We know the formula "wave velocity=frequency×wavelength" and the wave velocity for a standing wave is not zero. But, as the wave is "standing", so the wave velocity should be 0. Is it right? In the other hand, the energy transmitting along a standing wave is zero if we treat the standing wave as two traveling waves at opposite directions! Then it applies that the velocity of standing wave is zero. Who can give me the right energies of the standing waves in a clamped stretching string (the length is L). Obviously, the energies are discrete with a index number n. Then the energy E(n) is proportional to n, n^2 or n^3? Which power law is right?
The oscillation energy with the resonant mode n is proportional to n^2 and inverse proportional to the length L.
The energy transfer is absent in a standing wave, therefore, the provided formula is right if to represent the superposition of the two counter propagating waves.
Best regards,
A. Sizhuk
I am not professional in theoretical mechanics. This answer may be not accurate.
However, in my opinion, standing wave can be generated in two conditions. One is the moving of wave medium, in which the wave speed is equal to the moving speed of the medium (For example, some standing waves in the river). The other is the interference of two opposite wave, in which each the wave parameter (f v a etc.) of the two waves are the same (For example, the interference between incident waves and reflected waves) . In both case, if we select the medium as the reference system, the formula of wave velocity can be applied.
The standing wave in fluids is more complex nonlinear dynamic and the formula is incomplete,
https://www.youtube.com/watch?v=z3lVTRhOIEc
The standing wave is a composite wave which is composed from two equal and opposite waves every one has the same speed and wavelength.
So, mathematically it can be expressed by:
y(x,t)= A sin( 2 pi x/ lambda - wt) + A sin (2 pi x/ lambda + wt),
This form can be reduced to the form where time and position are separated:
y(x,t)= 2A sin (2 pi x/ lambda ) cos wt
Because of separation of the time and position, the wave is no longer propagating which means that it does not contain a phase relation like the travelling waves for example sin (2 pi x/ lambda - wt).
Accordingly one can not define for it a phase speed like the elementary wave. Here one can define for it a group velocity which describes the velocity of the envelope which in case of stationary wave is not moving resulting in zero group velocity meaning no net transfer of energy accompanying the stationary wave.
Best wishes
Abdelhalim abdelnaby Zekry
Though I am novice in this field, from the expression of standing wave (for perfect reflection, ) I may guess, Prof. Abdelhalim abdelnaby Zekry is right. The phase velocity of wave is defined as the speed of moving a particular constant phase in the direction of propagation. In standing wave the constant phase disappear, rather changes with time at particular point in the space.
My first concern is that as the group velocity of the wave is zero, can we call this phase velocity to be infinite? Because phase velocity x group velocity = c^2. To get some finite value, zero has to be multiplied with infinite value.
The second concern is that if the reflection is not perfect (magnitude of the reflection coefficient is less than 1), still the standing wave ratio will be there but with different SWR (less then infinity). Will this wave have some phase velocity? From the animations available over internet, in this case also those depicts same concept that constant phase does not exist.
If possible, I would like to have references from anyone regarding these concepts.
Regards
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