Reflection & Transmission
A fixed end flips the pulse; a free end doesn't.
Lesson at a glanceShow the key points
- At a fixed end the clamp sends back an inverted pulse; the incident and reflected displacements cancel at the end.
- At a free end (a ring sliding on a rod) the pulse reflects without inversion and the end overshoots.
- At a junction part is reflected, part transmitted.
- Entering a region of smaller wave speed (heavier string): the reflected wave is inverted. Entering a region of larger speed: not inverted. The transmitted wave is never inverted.
Send a pulse down a rope tied to a wall and it comes back. But it comes back upside down. Untie the rope and slip the end on a smooth pole, and it comes back the right way up. Why?
Reflection at a fixed end
Along the string, each element is pulled by the string on either side and moves to follow the pulse. At a clamped end the last element is pulled by the string on one side and by the clamp on the other. Since the end can’t move, the clamp’s force exactly opposes the string’s. That opposing force launches a pulse back along the string with the same shape but inverted.
At the end itself the incident pulse tries to lift the string and the reflected one pulls it down by the same amount, so the end never moves. You can find the string’s shape during reflection by adding to the real pulse an imaginary upside-down pulse approaching from beyond the wall.
Reflection at a free end
Now tie the end to a light ring that slides without friction on a vertical rod. When the pulse arrives, the string pulls the ring up and nothing pulls it back (the rod exerts no vertical force), so the end overshoots, rising higher than the pulse itself. Equivalently, a pulse of the same shape, not inverted, is reflected; at the end the two displacements add.
Partly reflected, partly transmitted
Most ends are neither perfectly fixed nor perfectly free. Join a light string to a heavier one and send a pulse along the light string: part of it is reflected (inverted) and part continues into the heavy string. Send it the other way, from heavy to light, and the reflected part is not inverted.
The rule in terms of speed
Waves are slower on the heavier string (v = √(F/μ)), so the rule can be stated using wave speed:
A fixed end is the extreme case of an infinitely heavy string (speed zero); a free end, of a massless one.
- 1Fixed end: the reflected pulse is inverted; the end stays still.
- 2Free end: the reflected pulse is upright; the end overshoots.
- 3At a junction: partial reflection and transmission.
- 4Into a slower medium: reflection inverted. Into a faster medium: not inverted. Transmission: never inverted.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.