Jufa
- 101
- 15
In basic optics, we are given the general solution of the wave equation (massless string of length L) as a linear combination of normal modes, that need to have some of the permitted frequencies due to boundary conditions. In laboratory, we observed that phenomenon. We generated a wave in a string that, as it reflexes at its end, ended up generating a standing wave. We did that for some number of harmonics.
My question (more than a question is a matter of discussion) is: We are often given the view of the harmonics as some sort of an orthogonal basis of the wave equation solutions space, in the sense that any of the solutions can be uniquely written as some linear combination of them and that the energy of the solution is equal to the sum of the energies of the harmonics of which it has been decomposed.
From this, I think there is a quite reasonable analogy between a standing wave and a free particle. Let me explain that. For a free particle, we have an infinite number of solutions for its movement. Every solution can be written as a straight movement with no acceleration and a certain velocity (let's fix the initial position, as we would fix where the standing wave was). I hope the analogy is clear until this moment. Now if I choose, for the particle, one solution as the movement along the "x" axis with a certain velocity and I do the same for the other orthogonal axis ("y" and "z") I obtain an orthogonal basis for the solutions in the same sense I obtained it for the standing wave (any solution will be a combination of this three and the energy resulting from the sum will be equal to the sum of energies. This is the analogy, and maybe the most remarkable differences are that for the particle we only have 3 degrees of freedom whereas that for the standing wave we have infinite. Now I would like to go further.
When speaking about the free particle I feel very comfortable talking about orthogonality of solutions because I associate it to the fact that I can "modify" the contribution of one element of the basis without disturbing the others. In other words, I can push the particle toward the "y" direction, which will increase its velocity in that direction, without modifying velocity in any of the other two directions. Now my intuition tells me that this sort of "non-disturbing modification" should have some analogy in the standing wave since it seems that analogies worked quite well between both systems until the moment. So I think that you should be able to modify the amplitude of one harmonic without disturbing others. Now is where I fail to figure how you can do that, how you "push" one single harmonic without interfering others. In the laboratory, we had a generator that made the string vibrate in a certain frequency. Once we reached a permitted frequency we could observe its corresponding normal mode but we never tried to create a superposition of them and, in fact, I have no clue of how could we make that.
Thank you for reading.
My question (more than a question is a matter of discussion) is: We are often given the view of the harmonics as some sort of an orthogonal basis of the wave equation solutions space, in the sense that any of the solutions can be uniquely written as some linear combination of them and that the energy of the solution is equal to the sum of the energies of the harmonics of which it has been decomposed.
From this, I think there is a quite reasonable analogy between a standing wave and a free particle. Let me explain that. For a free particle, we have an infinite number of solutions for its movement. Every solution can be written as a straight movement with no acceleration and a certain velocity (let's fix the initial position, as we would fix where the standing wave was). I hope the analogy is clear until this moment. Now if I choose, for the particle, one solution as the movement along the "x" axis with a certain velocity and I do the same for the other orthogonal axis ("y" and "z") I obtain an orthogonal basis for the solutions in the same sense I obtained it for the standing wave (any solution will be a combination of this three and the energy resulting from the sum will be equal to the sum of energies. This is the analogy, and maybe the most remarkable differences are that for the particle we only have 3 degrees of freedom whereas that for the standing wave we have infinite. Now I would like to go further.
When speaking about the free particle I feel very comfortable talking about orthogonality of solutions because I associate it to the fact that I can "modify" the contribution of one element of the basis without disturbing the others. In other words, I can push the particle toward the "y" direction, which will increase its velocity in that direction, without modifying velocity in any of the other two directions. Now my intuition tells me that this sort of "non-disturbing modification" should have some analogy in the standing wave since it seems that analogies worked quite well between both systems until the moment. So I think that you should be able to modify the amplitude of one harmonic without disturbing others. Now is where I fail to figure how you can do that, how you "push" one single harmonic without interfering others. In the laboratory, we had a generator that made the string vibrate in a certain frequency. Once we reached a permitted frequency we could observe its corresponding normal mode but we never tried to create a superposition of them and, in fact, I have no clue of how could we make that.
Thank you for reading.