Waves under Boundary Conditions

[Gear300]

For a string with one endpoint attached to a wall and the other to an oscillator (so that it is under boundary conditions), what is the character of waves that are not at a harmonic frequency?

 

[Gear300]

I was assuming that the waves would be somewhat similar in form to the stable standing waves (considering the forced nodal points), but not as resonant (out of phase from the nearest resonance frequency).

 

[Born2bwire]

If we had a rectangular waveguide, we could solve for the modes of the waveguide that arise due to the boundary conditions. These modes represent the possible solutions of waves that can travel through the waveguide. But since these modes are discrete, it can be a bit confusing on how waves of frequencies between two modes can propagate. What happens is that we discretize the possible wave numbers in say the x and y directions and allow the wave number in the z direction to be continuous. So as we change the frequency, the direction of the wave as it propagates changes slightly so that it bounces around while satisfying the boundary conditions.

The problem with the wave on a string as I see it is that we only have one dimension of freedom, along the length of the string. Thus, we do not have an extra dimension that we can vary freely to allow continuous solutions to the eigenvalue problem. That is to say, if we have two frequencies of waves that satisfy the boundary conditions (driven at one end and a nodal point at the other), then frequencies between those two modes cannot be a solution. Instead, they are evanescent modes, if you were to excite these modes on your string they would die out as they travelled.

You could show this explicitly by solving for the wave equations and you should arise with one that attenuates over time/distance.

 

[Gear300]

I see...so altogether, they are not stable enough to exist for the one dimensional case...though, if the oscillator were continuously running, would it sort of be the same as the condition for forced oscillations all across the wave?