h 2h 2h 4h
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2. Fourier Transform Examples 2 (a) See the figure below. −τ τ h (b) In the same manner as in problem 1(b), g 2 ( ω ) = -∞ f 2 ( t ) e - iωt dt = τ - τ b 1 - | t | τ cos ωt dt = 2 τ 0 b 1 - t τ cos ωt dt = 2 b τω 2 [1 - cos ωτ ] Note that f 2 ( t ) is an even function of t. (c) Now g 2 ( ω ) is real, so the imaginary part is zero. Refer to the following graph. Blue and red curves represent real and imaginary parts, respectively. 6 π τ 4 π τ 2 π τ 2 π τ 4 π τ 6 π τ b τ 3. Fourier Transform and differential equations (a)-(b) The integrals can be evaluated as before : G ( ω ) = -∞ F ( t ) e - iωt dt = - 1 F ( t ) e - iωt -∞ + -∞ f ( t ) e - iωt = g ( ω )
Image of page 2
g ( ω ) = -∞ f ( t ) e - iωt dt = F ( t ) e - iωt -∞ + -∞ F ( t ) e - iωt = iωG ( ω ) Here F ( t ) e - iωt -∞ = 0 because the magnitude of e - iωt is at most one but F(t) approaches zero fast enough. Although vague, mathematically strict definition can be given to ’fast enough’. (c) From the answers to Problem (1) and (2), we find that g 1 ( ω ) = iωg 2 ( ω ) provided that b/ τ =h, or f 1 ( t ) is the derivative of f 2 ( t ). We can check that the latter is true if b/ τ =h. Further note - we have just learned that differential relation in time domain is converted into simple algebraic manipulation (multiplication by in this case) in the frequency domain when Fourier transformed. Combined with the linearity of Fourier transform 1 , this simplifies the task of solving differential equations spectacularly. For example, let’s solve equation of motion for a driven simple harmonic oscillator, whose motion is described by a differential equation ¨ x ( t ) + ω 2 0 x ( t ) = cos ω d t, with driving frequency ω different from the natural frequency ω d . Then the Fourier transform(FT) of the equation is ( ) 2 X ( ω ) + ω 0 2 X ( ω ) = π [ δ ( ω -
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern