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Exercises

  1. Verify Equations (701)-(703). Derive Equations (704) and (705) from Equation (699) and Equations (701)-(703).
  2. Suppose that

    $\displaystyle F(x) =\exp\left(-\frac{x^2}{2\,\sigma_x^{\,2}}\right).$

    Demonstrate that

    $\displaystyle \bar{F}(k) \equiv \frac{1}{2\pi}\int_{-\infty}^{\infty} F(x) {\r...
...sqrt{2\pi \sigma_k^{ 2}}} \exp \left(-\frac{k^2}{2 \sigma_k^{ 2}}\right),
$

    where $ {\rm i}$ is the square-root of minus one, and $ \sigma_k=1/\sigma_x$ . [Hint: You will need to complete the square of the exponent of $ {\rm e}$ , transform the variable of integration, and then make use of the standard result that $ \int_{-\infty}^\infty {\rm e}^{-y^2}\,dy = \sqrt{\pi}$ .] Hence, show from Euler's theorem, $ \exp( {\rm i} \theta)\equiv\cos \theta + {\rm i} \sin \theta$ , that

    $\displaystyle C(k)$ $\displaystyle = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(x) \cos(k x) dx = \f...
...\sqrt{2\pi \sigma_k^{ 2}}} \exp \left(-\frac{k^2}{2 \sigma_k^{ 2}}\right),$    
    $\displaystyle S(k)$ $\displaystyle = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(x) \sin(k x) dx=0.$    

  3. Demonstrate that

    $\displaystyle \int_{-\infty}^{\infty} \frac{1}{(2\pi\,\sigma_k^{\,2})^{1/2}}\,\exp \left(-\frac{k^2}{2\,\sigma_k^{\,2}}\right)dk = 1.
$

  4. Verify Equations (728) and (729).
  5. Derive Equations (708) and (709) directly from Equation (707) using the results (727)-(729).
  6. Verify directly that Equation (746) is a solution of the wave equation, (730), for arbitrary pulse shapes $ F(x)$ and $ G(x)$ .
  7. Verify Equation (750).

  8. Consider a function $ F(t)$ that is zero for negative $ t$ , and takes the value $ \exp(-t/2\,\tau)$ for $ t\geq 0$ . Find its Fourier transforms, $ C(\omega)$ and $ S(\omega)$ , defined in

    $\displaystyle F(t) = \int_{-\infty}^\infty C(\omega) \cos(\omega t) d\omega + \int_{-\infty}^\infty S(\omega) \sin(\omega t) d\omega.
$

    [Hint: Use Euler's theorem.]

  9. Let $ F(t)$ be zero, except in the interval from $ t=-\Delta t/2$ to $ t=\Delta t/2$ . Suppose that in this interval $ F(t)$ makes exactly one sinusoidal oscillation at the angular frequency $ \omega_0 = 2\pi/\Delta t$ , starting and ending with the value zero. Find the previously defined Fourier transforms $ C(\omega)$ and $ S(\omega)$ .

  10. Demonstrate that

    $\displaystyle \int_{-\infty}^\infty F^{ 2}(t) dt= 2\pi\int_{-\infty}^\infty[C^2(\omega)+S^2(\omega)] d\omega,
$

    where the relation between $ F(t)$ , $ C(\omega)$ , and $ S(\omega)$ is as previously defined. This result is known as Parseval's theorem.

  11. Suppose that $ F(t)$ and $ G(t)$ are both even functions of $ t$ with the cosine transforms $ \bar{F}(\omega)$ and $ \bar{G}(\omega)$ , so that

    $\displaystyle F(t)$ $\displaystyle =\int_{-\infty}^\infty \bar{F}(\omega) \cos(\omega t) d\omega,$    
    $\displaystyle G(t)$ $\displaystyle =\int_{-\infty}^\infty \bar{G}(\omega) \cos(\omega t) d\omega.$    

    Let $ H(t)=F(t)\,G(t)$ , and let $ \bar{H}(\omega)$ be the cosine transform of this even function, so that

    $\displaystyle H(t)= \int_{-\infty}^\infty \bar{H}(\omega) \cos(\omega t) d\omega.
$

    Demonstrate that

    $\displaystyle \bar{H}(\omega) = \frac{1}{2}\int_{-\infty}^\infty \bar{F}(\omega')\left[\bar{G}(\omega'+\omega) + \bar{G}(\omega'-\omega)\right] d\omega'.
$

    This result is known as the convolution theorem, since the above type of integral is known as a convolution integral. Suppose that $ F(t)=\cos(\omega_0\,t)$ . Show that

    $\displaystyle \bar{H}(\omega) = \frac{1}{2}\left[\bar{G}(\omega-\omega_0) + \bar{G}(\omega+\omega_0)\right].
$

  12. Demonstrate that

    $\displaystyle \psi({\bf r}, t)= F(v\,t-{\bf n}\cdot{\bf r}),
$

    where $ F$ is an arbitrary function, and $ {\bf n}$ a constant unit vector, is a solution of the three-dimensional wave equation, (537). How would you interpret this solution?
  13. Demonstrate that

    $\displaystyle \psi(r,t)= \frac{F(v t-r)}{r},
$

    where $ F$ is an arbitrary function, is a solution of the spherical wave equation, (540). How would you interpret this solution?

next up previous
Next: Dispersive Waves Up: Wave Pulses Previous: Bandwidth
Richard Fitzpatrick 2013-04-08