2

Often there are exactly two common ways for describing the amplitude of a certain type of wave:

sound waves: pressure or displacement

electromagnetic waves: E or B

waves on a string: transverse displacement, transverse velocity

Of course we could make up others, like the transverse acceleration of waves on a string. But I don't think it's a coincidence that there are so often two of them that people really care about. My argument for this comes from considering two equal-amplitude waves that collide head-on and superpose. Energy has to be conserved, and almost always the way that plays out is that there are two types of energy in the wave, and even if one goes down because that amplitude cancels, the other goes up to compensate because the other amplitude interferes constructively.

It seems like we also see this sort of two-variables thing playing out over and over again in expressions for impedances and phase velocities.

Is this analysis reasonable? Is there a term for these pairs of amplitude variables, like "conjugate amplitudes" or something? Can anyone give a sketch of the physics, or point me to a good open-access description that talks about this?

What I seem to find when I look at various sources is detailed analyses of specific types of waves, but nothing about the generalization.

  • I'm not sure this is really leading anywhere except that in Lagrangian mechanics, kinetic energy is a function of velocity and potential energy is a function of position. Interference patterns etc are just special cases. There is interchange of KE and PE in any wave motion. (The analogy with EM waves is left as an exercise!) – alephzero Dec 24 '19 at 23:44
  • This is mostly due to physics being second order in time. – Gabriel Golfetti Dec 25 '19 at 00:23
  • @alephzero: That sounds plausible, but not trivial to work out, and the analogies don't always seem to come out right. For instance, the ratio of E to H in an electromagnetic wave is the impedance, but for a wave on a string, the ratio of transverse displacement $y$ to transverse velocity $v_y$ is independent of the medium for a fixed frequency. This is the kind of thing I'd like to see worked out clearly. For the string, perhaps the variables should be $v_y$ and transverse force per unit length, but then those aren't conjugate variables in the sense of Lagrangian mechanics. –  Dec 25 '19 at 16:28
  • somewhat related: https://physics.stackexchange.com/q/521121/45664 – user45664 Dec 25 '19 at 17:15

0 Answers0