What would happen if the boundary value for $u_{tt}=a^2u_{xx}$ is that $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$

During the discussion of non-homogenous boundary values for the one-dimensional wave equation $$u=u(x,t),\;\frac{\partial ^2u}{\partial t^2}=a^2 \frac{\partial ^2u}{\partial x^2}$$ where the boundary values that were discussed are $$u(0,t) = 0, \; u(l,t) = \sin \omega t \quad \color{blue} {(\omega \ne \frac{n\pi a}l, n=1,2,\ldots)} \\ u(x,0) = 0, \; \frac{\partial u}{\partial t}(x, 0) = 0$$ I am particularly curious about the restriction that $$\omega \ne \cfrac{n\pi a}l$$ for any positive integer $$n$$, as the book didn't wrote anything about that (it simply put that restriction there).

The solution given in the textbook is to split $$u = v + w$$ and let $$v(x, t) = X(x)\sin \omega t$$. Then it can be worked out that $$X = \cfrac{\sin\frac{\omega x}a}{\sin\frac{\omega l}a}$$. It is apparent that when $$\omega = \cfrac{n\pi a}l$$ there'll be $$\sin\frac{\omega l}a = \sin n\pi = 0$$ and therefore $$X(x) \rightarrow \infty$$, but it's not immediately clear what the physical meaning is. Can anyone give an explanation about what would happen here?

1 Answer

Each $$\omega = \frac{n\pi a}{l}$$ is a Natural frequency of the system, with any frequency which come close to one of these frequencies causing Resonance. $$X(x) \rightarrow \infty$$ mainly because there's no maximum of the potential amplitudes as a frequency approaches a natural frequency. A common example cited about showing this effect is the dramatic collapse of the Tacoma Narrows Bridge (1940), but apparently the overall cause was more complicated according to another page's Tacoma Narrows Bridge section. Nonetheless, just shortly further down in the same page, the Types of resonance section states:

Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft.

• Very correct! See my images created with Wolfram Mathematica: n=1/2, n=1, n=2, n=4. – iBug May 12 at 14:01