What crosses, what flips, what gets filtered

Count wavefronts arriving at a boundary each second. The same number has to leave, because they cannot pile up at the surface. So frequency is the property that carries across unchanged.

The new medium sets a new speed, and the wavelength does the adjusting:

$$f \ \text{unchanged}, \qquad v \ \text{set by the medium}, \qquad \lambda = \frac{v}{f}.$$

Watching a wave slow down and cutting its frequency instead is the error, and its consequences are audible and visible. Light entering glass would change colour; sound entering water would change pitch. Neither happens.

The clean statement of the division of labour: frequency belongs to the source, speed belongs to the medium, so wavelength is what gives.

A pulse arriving at a boundary splits: part reflects, part transmits. Whether the reflected part comes back inverted is decided by what the wave meets:

1. Slower on the far side (heavier string), or an end held fixed: the reflection is inverted.
2. Faster on the far side (lighter string), or a free end: the reflection is upright.

Memorising the fixed-end demonstration and applying it everywhere is what makes every reflection invert. The question to ask first is what happens to the wave speed on the far side.

The other half of the answer gets left out routinely: there is a transmitted pulse too, and it is never inverted. A boundary splits the energy between the reflected and the transmitted pulse, so a complete answer draws both. A fixed end is the limiting case where the far side is infinitely heavy and nothing is transmitted at all.

Polarization means restricting the oscillation to a single plane perpendicular to the direction of travel. That requires such a plane to exist in the first place.

A transverse wave oscillates across its travel direction, so there is a whole plane of possible directions to choose from, and a filter can select one. A longitudinal wave oscillates along its own direction of travel, so there is no sideways plane available. Sound cannot be polarized, and neither can any other longitudinal wave, whatever a slotted filter is doing.

So before reaching for polarization at all, check the oscillation direction. Light, string waves and other transverse waves can be polarized; sound compresses and rarefies air along its own line of travel and has nothing to select from.

That also makes polarization a genuine test: the fact that light can be polarized is direct evidence that light is a transverse wave.

Two different rules apply, and the first filter is the odd one out.

1. The first filter, meeting unpolarized light: it passes half the intensity and fixes the polarization direction to its own axis.
2. Every later filter: what gets through depends on the angle between its axis and the polarization now arriving. Parallel axes pass everything; perpendicular axes pass nothing.

So the light carries a direction with it, and that direction is rewritten at every filter it passes. Track it filter by filter.

Two failure modes have obvious tells. Halving at every filter regardless of orientation gives a nonzero answer for crossed filters, which is wrong: crossed polarizers are black. Subtracting the two angles and treating the difference as a fraction of the intensity gives zero for parallel filters, which is wrong the other way: parallel filters pass everything the light arrived with.