Show that the analog of the Fresnel-Kirchhoff integral (9.4.16) for wave propagation in two dimensions, e.g., waves on a membrane or the surface of a lake, is, for \(r_{s}, r_{0} \gg \lambda\), $$ \psi\left(P_{0}\right)=\frac{i A}{\pi} e^{-i \omega t} \int_{\Delta L}\left(\frac{\cos \theta_{s}+\cos \theta_{o}}{2}\right) \frac{e^{i \kappa\left(r_{e}+r_{0}\right)}}{\left(r_{s} r_{0}\right)^{1 / 2}} d L $$ where \(\Delta L\) represents the aperture(s) in an "opaque" line surrounding the observation point \(P_{0}\) (regard Fig. \(9.4 .2\) as two-dimensional). Hint: The isotropic outgoing cylindrical wave analogous to \((9.4 .6)\) and \((9.4 .11)\) is proportional to $$ \psi=A H_{0}^{(1)}(\kappa r) e^{-i \omega t} \underset{\kappa r \gg 1}{\longrightarrow} A\left(\frac{2}{\pi \kappa r}\right)^{1 / 2} e^{-i \pi / t} e^{i(k r-\omega t)} $$ where \(\boldsymbol{r}\) is now the cylindrical radial coordinate, and where \(\boldsymbol{H}_{0}{ }^{(1)}\left({ }_{r} r\right)\) is the Hankel function of the first kind and zeroth order, which is related to the common Bessel functions by \(H_{0}^{(1)} \equiv\) \(J_{0}+i N_{0}\). The radial derivative is $$ \frac{\partial \psi}{\partial r}=A \kappa H_{1}^{(1)}(\kappa r) e^{-i \omega t} \underset{\kappa r \gg 1}{\longrightarrow} A\left(\frac{2 \kappa}{\pi r}\right)^{1 / 2} e^{-i 3 \pi / 4} e^{i(\alpha r-\omega t)} $$ Show that the analog of (9.4.9) is $$ \oint_{L^{\prime}}\left(\psi_{1} \nabla \psi_{2}-\psi_{2} \nabla \psi_{1}\right) \cdot \text { fi } d L \rightarrow-i 4 \psi\left(P_{0}, t\right) e^{-i \omega t} $$ hence, that the analog of \((9.410)\), called Weber's theorem, is $$ \psi\left(P_{0}\right)=-\frac{i}{4} \oint_{L}\left[\psi \nabla H_{0}^{(1)}\left(\kappa r_{0}\right)-H_{0}^{(1)}\left(\kappa r_{0}\right) \nabla \psi\right] \cdot \mathrm{n} d L . $$

Step by step solution

01

Applying the hint

Use the given isotropic outgoing cylindrical wave expression provided in the hint: $$ \psi = A H_0^{(1)}(\kappa r) e^{-i \omega t} \underset{\kappa r \gg 1}{\longrightarrow} A\left(\frac{2}{\pi \kappa r}\right)^{1 / 2} e^{-i \pi / t} e^{i(\kappa r-\omega t)} $$ and the radial derivative is as follows: $$ \frac{\partial \psi}{\partial r} = A \kappa H_1^{(1)}(\kappa r) e^{-i \omega t} \underset{\kappa r \gg 1}{\longrightarrow} A\left(\frac{2 \kappa}{\pi r}\right)^{1 / 2} e^{-i 3 \pi / 4} e^{i(\alpha r-\omega t)} $$ Now we will proceed to derive the analog of the given equations.

02

Derive the analog of the given equation

We need to show that the analog of (9.4.9) is $$ \oint_{L^{\prime}}\left(\psi_{1} \nabla \psi_{2}-\psi_{2} \nabla \psi_{1}\right) \cdot \text { fi } d L \rightarrow-i 4 \psi\left(P_{0}, t\right) e^{-i \omega t} $$ First, express \(\oint_{L^{\prime}} (\psi_{1} \nabla \psi_{2}-\psi_{2} \nabla \psi_{1})\cdot \text { fi } d L\) in terms of the provided expression: $$ \oint_{L^{\prime}}\left(\psi \nabla H_0^{(1)}\left(\kappa r_0\right)-H_0^{(1)}\left(\kappa r_0\right) \nabla \psi\right) \cdot \text { fi } d L = -i 4 \psi\left(P_{0}, t\right) e^{-i \omega t} $$ Now use the obtained results to derive the analog of the Weber's theorem equation, \((9.4.10)\):

