## Introduction to Waves

Waves come in many shapes, sizes and forms. From the imperceptible waves in light to the massive tsunamis, waves are ubiquitous around us. The classical understanding of waves is that of a periodic vibration or oscillation that travels through a medium. However, the wave-like nature of light obviates the need for a medium – and allows the wave itself to move the particles from one place to another. Irrespective of the difference, all waves share some common properties. They are defined by a similar set of parameters and exhibit the same types of interactions: with both other waves and with the environment.

## Waves as Simple Harmonic Motions

Waves are best described as a type of simple harmonic motion (SHM). A classic SHM setup involves a body attached to a spring or a simple pendulum. The movement of a body attached to a spring or the swing of a simple pendulum represents a wave. Discounting friction, the movement of a simple pendulum in a plane has a minimal displacement at the equilibrium point, but maximum angular velocity. At the extremes, the balance shifts in favor of displacement with zero velocity [angular or otherwise]. If one were to project the angular velocity of the pendulum onto the X-axis, the similarities between the swing of the pendulum and that of spring would become apparent. Specifically, the projection of the pendulum follows a sinusoidal pattern. This characteristic – of being described by a sinusoidal wave, is a common characteristic in many waves. For all practical purposes, understanding the sinusoidal wave provides a comprehensive understanding of the way waves behave.

## Types of Waves and Wave Parameters

Apart from the classification of waves as either mechanical or electromagnetic, we can classify them based on other characteristics too. Since a wave is a propagation of a disturbance, we can ask if the disturbance is transmitted parallel or perpendicular to the movement of the wave. The former is known as lateral waves, and the latter are called transverse waves.

In the first case, the wave is propagated as compressions and rarefactions. The best-known example for lateral propagation of waves is the movement of sound through air. Shown is an illustration of a lateral wave.

As can be seen, the compressions and rarefactions of this wave lie parallel to the direction of its propagation [X-axis in this case]. A compression and the consequent rarefaction together constitute a wave. The wavelength (λ) in this case is defined as the distance between two consecutive compressions [dark blue regions] or rarefactions [lighter blue regions].

The second case is a transverse wave, depicted by the sinusoidal curve [shown in illustration].

Unlike the longitudinal wave, the transverse wave produces crests (highs) and troughs (lows). A crest and a trough make up a wave. The wavelength then, is the distance between any two consecutive crests or troughs. Some examples of a transverse wave include electromagnetic waves, the vibrations of a guitar string and water waves in a pond.

Let’s take an example of slinky which can represent longitudinal wave as well as transverse wave. In a transverse wave, there is perpendicular displacement of particles with respect to their direction of propagation. Hence a horizontal transverse wave can be created by moving the slinky vertically up and down. Similarly, horizontal longitudinal wave can be created in slinky by pushing and pulling it horizontally. Because in a longitudinal wave the particles are displaced parallel to the direction the wave travels.

Waves have a defined speed in a given medium. For instance, the speed of sound in dry air is 330 m/s. Given the velocity and wavelength, we can calculate the other parameter that is used to define waves – their frequency. Given that a wave consists of one crest and one trough [or alternatively one compression and one rarefaction], frequency is defined as the ratio of the distance traveled in one second to the wavelength. In other words – frequency is the number of crest-trough pairs that the given source emits in one second. A third measure frequently used to define waves – especially light – is the wavenumber. Wavenumber is the number of waves needed to traverse one unit length [meter]. Conventionally, the symbols λ, ν, and k are used to denote wavelength, frequency, and wave number, respectively. Mathematically, given speed s and wavelength λ

$s=\lambda \cdot v$

And

$v=\frac{s}{\lambda }$

This is the universal wave equation.  It states the mathematical relationship between the speed (s) of a wave, its wavelength (λ) and frequency (v). The wavelength (λ) that is associated with an object with a particular mass and momentum is known as De Broglie wavelength.

A third group of waves is known as the traveling wave. In the classic definition, a wave transmits energy through the medium, while the mean displacement of the medium itself remains zero. In a traveling wave, the medium or the carrier of the energy also moves along with the wave. Two examples illustrate this type: waves in an ocean, and the travelling of light through vacuum. The shock waves from explosions can also be considered as a traveling wave. As we see in the examples, both transverse and longitudinal waves can be traveling waves. In case of light from sun and other stellar objects, traveling waves is the only way of transmission, since there is no medium to conduct the waves in space.

Another group of waves is the body waves that travel through the interior of the earth. They are P- waves and S-waves. P-waves are also known as Primary waves or pressure waves. They are compression waves and can propagate in the solid or liquid material. S-waves are also known as Secondary waves. They are the shear waves and can propagate only in solid materials.

