Nodes and antinodes in stationary waves

[arvindsharma]

Dear Friends,

in my textbook it is written that when a stationary wave is formed in a string which is clamped at one end and free at the other end then a node is produced at the clamped end and an Anti-node is produced at the free end.the explanation given is that since at clamped particle can't move so a node is formed(i am also agree with the mechanism of node formation)) and at free end since the particle can move so an anti-node is formed.my doubt is that why anti-node is formed at free end,why can't there be a situation where particle can move but does not achieve the maximum amplitude.i mean why can't there be a situation where particle at free end behaves like in-between node and Anti-node positions.please explain me the whole mechanism.

Thanks

Arvind

 

[sophiecentaur]

Hi and welcome to PF.
That's a good question and there are possible answers at various levels. The answer that I like best is as follows.

Firstly, let's just go over how a standing wave is formed. It's basically what happens when a wave in one direction encounters another wave (at the same frequency) from the other direction. The resulting interference gives you a stationary pattern of nodes and antinodes. The traveling wave that is approaching the clamped end cannot couple any energy to the clamp because the clamp is too massive and rigid. All the wave energy has to go somewhere and the only possible place it can go is in a 100% reflected wave, back down the string with a node at the clamp. A wave approaching the free end will actually a small amount of energy into wafting the air up and down. Most of the energy will go back from the end in the form of a reflected wave. But the free end is not, actually, a perfect reflector ; some energy can be lost to the air, so the wave at this point is not actually a perfect antinode.
You are therefore, right about not having a true antinode at the far end - good thinking.

Wind instruments have a hole / horn at the end and there is a massive 'end effect' as the energy delivered as musical sound forces a different actual frequency than would be suggested b y the physical length of the pipe. Quartz crystals, as used in oscillators, are clamped at two points, not quite at the ends, rather than right at the end because of a similar 'end effect'. Also, look at a child's glockenspiel bars, they are also mounted on nodes but there is always an overlap at each end.