Understanding Basic Traveling Waves

After learning the basics of periodic motion, it's time to take up the study of oscillations to the next level. By the next level, I mean the dependence of the motion of the considered particle on not just the time variable, but also on the distance variable. Let's dive right into the topic by first differentiating waves.

We all have heard about various kinds of of waves traveling around us. Starting from the most familiar waves of light and sound, to the complex matter waves, all of them follow a common feature, i.e. oscillation of energy.

That is what a wave is.

A wave is just the phenomenon of oscillation of energy, using various properties of a medium such as physical, electro-magnetic, etc. For example, sound waves and light waves are both the carriers of energy, but a sound wave propagates through pressure variations, whereas a light wave travels by making use of electro-magnetic phenomena, which we'll discuss shortly.

Differentiating waves on alignment of propagation with respect to oscillation:

• Longitudinal waves: The type of waves that move in such a way that the oscillation of energy is along the direction of motion of the wave is defined as a longitudinal wave. Such kind of wave can be viewed by imagining two friends, who are walking forward, one ahead of the other, such that they continue tossing a ball between each other. In this, their motion can be seen as the motion of a wave and the ball acts as the packet of energy. These waves are also known as pressure waves.

• Transverse waves: The type of waves that move in such a way that the oscillation of energy is perpendicular to the direction of the motion of the wave is defined as transverse wave. Such kind of wave can be pictured by imagining the same two friends, walking forward, but one beside another, such that they continue tossing a ball between each other. In this, their motion can be seen as the motion of a wave and the ball acts as the packet of energy.

Imagine stretching a string and fixing both its ends on two points. Now grab the mid point of the string and pull it down, before letting it go. Chances are that you'll see the mid point of the string oscillating with an amplitude, with the end points fixed at their respective positions. These kind of waves are what we call the standing waves.

Now, for the next experiment, get into a hall with your friend and call out to him. If you shout loud enough and your friend hears well, chances are that he'll hear your call. Your voice reached him by the motion of sound waves, i.e. travelling waves. Had the sound waves been stationary, your voice would have never reached him.

• We all have read about the basic mathematical equation that governs the trajectory or the position of a particle in SHM, or in other words, a particle belonging to a stationary wave. It is given by \(y(t)=a\sin \omega t \), where \(a\) represents the maximum displacement of the particle from the mean position (amplitude), and \(\omega\) represents the angular frequency for the SHM.

• Now, since a travelling wave also moves forward while changing with time, a similar equation in case of a travelling wave must definitely include a function of both the direction of propagation (let it be \(z\)) and time. So, we get: \(y\left( z,t \right) =a\sin [z,t]\), where the symbols have their usual meanings.

To find this function that will explain the oscillation of the particle, we use the basic property of linearity of the function, i.e. the function enclosed inside the \(\sin\) block must be a linear function of \(z\) and \(t,\) or else the oscillating graph would lose its shape and the wave would compress or stretch out in different locations.

So, let that be given by \(\Phi \left( z,t \right)=\alpha z+\beta t\), but since we assume the wave to be travelling towards \(z=+\infty\), \(\alpha\) and \(\beta\) must have opposite signs. Therefore,

\[\Phi \left( z,t \right)=|\alpha |z-|\beta |t.\]

Also, we already know that this function must have the dimensions of radians, so \(\beta\) has the dimensions of \(T^{-1}\) and \(\alpha\) has the dimensions of \(L^{-1}\). As it turns out, \(\alpha\) is given by a constant \(k\) that is known as the wave-number for the wave, and equals \(\frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength of the wave. On the other hand, \(\beta\) is our age old friend \(\omega\) which equals \(2\pi\nu\), where \(\nu\) is the frequency.

So, we end up with: \(y\left( z,t \right) =a\sin ({k z-\omega t}). \)

Note: If the wave travels towards \(-\infty\), the function would change to \(y\left( z,t \right) =a\sin ({k z+\omega t}) .\)