Waves under Boundary Conditions

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Discussion Overview

The discussion revolves around the behavior of waves on a string under specific boundary conditions, particularly focusing on the nature of waves that are not at harmonic frequencies. It explores theoretical aspects, potential solutions, and implications of boundary conditions in both one-dimensional and multi-dimensional contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the characteristics of waves on a string with one end fixed and the other connected to an oscillator, specifically regarding non-harmonic frequencies.
  • Another participant suggests that non-harmonic waves may resemble stable standing waves but lack resonance, being out of phase with the nearest harmonic frequency.
  • A different perspective introduces the concept of a rectangular waveguide, explaining that modes arise from boundary conditions and that discrete modes can lead to confusion regarding the propagation of frequencies between modes.
  • This participant notes that in a one-dimensional string, the lack of an additional dimension limits the existence of continuous solutions, leading to the conclusion that frequencies between two modes result in evanescent modes that decay over distance.
  • One participant reflects on the implications of continuous oscillation from the oscillator, questioning whether this would create a condition similar to forced oscillations throughout the wave.

Areas of Agreement / Disagreement

Participants express varying views on the nature of non-harmonic waves and their stability, with some suggesting similarities to forced oscillations while others emphasize the limitations of one-dimensional systems. The discussion remains unresolved regarding the implications of continuous oscillation and the existence of evanescent modes.

Contextual Notes

The discussion highlights limitations related to the dimensionality of the system and the nature of boundary conditions, which affect the existence of certain wave solutions. The implications of these factors on wave behavior are not fully resolved.

Gear300
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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?
 
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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).
 
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.
 
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?
 
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