Waves are ubiquitous and especially central to atmospheric sciences. Vibration on any material with restoring forces can generate waves. The existence of restoring force implies the existence of 2nd-order derivatives in time. One of the most well-known wave equation is Hooke’s Law
where the restoring force is equivalent to how much the material is stretched except with an opposite sign. This suggesting the force is always going against to the stretched direction. It’s not hard to find the solutions share a form of
(177)#\[\begin{align}
x = a_n \cos(\sqrt{k}t)+ b_n \sin(\sqrt{k}t)
\end{align}\]
The coefficients in (175) will be determined by the given initial values of \(x\) and \(x_t\). This is the general setup of wave solutions, where only the temporal structure is considered.
Here we will consider a slightly more complicated case where the propagating over a space (i.e., we are observing wave at both space and time.). Therefore, the original equation can be rewritten as
Suppose a string with fixed ends at \(x=0\) and \(x=\pi\), with initial conditions of
\[\begin{split}f(x) = \begin{cases}
x \textrm{ for } 0\leq x \leq \frac{\pi}{2}\\
\pi-x \textrm{ for } \frac{\pi}{2}< x \leq \pi\\
\end{cases} \end{split}\]
and
\[g(x) = x (1-\cos(x))\]
We also assume \(c=1\), which represents the propagating speed of waves
D’Alembert’s Solutions, Characteristic Lines, and Dispersion Relationship#
Jean Le D’Alembert (1717-1783) developed a simple method to approach the wave equations, known as the D’Alembert’s Wave Solutions. In optical physics and atmospheric wave dynamics, it has a special name dispersion relationships. i.e., the relation between wave length and wave speed.
Specifically, the D’Alembert’s wave solution describes
(206) is an ODE for \(y\), which can be solved with direct integration. One can easily find y is independent of \(\xi\) and \(\eta\). That means we gonna apply certain linear transform from \(x,t\) plane to \(\xi,\eta\) plane where \(y\) is a constant along \(\xi\) and \(\eta\).
To find the corresponding Jacobian matrix, we can first observe the original equation.
(205) can be written as
\(y_x\) and \(y_t\) represent how \(y\) changes with respect to \(x\) and \(t\). If we assign the first coordinate as \(x\) and the second coordinate as \(t\), the first equation above indicates that \(y\) is constant along the direction of \((-c,1)\) or \(y\) is conserved along each line of \(x=-ct+\eta\). As long as \(\eta\) is given, we know the solution of \(y\). This also implies that \(y\) is a function of \(\eta\). The same idea can be applied to \(\xi\).