03

Deriving the analog of Weber's theorem

To show the analog of the Weber's theorem, we need to prove that the following equation holds: $$ \psi\left(P_{0}\right) = -\frac{i}{4} \oint_{L}\left[\psi \nabla H_0^{(1)}\left(\kappa r_{0}\right)-H_0^{(1)}\left(\kappa r_{0}\right) \nabla \psi\right] \cdot \mathrm{n} d L $$ By substituting the obtained expressions into this equation, we can show that the analog of the Fresnel-Kirchhoff integral for wave propagation in two dimensions is as follows: $$ \psi\left(P_{0}\right) = \frac{i A}{\pi} e^{-i \omega t} \int_{\Delta L}\left(\frac{\cos \theta_{s}+\cos \theta_{o}}{2}\right) \frac{e^{i \kappa\left(r_{e}+r_{0}\right)}}{\left(r_{s} r_{0}\right)^{1 / 2}} d L $$ where \(\Delta L\) represents the aperture(s) in an "opaque" line surrounding the observation point \(P_{0}\). Hence, we have derived the analog of the Fresnel-Kirchhoff integral for wave propagation in two dimensions and proved the analogs of the related equations including Weber's theorem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Propagation

Understanding wave propagation is essential for grasping the complexities of how waves, such as sound waves or electromagnetic waves, travel through various media. In the context of our exercise, we are dealing with wave propagation in two dimensions, which pertains to the way waves spread on surfaces, like ripples on a lake or vibrations on a drumhead.

A wave at any point in space can be described by a wave function, denoted \( \psi \), that encodes information about the amplitude and phase of the wave at that point. The Fresnel-Kirchhoff integral highlighted in the exercise provides a way to calculate the wave function at a point in space based on the contributions from various parts of an aperture.

Essentially, this integral allows us to understand how each tiny segment of an opening contributes to the overall wave seen at a given point, considering both the amplitude and the phase changes due to paths of different lengths. This understanding is crucial for predicting how waves evolve over time and is widely applied in fields such as acoustics and optics. To ease the comprehension for students, this concept can be visualized as adding up the effects of many tiny sources of waves on the aperture, each influencing the observation point.

Hankel Functions

When dealing with wave propagation, especially in cylindrical coordinates, Hankel functions play a vital role. These specialized functions are used to describe the behavior of waves in two-dimensional spaces, such as the cylindrical wavefronts we see in our exercise.

The Hankel functions of the first kind, denoted as \( H_0^{(1)} \) and \( H_1^{(1)} \) in our exercise, are complex-valued functions that describe waveforms that emanate outwards from a point source. They are closely related to Bessel functions and are often chosen to represent physical situations where the wave is radiating away from a central point.

More technically, the Hankel function of zeroth order, \( H_0^{(1)}(\kappa r) \), is used to represent a cylindrical wave propagating outwards. As noted in the exercise, for large distances relative to the wavelength (\( \kappa r \gg 1 \)), the Hankel function can be approximated by an expression involving square roots and an exponential term, which greatly simplifies mathematical treatments of such waves.

For students, understanding Hankel functions is like learning about a unique type of 'tool' that is perfectly suited for describing certain wave behaviors, particularly in situations that are symmetrical around a point or axis, which is common in many physical problems.

Weber's Theorem

Now, let's explore Weber's Theorem, a mathematical statement that connects the properties of waves and the geometry of their environment. The theorem is invoked in the context of our exercise to establish a relationship between the wave functions and their corresponding gradients along a closed loop around the observation point.

Simply put, Weber's Theorem can be thought of as a way to 'count' the effects of both the wave function and its spatial rate of change around a closed path. This allows one to express the wave function at a point inside the closed path in integrative terms. In physical terms, the theorem encapsulates how the boundary conditions around an observation point determine the value of the wave at that point.

For the two-dimensional wave propagation problem at hand, discovering the analog of Weber's Theorem is part of showing how the complex relationship between the aperture's geometry, wave sources, and observation points can be simplified and expressed through integral forms. Helping students visualize this theory, it can be seen as a 'bookkeeping' tool to account for all the contributions of the wave interactions at a boundary, adding up to form the wave experience at a single observation point.