Lesson 7 of 10 · 8 min
Reflection and standing waves
NCERT §14.6
Riya ties one end of the bead string to a heavy peg in the wall and flicks the free end. The hump runs to the peg and comes back, but upside down. Then Mamaji threads the far end through a ring sliding on a smooth rod. This time the hump comes back the right way up.
The lesson in notes
In short
A wave that meets a boundary is reflected, as an echo is. At a boundary between two media part of it is reflected and part transmitted; the transmitted part obeys the laws of refraction and the reflected part the laws of reflection.
Rigid boundary (a string tied to a wall): the end cannot move, so the wall pulls the string the opposite way (Newton's third law). The pulse comes back inverted, a phase change of π. If yi = a sin(kx − ωt), then yr = a sin(kx − ωt + π) = −a sin(kx − ωt).
Free boundary (a string tied to a ring that slides freely on a rod, or the open end of an organ pipe): the pulse comes back upright, with no phase change, and the displacement at the end becomes twice the pulse's.
A wave reflected at a boundary overlaps the incoming wave. Two identical waves moving in opposite directions, a sin(kx − ωt) and a sin(kx + ωt), add to y = 2a sin kx cos ωt.
This is a standing wave: kx and ωt appear separately, so the pattern does not travel. Each point does SHM with its own amplitude 2a sin kx.
Nodes, where sin kx = 0, never move: x = nλ/2 for n = 0, 1, 2, …
Antinodes, where |sin kx| = 1, swing with the largest amplitude 2a: x = (n + ½)λ/2.
Neighbouring nodes are λ/2 apart, and so are neighbouring antinodes; a node and the next antinode are λ/4 apart.
All particles between two neighbouring nodes move in phase, with different amplitudes; particles on opposite sides of a node are in opposite phase.