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Waves are an important topic taught is secondary school physics classes. What students learn from this subject can be applied not just to waves in nature, such as ocean waves and sound waves, but also to mathematics, engineering, and more complex physical theories. Waves can often be a 'dry' topic for some high school students, in which case I think it is important to be able to demonstrate the 'cool' physical properties and capabilities of them to inspire young minds. For some, the math itself is enough to excite a person, but for others, its what that math can accomplish and produce in our real world that brings them wonder.

One of the best ways to communicate how waves can interact with us in our daily lives is through sound. Many students, when they first think of waves, think of ocean waves. However, sound waves, being invisible to our naked eye, are something that many high school students might not realize have equal or more interesting properties. By investigating sound waves properties, we can also learn about how pressure works in air, and thus, can understand more about how things such as airplanes fly. By investigating waves, sound, and air pressure with a high school level class through demos like acoustic levitation, we can effectively and visually teach new, but simple, topics to students, while also inspiring them with the world of physics.

Theory Behind Waves

There are two types of forms physical waves come in. There are transverse waves, where the surface oscillates perpendicular to the direction of the waves travel, and longitudinal waves, which consist of patterns of alternating compressions and rarefactions travelling along the waves path. Transverse waves have several notable characteristics. Their amplitude (A) which is the height of their highest point. The wavelength (λ), which is the distance it takes for the waves pattern to repeat, and is often best measured from peak to peak or trough to trough. A wave's frequency (υ) is the most complicated of the transverse wave's characteristics to measure. It is defined as number of waves that have passed a specific point in an amount of time, often measured in Hertz (Hz), which is the number of waves passed per second. They can be more easily measured by finding the waves period (T), which is the amount of time it takes one wavelength to fully traverse a point. The inverse of the period gives the frequency.

υ = 1⁄T

Another way to measure the frequency is to take the speed the wave is travelling at (v) and divide by it's wavelength of the wave.

υ = v⁄λ

When two waves come across each other in the same location, they go through something called interference. At interference, the amplitudes of the waves at each point sum together to create either constructive interference where the net amplitude is greater than the individual ones, or destructive interference where the net amplitude is less that the individual ones. If the waves are travelling in opposite directions, once they pass each other they will regain their individual amplitudes and waves speeds, as neither of the waves energy is lost.

If consecutive wave patterns cross paths, the total interference can be a combination of constructive and destructive interference, if the frequencies or waves speed is different. This interference can create wave patterns that look strange, and do not look like conventional sinusoidal waves. By changing the frequency, amplitude, wavelength, or wave speed of any of the subsequent waves, the net wave pattern can change drastically.

A particularly interesting physical phenomenon, resulting from interference, is generated from two waves, equal in wave speed and frequency, moving in opposite directions, with their x-intercepts matching in location. This creates a standing wave, which is a wave that oscillates vertically, but not horizontally. Standing waves have two important positions, they have nodes, which are locations that do not oscillate vertically at all, and anti-nodes, which are positions on the waves that have the maximum oscillation. The distance between nodes is half a wavelength of the subsequent waves. Typically, standing waves can be seen in the physical world in the form of resonance frequencies, when waves are reflected off a surface at a point where the waves amplitude is zero.

Standing waves are not just a transverse wave phenomenon, they also exist in longitudinal waves, the same waves that travel in air and through slinkies. If we compare a transverse standing wave and a longitudinal standing wave in air, as seen in figure below, pressure nodes are created, where the pressure does not oscillate.

Pressure Nodes

Objects in mediums tend to prefer being in areas of low air pressure. This same physical phenomenom is how air planes fly in the air, using Bernoulli's principle, where fast moving air create areas of lower air pressure.

In pressure antinodes, the pressure oscillates rapidly from high to low. Light objects will tend towards volumes of low air pressure and stability, hence why light enough objects can be suspended in pressur nodes. If multiple standing waves are created with their nodes aligned at a single location, constructive interference occurs, increasing the amplitude (or pressure oscillations) in the wave, making it harder for an object to escape the pressure node.