# 12 Superposition

## Noise-cancelling AirPod Pro

The recently launched Apple AirPod Pro presents a wonderful opportunity to relate an A-level concept to a real-world example – how noise-cancelling earphones work.

Apple’s website explained it in layman terms that seem to make sense. Let your students attempt to do a better job of explaining how destructive interference of waves is applied.

I probably won’t spend SGD379 on it though.

## Phase Difference GeoGebra Apps

I created a series of GeoGebra apps for the JC topics of Waves and Superposition, mainly on the concept of Phase Difference. The sizes of these GeoGebra apps are optimised for embedding into SLS. When I have time, I will create detailed instructions on how to create such apps. Meanwhile, feel free to use them.

Instructions on how to embed the apps into SLS can be found in the SLS user guide.

Phase difference between two particles on a progressive wave. Move the particles along the wave to see the value.

Phase difference between two particles on a stationary wave. Move the particles along the wave to observe how their velocities are different or similar.

Observe velocity vectors of multiple particles on a progressive wave.

A very good explanation of standing waves on Chladni plates. Watch out for the 3-Dimensional standing wave at 3’11”.

## Microwave Standing Waves

In the last tutorial, we were talking about the typical wavelength of different categories of electromagnetic waves. To help us remember the typical wavelength of microwaves, I suggest that we familiarise ourselves with a popular science experiment involving stationary microwaves in an oven.

Watch the following video from 2 min 20 sec to see how the experiment is conducted and how the wavelength of microwave can be measured after determining the distance between two adjacent nodes (the wavelength will be twice that distance). Therefore, the typical wavelength of microwaves will be of the order of magnitude of several centimetres.

## Pressure Variation in Stationary Sound Waves

For sound waves, we learnt that the compressions (position of maximum pressure) and rarefactions (minimum pressure) occur at the equilibrium position of the displacement of particles. This suggests that the pressure would vary the most in a stationary wave at the nodes of displacement. Right in the middle between two adjacent displacement nodes is the displacement antinode and we should expect the pressure variation to be the minimum there.

A displacement node is a pressure antinode.
A displacement antinode is a pressure node.

The standing waves associated with resonance in air columns can, therefore, be visualized in terms of the pressure variations in the column. Daniel A. Russell from The Pennsylvania State University made a wonderful animation showing how the variation of pressure occurs along an air column. (Link here)

It is a common misconception, even among physics teachers, that if a microphone is moved along the air column, it will pick up the loudest sounds at the displacement antinodes. However, according to Young & Geller (2007), College Physics 8th Edition, Pearson Education Inc. (pg 385), microphones and similar devices usually sense pressure variations and not displacements. In other words, the position within a stationary sound wave at which the loudest sound is picked up is at the displacement nodes which are the pressure antinodes.

Check out my own animation for a progressive longitudinal wave.

## Single Slit Diffraction using Fingers

This demonstration requires no material other than your own fingers. Hold your index and middle fingers close to each other, leaving a small slit between them about 1 mm in width.

Look through the slit into a source of light such as the window or a lamp. You will need to look with one eye up close to the slit. Warning: do not look directly at the sun.

You will be able to see a number of vertical dark lines between the fingers.

Science Explained

So where do these vertical lines come from? They are dark fringes caused by destructive interference of light when it diffracts through your finger tips.

This phenomenon can be explained using Huygens’ principle. Huygens pictures every point on a primary wavefront as a source of secondary wavelets and the sum of these secondary waves determines the form of the wave at any subsequent time. Hence, each of these secondary wavelets can interference with one another.

Constructive interference takes place when the difference in path lengths between two coherent waves is an integer multiple of the wavelength. This is when the resultant wave is the brightest. Destructive interference occurs when that difference in path length is a half-integer of the wavelength (e.g. $\frac{1}{2}\lambda$, $\frac{3}{2}\lambda$, $\frac{5}{2}\lambda$, etc.) and gives a dark fringe.

The alternating bright and dark fringes is a diffraction pattern, which becomes observable by the eye looking through the slit.