ps6 - ω 2 x t = f t/m where ω> α> 0 m = 1 and f t =...

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Problem Set 6 February 23, 2007 Due March 2, 2007 ACM 95b/100b 3pm in Firestone 303 Niles A. Pierce (2 pts) Include grading section number 1. (2 × 5 pts) Shifting properties for the Fourier transform. a) Derive an expression for F{ f ( x - a ) } in terms of F ( k ) ≡ F{ f ( x ) } . b) Derive an expression for F - 1 { F ( k - ib ) } in terms of f ( x ) ≡ F - 1 { F ( k ) } . 2. (3 × 5 pts) Suppose f ( x ) is real. Show the following: a) F ( k ) = F ( - k ) b) If f ( x ) is even, F ( k ) is real c) If f ( x ) is odd, F ( k ) is pure imaginary 3. Calculate and sketch the Fourier transform for the following functions a) (10 pts) f ( x ) = H ( x + a ) - H ( x - a ), a > 0, H ( x ) is the Heaviside step function b) (24 pts) f ( x ) = a x 2 + a 2 , a > 0 4. Consider a damped harmonic oscillator under external force f ( t ) governed by the differential equation ¨ x ( t ) + 2 α ˙ x ( t ) +
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Unformatted text preview: ω 2 x ( t ) = f ( t ) /m where ω > α > 0, m = 1 and f ( t ) = H ( t + τ )-H ( t-τ ) for τ > 0. Assuming x ( ±∞ ) = ˙ x ( ±∞ ) = 0, use the Fourier transform to find x ( t ) for: a) (20 pts) t > τ b) (10 pts) t <-τ c) (20 pts) | t | < τ 5. (25 pts) Consider the temperature in an infinitely long insulated metal rod governed by the heat equation u t = α 2 u xx ,- ∞ < x < ∞ , t > with initial condition u ( x, 0) = 1 x 2 + 1 ,- ∞ < x < ∞ . Use Fourier transform methods to find a convolution representation for the time-varying tempera-ture u ( x, t ). Total points: 136...
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This note was uploaded on 11/23/2009 for the course ACM 95b taught by Professor Nilesa.pierce during the Winter '09 term at Caltech.

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