# Characteristics of Waves

When studying simple harmonic motion, we have considered one or two bodies that may feel external forces, or may exert forces on each other. The motion of the system was found by analyzing the objects individually, writing down the different forces that act on each object separately, using Newton's Second Law to arrive at the equations of motion, and finally solved these equations. Ultimately, we were after the position as a function of time, \( x(t) \).

What happens if we have many bodies, all interacting with each other? Writing down a position function of each body would be too difficult, with an uncountable number of equations associated with all the different bodies. Many important and interesting systems in nature are like this.

If the number of bodies is so large that they can be treated collectively as some sort of medium, then indeed there is much we can learn about the medium. It turns out that such systems often exhibit collective behaviour that can be analyzed and understood without having to understand all the internal interactions between the different bodies making up the medium.

For example, a tight string is composed of many atoms, but we may just want to describe its displacement away from its stretched position. In that case, we can do so with a function \( y(x,t) \), where \( x \) denotes the position along the string, \( t \) is the time, and \( y(x,t) \) denotes the displacement at position \( x \) and at time \( t \). To solve for the explicit form of \( y(x,t) \).

When dealing with a single body, in order to solve for the explicit form of \( x(t) \), we drew a force diagram, used Newton's Second Law to write down the equation of motion for \( x(t) \), and finally solved the equation of motion. It is possible to follow similar steps, applying Newton's laws to every point in the medium, to arrive at the equation of motion for \( y(x,t) \). The resulting equation for the case of a string takes the form,

$$ T \frac{\partial^2 y(x,t)}{\partial x^2} = \mu \frac{\partial^2 y(x,t)}{\partial t^2} $$

Where \( T \) is the string's tension, and \( \mu \) is the mass density of the string (mass density if equation to the mass of the string divided by the length of the string). You can think of the right hand side as the usual \( m d^2 x(t) / d t^2 \) term in Newton's Second Law, and the left hand side is the force, that is, \( F = m a \). But, this is now applied at every point along the string, \( x \). This equation is known as the wave equation and it finds application in many fields of science and technology. The solutions of these are known as waves.

### Characteristics of Waves

There are many types of media that support waves, and what is "waving" in each one could be different. Waves are disturbances that propagate through the medium about some unperturbed state. Water waves are disturbances in the height of the water's surface. Sound waves are disturbances in the local density of materials. Waves in a string are disturbances in the local strength of the electromagnetic field. In each of these cases, the function \( y(x,t) \) represents something different, such as height, displacement, density, and so on, but the mathematical description is the same.

The most general solution to the wave equation is:

$$ y(x,t) = f(x - v_w t) + g(x + v_w t) $$

where \( f(x - v_w t) \) is any continuous function of the single variable \( x - v_w t \) and \( g(x + v_x t) \) is any continuous function of the variable \( x + v_w t \). The constant \( v_w \) has units of velocity and it is given by,

$$ v_w = \sqrt{T / \mu} $$

The first function, \( f( x - v_w t ) \) represents waves traveling in the positive x-direction. Notice that is we simultaneously translate time by some amount \( \Delta t \) and distance by an amount \( \Delta x = v_w \Delta t \) then \( f((x + \Delta x) - v_w (t + \Delta t)) = f(x - v_w t) \). Similarly, the function \( g(x + v_w t) \) represents waves traveling in the negative x-direction. Therefore, a general wave in one dimension is propagating through the medium in either one way or the other with velocity \( v_w \). This is the physical importance of the constant \( v_w \). The velocity of propagation of waves through the medium depends on the properties of the medium.

What is vibrating also depends on the medium. In the case of a stretched string for example, the displacement of the string is vibrating about its stretched position. In water waves, the height of the water is vibrating. Notice that in both of these cases, the vibrations are perpendicular to the direction of the motion of the wave itself. Such waves are known as transverse waves, since the vibrations at every point in the medium are transverse to the direction of the wave's motion. Importantly, we still call it a one-dimensional wave because the propagation of the wave is confined to one dimension. Its vibration, however, is happening in a transverse medium. For example, a water wave in a narrow canal can be described as moving in the horizontal direction, but the water's displacement is in the vertical direction.

In contrast, the sound waves in a material are associated with compression and rarefaction of the material's density. The vibration in this case is along the direction of motion of the wave itself.

Such waves are known as longitudinal waves. The vibrations induced in a long spring are another example of longitudinal waves. Earthquakes produce different type of waves, but the primary ones (p-waves) and least destructive are longitudinal waves that travel fastest and therefore are first to arrive. These correspond to compression and rarefactions in the earth's density.

### Sound

Sound waves are the compression and rarefaction of the local density of materials and as such they are longitudinal waves. You are likely familiar with sound as it travels through air, a gas. As you might know, for a fixed temperature, the density of a gas is related to its pressure. So the compression and rarefaction of density associated with sound waves in air result in changes of pressure, which is what your ear drum senses as sound. The speed of sound in dry air at \( 20^{\circ} \) is

$$ v_s = 343 m/s $$

Note the particular way I quoted this velocity as depending on the temperature of the air as well as its humidity. This is an example of the very general point mention above, namely that the speed of wave in a medium depend son the properties of the medium. In the case of gases, the general expression for the speed of sound is given by an expression similar to the speed of waves in a string, with the string's tension replaced by the gas' average pressure.

A gas is not he only medium to support sound waves. A liquid or a solid can also serve as media where density fluctuations result in the propagation of sound waves. Considering the very different average densities and pressures found in liquids and solids, it is no surprise that sound waves in liquids and solids propagate at very different speeds as compared to sound speed in air. The dependence of the speed of the properties of the medium can actually be of use in learning about the interior of maters. For example, p-waves, which are a type of sound-wave produced by an earthquake, are used to study the Earth's composition.

### Light

One of the triumphs of \( 19^{th} \) century physics is the realization by James C. Maxwell that his theory of electric and magnetic fields predicts the existence of light. He worked out the speed of waves in the electromagnetic field and found to his amazement that it is the speed of light in a vacuum,

$$ c = 299,792,458 // m/s $$

This helped identifying light as an electromagnetic wave. Light is a transverse wave, but this fact is by no means obvious. It is the oscillations of magnetic and electric fields in a plane perpendicular to the direction of motion. But, unlike other waves where the medium is manifest (water for water waves, air for sound waves, etc.), light requires no medium to propagate. The absence of any clear mechanical medium associated with light has perplexed \( 19^{th} \) century physicists who attempted to introduce such a medium, known as the ether. These attempts, and the ether, were abandoned with the advent of Einstein's special theory of relativity.

When light travels through a medium other than a vacuum, it slows down. The ratio of its velocity in vacuum, to its velocity in the medium is known as the index of refraction

$$ n = \frac{c}{v} $$

Where \( v \) is the light velocity in the medium. The index of refraction depends on the properties of the medium as well as the wavelength of light.