Exercises

1. (a) Verify Equations (8.4)–(8.6). (b) Derive Equations (8.7) and (8.8) from Equation (8.2) and Equations (8.4)–(8.6).

2. Suppose that
   $$F(x) =\exp\left(-\frac{x^{\,2}}{2\,\sigma_x^{\,2}}\right).$$
   Demonstrate that
   $$\bar{F}(k) \equiv \frac{1}{2\pi}\int_{-\infty}^{\infty} F(x)\,{\r... ...{2\pi\,\sigma_k^{\,2}}}\,\exp \left(-\frac{k^{\,2}}{2\,\sigma_k^{\,2}}\right),$$
   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

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

   $$S(k) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(x)\,\sin(k\,x)\,dx=0.$$

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

4. Verify Equations (8.31) and (8.32).
5. Derive Equations (8.11) and (8.12) directly from Equation (8.10) using the results (8.30)–(8.32).
6. Verify directly that Equation (8.49) is a solution of the wave equation, (8.33), for arbitrary pulse shapes $F(x)$ and $G(x)$.
7. Verify Equation (8.53).

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 by
   $$F(t) = \int_{-\infty}^\infty C(\omega)\,\cos(\omega\,t)\,d\omega + \int_{-\infty}^\infty S(\omega)\,\sin(\omega\,t)\,d\omega.$$
   [Hint: Use Euler's theorem.]

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

10. Demonstrate that
    $$\int_{-\infty}^\infty F^{\,2}(t)\,dt= 2\pi\int_{-\infty}^\infty\left[C^{\,2}(\omega)+S^{\,2}(\omega)\right]d\omega,$$
    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

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

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

    Let $H(t)=F(t)\,G(t)$, and let $\bar{H}(\omega)$ be the cosine transform of this even function, so that
    $$H(t)= \int_{-\infty}^\infty \bar{H}(\omega)\,\cos(\omega\,t)\,d\omega.$$
    1. Demonstrate that
       $$\bar{H}(\omega) = \frac{1}{2}\int_{-\infty}^\infty \bar{F}(\omega')\left[\bar{G}(\omega'+\omega) + \bar{G}(\omega'-\omega)\right] d\omega'.$$
       This result is known as the convolution theorem, because the previous type of integral is known as a convolution integral.

    2. Suppose that $F(t)=\cos(\omega_0\,t)$. Show that
       $$\bar{H}(\omega) = \frac{1}{2}\left[\bar{G}(\omega-\omega_0) + \bar{G}(\omega+\omega_0)\right].$$

12. Demonstrate that
    $$\psi({\bf r}, t)= F(v\,t-{\bf n}\cdot{\bf r}),$$
    where $F$ is an arbitrary function, and ${\bf n}$ a constant unit vector, is a solution of the three-dimensional wave equation, (7.9). How would you interpret this solution?
13. Demonstrate that
    $$\psi(r,t)= \frac{F(v\,t-r)}{r},$$
    where $F$ is an arbitrary function, is a solution of the spherical wave equation, (7.12). How would you interpret this solution?