In this post we derived the wave equation for a one-dimensional object moving in two-dimensional space (namely, a string vibrating up and down). This time we’re going to derive a family of solutions for this one-dimensional wave equation.
Archive for March, 2008
26 Mar
PDEs: Even More on Characteristics
So far, we’ve found the general solutions for two types of linear, homogeneous, first order PDEs: those with constant coefficients, and those with varying coefficients. It turns out that you can also extend the method of characteristics for linear, inhomogeneous, first order PDEs. In fact, the procedure is almost exactly the same.
26 Mar
PDEs: The Wave Equation (Long Post..)
So far we’ve derived and found solutions for a simple, first order, linear PDE describing the transport of a chemical through a fluid. Today we’re going to derive our first second-order equation, namely the wave equation. This equation is actually rather famous, and applies to a crap-ton of different physical phenomena. You’ve no doubt heard [...]
24 Mar
PDEs: Extending the Method of Characteristics
Last time we derived the general solution of a linear, homogeneous PDE with constant coefficients: . Physically, we can interpret such an equation as describing the flow of some chemical through water moving at a constant speed . Suppose, however, that the water does not move at a constant speed. Suppose that and [...]
23 Mar
PDEs: Solving Simple PDEs Using Characteristics
Last time we derived a simple PDE describing the process of non-diffusive advection in a constantly flowing stream of water. That PDE looked like this: , where was the constant speed of the water. The solution of this PDE is a function of the form .
This PDE follows a more general form that looks [...]
22 Mar
PDEs: Deriving a Simple Advection Equation
Suppose there’s a pipe with water flowing through it at a constant rate , and that a chemical of some kind is suspended in and moving with the water. This process is called advection. Suppose also that the chemical does not diffuse (i.e. spread out) as it moves. In other words, each particle of the [...]
21 Mar
PDEs: L, H, and O
One of the math classes I’m taking this semester is on partial differential equations (or PDEs). In lieu of actually studying this troublesome topic, I think I’m going to write a few posts about it. Which will likely take more time, and which will count as studying anyway. Why am I so masochistic? Meh.
Anyways. So [...]
21 Mar
Sciencin’ Ain’t Easy
The Discovery Institute’s Casey Luskin is miffed because smart Finnish kids understand modern science. No really. An international study measuring education in 57 countries found that Finnish 15-year-olds are precocious little beasts. They scored first place in science, and near the top in both reading and math.
But Luskie is all QQ:
However, part of the [...]
15 Mar
Grand
Despite being a pianist of about twelve or so years, my repertoire is slightly cringe-worthy as of late. Aside from the fact that I haven’t had access to anything but a cheap keyboard for a few months (and honestly, they just don’t suffice–no dynamics, no key pressure, no pedal, and that ever-present clackety-clackety-clack!), I just [...]
Recent Comments