wave patterns
📄P411, Sec 12.4
For a point source \(P\), the characteristics carrying disturbances pass through \(P\) and we have a centered wave \[ \frac{x_2}{x_1}=-f^{\prime}\left(k_2\right) ; \]
Centered wave means the line passes through the origin
For a point source at \(P=(0,0)\), each characteristic line is assumed to emanate from \(P\). Hence the "integration constant" must be zero (it goes through \(\left(x_1, x_2\right)=(0,0)\) ):
\[ x_2(0)=0 \Longrightarrow x_2=-f^{\prime}\left(k_2\right) x_1 . \]
Thus, at any point along that characteristic,
\[ \frac{x_2}{x_1}=-f^{\prime}\left(k_2\right) \]
Geometrically, \(\frac{x_2}{x_1}\) is just the slope of that line in the plane.
📄P415, Eqn(12.38)
\[ \begin{aligned} & \frac{x_2}{x_1}=\tan \xi=\frac{G_{k_2}}{G_{k_1}} \\ & G_{k_1}=U+\left(\frac{T h}{\rho}\right)^{\frac{1}{2}} 2 k \frac{\partial k}{\partial k_1} \\ & k=\sqrt{k_1^2+k_2^2} \\ &\frac{\partial k}{\partial k_1}=\frac{1}{2}\left(k_1^2+k_2^2\right)^{-\frac{1}{2}} 2 k_1 \\ &=k_1 / k \end{aligned} \]
\[ \begin{aligned} & G_{k_2}=\left(\frac{T h}{\rho}\right)^{\frac{1}{2}} 2 k \frac{k_2}{k} \\ & \frac{x_2}{x_1}=\frac{\left(\frac{T h}{\rho}\right)^{\frac{1}{2}} 2 k_2}{U+\left(\frac{T h}{\rho}\right)^{\frac{1}{2}} 2 k_1} \end{aligned} \]
pre:
\[ \begin{aligned} & k_x=-k \cos \psi \quad k_y=k \sin \psi \quad\left(\frac{T h}{\rho}\right)^{\frac{1}{2}}=\alpha \\ & G=U k_1+2 k^2=3 \end{aligned} \]
Then
\[ \begin{aligned} & U k \cos \psi=\alpha k^2 \\ & k=U \cos \psi / \alpha \end{aligned} \] $$ \[ k_2=\frac{1}{2}(U \cos \psi \sin \psi), k_1=-\frac{1}{2}(U \cos \psi \cos \psi) \] \(\Rightarrow\) \[ \begin{aligned} & \tan \xi=\frac{2 \alpha \frac{1}{2}(U \cos \psi \sin \psi)}{U+2 \alpha\left(-\frac{1}{a} U \cos ^2 \psi\right)} \\ &=\frac{2 U \sin \psi \cos \psi}{U-2 U \cos ^2 \psi} \\ &=\frac{2 \sin \psi \cos \psi}{\sin ^2 \psi-\cos ^2 \psi} \\ &=\frac{2 \tan ^2 \psi}{\tan ^2 \psi-1} \\ &=-\tan(2 \psi) \end{aligned} \] \(\Rightarrow\) \[ \xi = \pi - 2\psi \]
wave vector vs wave front vs rays
Wave Fronts
Definition: A wave front is a surface (in three dimensions) or a curve (in two dimensions) along which the wave has a constant phase.
Geometrical Interpretation: If you freeze a wave at a given instant in time, the loci of points that share the same phase (for example, all the peaks or all the troughs) form a wave front.
Examples:
Plane waves: The wave fronts are infinite parallel planes.
Spherical waves: The wave fronts are spheres expanding outward from a point source.
Wave Number Vector
Definition: Often denoted by \(\mathbf{k}\), the wave number vector (also called the wave vector) encodes both the direction and the "spatial frequency" of the wave.
Magnitude: The magnitude of \(\mathbf{k}\) is the wave number \(k=\|\mathbf{k}\|=\frac{2 \pi}{\lambda}\), where \(\lambda\) is the wavelength.
Direction:
\(\mathbf{k}\) is perpendicular (normal) to the wave fronts in many idealized situations (e.g., in homogeneous media with no refraction).
It points in the direction of wave propagation-the direction in which the phase advances most rapidly.
Essentially, if we draw the wave fronts, \(\mathbf{k}\) will point "straight out" of those surfaces or lines.
Rays
Definition: A ray is a line (or curve) that shows the direction along which energy or wave propagation is moving. In many cases (e.g., in homogeneous media and in ray/optical approximations), the rays are drawn normal to the wave fronts.
Geometrical Optics & Ray Theory:
Ray diagrams are often used in geometrical optics or the high-frequency (short-wavelength) limit, where diffraction effects can be neglected.
In this regime, the wave propagation is effectively tracked by following these rays.
📄P415, Eqn(12.40)
\[
\theta=\int_0^{\vec{r}} \vec{k} \cdot d \vec{r}
\] 
\(\Rightarrow\) \[ \begin{aligned} \vec{k} \cdot d \vec{r} & =\vec{k} \cdot \vec{e}_r d r \\ & =k \cos \mu d r \end{aligned} \] \(\Rightarrow\) \[ \begin{aligned} \theta & =\int_0^{\vec{r}} \vec{k} \cdot d \vec{r} \\ & =\int_0^r k \cos \mu d r \\ & =k \cos \mu \int_0^r d r \\ & =k \cos \mu r \end{aligned} \] \(\xi \rightarrow\) constant, from \((12.30)\) \[ \begin{aligned} & \tan \mu=\frac{1}{k} \frac{\partial G / \partial k}{\partial G /(k \partial \psi)} . \\ & G=-U k \cos \psi+2 k^2=0 \end{aligned} \]
\[ \left\{\begin{array}{l} \xi=\pi-\mu-\psi \\ \xi=\pi-2 \psi \end{array}\right. \]
\(\Rightarrow\) \[ \mu=\psi=\frac{\pi}{2}-\frac{\xi}{2} \]
\[ \begin{aligned} k \cos \mu & =k \cos \left(\pi-\frac{\xi}{2}\right) \\ & =k \sin \frac{\xi}{2} \end{aligned} \]
we already have \[ u \cos \psi=\alpha k \] \(\Rightarrow\) \[ k=\frac{U \cos \psi}{\alpha} \]
\[ \begin{aligned} k \cos \mu & =\frac{U \cos \psi}{2} \sin \frac{\xi}{2} \\ & =\frac{U}{\alpha} \cos \left(\frac{\pi}{2}-\frac{\xi}{2}\right) \sin \frac{\xi}{2} \\ & =\frac{U}{\alpha} \sin ^2 \frac{\xi}{2} \end{aligned} \]
therefore \[ \begin{aligned} \theta & =k \cos \mu r \\ & =\frac{U}{\alpha} r \sin ^2 \frac{\xi}{2} \\ & (=\text { constant }) \end{aligned} \]