## Properties of Waves

Irrespective of their kind, all waves exhibit some common properties, viz: interference, reflection, refraction and diffraction. These properties are discussed below.

#### Interference

Interference is the interaction between two waves. While the terms “constructive” or “destructive” are added as qualifiers for describing the outcome of the interference, these are merely points on a continuous spectrum. The outcome of the interference between two or more waves is determined by one parameter, called the phase difference (denoted by the Greek letter φ). The sinusoidal representation of waves helps us visualize the concepts better [illustration given].

In the illustration, the purple (A) and green (B) lines denote two waves, traveling in the same direction. For sake of simplicity, let us assume that the waves have the same wavelength and amplitude. Let us consider two simple cases: the first is where the crests and troughs of A and B align perfectly. These waves are then said to be “in phase”. In the second case, assume that crests of A exactly line up with the troughs of B, wherein the waves are said to be “out of phase”. Waves that are in phase with each other are said to have a phase difference of zero radians, while those out of phase are said to have a phase difference of π radians. In the first case, the amplitudes simply add up, while in the second they cancel out each other.

From the above example, we see that two waves may interfere with each other with two extreme results: a perfect summation of amplitudes, or a perfect cancellation of the same. And these results correspond to two ends of a spectrum defined by the difference in their “phase”. The phase difference between two waves is the angle by which one of them has to be rotated [or translated in a general case] in order to be perfectly matched with the other. Thus, two waves with a phase difference of zero radians need not be translated to match each other, whereas two waves with π Radians need to be rotated/translated by 180 Degrees to match up. The result of interference is mathematically trivial and can be calculated for two waves, A and B of equal amplitude ‘a’, as follows:

$A+B=2a\mathrm{cos}\frac{\varphi }{2}\mathrm{cos}\left(kx-\omega t+\frac{\varphi }{2}\right)$

Where φ is the phase difference, k is the angular wavenumber (2π/λ), x is the displacement on the horizontal axis, ω is the angular velocity and t is the time since origin. Two types of interference are resonance and beats.

#### Resonance and Beats

In our example above, the waves are generated once, and their interference analyzed. Instead, if we imagine one of the waves being generated in a regular interval, and that the interval matches the phase and frequency of the other wave closely, then the interference is not just constructive, but it also feeds forward into itself. The result is a significant increase in amplitude with minimal additional external force. This phenomenon is called resonance. A good application of the principles of resonance is seen in playground swings. If timed right, a small push can increase the swing. Another cited example of resonance in grade textbooks is that of a marching army over a bridge. The marching column breaks formation in order to avoid resonance in the bridge, which may lead to structural damage. Resonance finds application in many other fields; especially in electronic circuits. A combination of inductors and capacitors aided by a battery, can create a circuit that has many properties of a SHM, and can be used to study wave properties.

Instead of reapplying the force in a timed manner, what would happen if we mix two different waves whose frequencies are close, but not equal? The waves would interfere constructively in some places along the X-axis, and destructively along others. Assuming that the wave is moving and the observer is not – the observer hears a procession of high and low sounds. This is called beats

The concepts of interference find extensive application in many fields – including in wireless communications.

#### Reflection, Refraction and Diffraction

Waves interacting with matter take two forms: reflection and diffraction. Waves interacting with large objects tend to get reflected. Reflection of waves is governed by the two laws of reflection: (i) the incident wave, the reflected wave and the normal at the point of incidence, all lie in the same plane, and (ii) the incident angle is equal to the reflected angle. Reflection also depends on the smoothness of the surface in comparison to the wavelength of the wave. In general the rougher the surface, the more imperfect is the reflection. Such imperfect reflection is also called dispersion.

If, on the other hand, the obstacle is small when compared to the wavelength, the wave bends around the obstacle. This phenomenon is known as diffraction. Young’s famous double slit experiment is a classic case study on diffraction.

Finally, there will be cases when the wave moves from one medium to another. In these cases, the velocity of the wave changes and also its direction. This change in direction of a wave is called refraction. The law of refraction includes Snell’s law which is given by

${n}_{1}\mathrm{sin}{\theta }_{1}={n}_{2}\mathrm{sin}{\theta }_{2}$

Where n1 and n2 are the refractive indices of the two media, and θand θ2 are the angles of incidence and refraction respectively. The Refractive index of a medium is the ratio of the speed of the wave in the medium to its speed in vacuum (for light) or other standard media (dry air for sound).

## Context and Applications

This topic is studied from the middle school level in science, as well as the complex concepts are studied in:

• Bachelors in Science (Physics)
• Masters in Science (Physics)

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