Wave Behavior

Reflection

fixed ends flip, free ends do not

A wave reflecting off a fixed end comes back inverted; off a free end it comes back upright.

waves A qualitative rule: only whether the end is held or free decides if the reflection flips.

Example

A wave reflecting off a fixed end comes back inverted; off a free end it comes back upright. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A fixed end flips the wave over

When a wave reaches an end that is held fixed, it bounces back inverted — turned upside down. The fixed point cannot move, so the returning wave must pull the opposite way to keep it still. A crest comes back as a trough.

fixed end: reflection is inverted\text{fixed end: reflection is inverted}
Reflection at a fixed end is upside downAn upward pulse on a string and, after the fixed wall, its reflection coming back as a downward pulse.incomingreflected: flipped

A free end reflects right side up

When the end is free to move up and down, the wave bounces back the same way up — not inverted. A crest comes back as a crest. So whether a reflection flips depends only on whether the end is held or free.

free end: reflection is upright\text{free end: reflection is upright}
Reflection at a free end keeps its shapeAn upward pulse and, after a free end shown as an open ring, its reflection coming back still an upward pulse.incomingreflected: uprightfree end

Reflection is a sign rule at the end

A fixed end behaves like a sign flip: multiply the incoming displacement by minus one. A free end keeps the sign. This is why a fixed-end crest returns as a trough, while a free-end crest returns as a crest.

Ryinyref−11 m−1 m−1−1 m1 m11 m1 m\begin{array}{c|c|c}R & y_{\text{in}} & y_{\text{ref}} \\ \hline -1 & 1\ \text{m} & -1\ \text{m} \\ -1 & -1\ \text{m} & 1\ \text{m} \\ 1 & 1\ \text{m} & 1\ \text{m}\end{array}