Week 12: Wave Equation#
Problem Setup and Initial Conditions#
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
The coefficients in (175) will be determined by the given initial values of
Wave Solution with Space and Time Structures#
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
with initial and boundary conditions
Using separation of variable, one can expect that the solution shares a form of (leave this practice to readers)
In most cases, we can satisfy the initial condition with finite sum of
using initial condition, we know
indicating
Example 1
Suppose a string with fixed ends at
and
We also assume
From (181), we know the solution has a form of
where
and
Now considering a more complicated case, where we have external forcing. i.e.,
Similar to how we deal with heat equation with external forcing, we can assume the solution shares a form of
Substitute (185) back to (184), we can have a homogeneous equation for
and an 2nd-order ODE for the forcing term i.e.,
Integrating (187) twice, we have
Now, look at boundary conditions,
To make (189) a homogeneous problem, we choose
Let
we can have the same homogeneous equation at the second boundary.
(191) indicates that
With the result above, we can rewrite the entire set of wave equation to
We know how to solve this problem.
Wave Motion in an Unbounded Medium#
Consider the problem
Similarly, using separation of variable, we have
The first equation has a solution of
where
Here we make the increment between two adjacent wave numbers small (
We can rewrite (196) into
and
(readers can think about why T has a constant multiple of Fourier consine function as its solution?)
According to (199), we know
and the completed equation is
Example 2
Solve the problem on a real line with initial displacement of
All we have to do is calculate the Fourier integral for
and
because
The solution is
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
where
To show that, we first observe (195) and we can find the following two characteristic equations
while (205) is a slightly more complicated form, we can try to convert it into a simpler question
(206) is an ODE for
To find the corresponding Jacobian matrix, we can first observe the original equation. (205) can be written as

Fig. 16 The D’Alambert’s wave solution.#
Combining both, we have
Here we demonstrate how to derive the D’Alambert’s wave solution. According to the initial condition of the wave position, we know
Here if we properly choose an integral for the second equation (I will show that in a seconds), we will have
Then we can represent
Adding both together, we have solutions