# Waves at a Boundary

Thursday, February 17, 2022

## What is a Wave Boundary?

A **boundary** is where a wave moves from one medium to another, such as light waves move from air to glass. At every boundary, some of the wave intensity is transmitted into the second medium and some is reflected back into the first medium.

After passing through a boundary, the wave's amplitude will be lessened due to the partial reflection at the boundary.

### Penetration of the Reflected Wave

When a wave is completely reflected by a boundary, an exponentially decreasing wave (the *evanescent wave*) penetrates into the second medium. Since 100% of the wave intensity is reflected however, the evanescent wave caries no energy and cannot directly be observed. However, making the second medium very thin can result in the wave penetrating through it and emerging on the other side.

The same is true for de Broglie waves, such as those of electrons. Electron waves can penetrate into a forbidden region but cannot be observed as being there, nor can their speed be measured. This can be attributed to the uncertainty principle (since the energy of electrons cannot be completely known, it is possible they can penetrate the forbidden region).

### Continuity at the Boundaries

When a wave crosses a boundary, the mathematical function describing it must meet the following two criteria:

- The wave function must be continuous
- The slope of the wave function must be continuous, except where the boundary height is infinite

Across any non-infinite boundary, the wave must be smooth (no gaps in the slope). To find the undetermined parameters of a wave function meeting another wave function, set the values equal to each other at the point of interaction and set the first derivative of each function equal to each other to ensure the curve is smooth.

### Infinite boundaries

For some boundary that will apply infinite acceleration (in theory; these are used as simplifications), the smooth curve property can be ignored. For instance, a perfectly elastic collision produced infinite acceleration, and would result in the velocity changing instantly. However, since this is an approximation of a very high acceleration, it is often accepted as allowed, and will be used in further discussions of quantum mechanics.

## Wave Properties

Classical waves and quantum waves have the following properties:

- When a wave crosses a boundary between two regions, part of the wave intensity is reflected and part is transmitted
- When a wave encounters a boundary to a region from which is it is forbidden, the wave will penetrate perhaps by a few wavelengths before reflecting
- At a finite boundary, the wave and its slope are continuous. At an infinite boundary, the wave is continuous but its slope is discontinuous