Videos on Traveling Waves and Standing Waves


We start by studying the wave equation on the real line. The wave equation is $$ \frac{\partial^2 f(x,t)}{\partial t^2} = v^2 \frac{\partial^2 f(x,t)} {\partial x^2},$$ where the constant $v$ gives the velocity of a wave. Note that (for now) we're allowing $x$ to run from $-\infty$ to $\infty$. If $f(x,t)=h_1(x-vt)$, or if $f(x,t)=h_2(x+vt)$, then it's easy to check that $f$ satisfies the wave equation. These kinds of solutions are called traveling waves because they keep their shape and just move forwards (for $h_1(x-vt)$) or backwards (for $h_2(x+vt)$) at speed $v$.

It turns out that every solution to the wave equation can be written as a sum of forward and backward traveling waves: $$f(x,t) = h_1(x-vt) + h_2(x-vt).$$ We compute the functions $h_1$ and $h_2$ from our initial conditions. If $f_0(x)=f(x,0)$ and $g_0(x) = \partial_t f(x,0)$, then \begin{eqnarray*} f_0(x) & = & h_1(x) + h_2(x) \cr g_0(x) & = & -v h_1'(x) + v h_2'(x) \cr \int g_0(x) dx & = & -v h_1(x) + v h_2(x) \cr h_1(x) & = & \frac{1}{2} \left [ f_0(x) - \frac{1}{v}\int g_0(x) dx \right ] \cr h_2(x) & = & \frac{1}{2} \left [ f_0(x) + \frac{1}{v}\int g_0(x) dx \right ] \end{eqnarray*} This is explained, and an example is worked out, in the first video.

We can also solve the wave equation on the interval $[0,L]$ with Dirichlet boundary conditions $f(0,t) = f(L,t) = 0$. That can still be written as a sum of traveling waves, but now the boundary conditions restrict the choices of $h_1$ and $h_2$: $$ h_1(x+2L) = h_1(x); \qquad h_2(x+2L)=h_2(x); \qquad h_2(x) = - h_1(-x).$$

But we already had a solution to the wave equation on $[0,L]$ in terms of standing waves: $$ f(x,t) = \sum_{n=1}^\infty \sin(n \pi x/L) [a_n \cos(n \pi vt/L) + b_n \sin(n \pi vt/L)].$$ In the second video, we show that the traveling wave and standing wave descriptions are equivalent. Using trig identities and the relations between periodic and sine Fourier series, we see how to write a standing wave as a sum of traveling waves, and how to write traveling waves as sums of standing waves.

So which is better? That depends on how smooth our data is. If $f_0(x)$ and $g_0(x)$ are smooth and spread out, then they can be approximated with only a few terms of a Fouries series, and we are best off using standing waves. If $f_0(x)$ and $g_0(x)$ are rough and/or localized, then standing waves are inefficient and you get a lot more insight using traveling waves.