FHSST Physics/Waves/Definition

What are waves?

Waves are phenomena that everyone experiences constantly; water waves, sound waves, light waves, human waves when the home team scores... the list goes on. When asked what makes a wave a wave, the most common responses would probably be that a wave is something that moves, or propagates, or perhaps that it is something that repeats over and over again. These properties do capture the essential qualities of waves. Now we must determine these properties quantitatively, and discover what governs their behavior.

Generally, a wave is defined as any phenomenon which can be modeled by a function of the form $f({\bar {k}}{\bar {r}}-\omega t)$ where the $r$-vector represents a position in space, and $t$ represents a time, and the $k$-vector and omega are both constants. Don't be intimidated by the vectors in the argument - most of our time at first will be spent on one-dimensional waves. If the wave is in only one spatial dimension 'x', for instance a wave travelling on a taut string, it can be written simply as $f(kx-wt)$ .

Any function of this form "propagates" along the ${\bar {k}}$ direction over time. As time increases, the argument of the function increases; over time the form of the function effectively advances through space. Try coming up with functions of this form, and plot them at time $t=0$ , then plot them again at a later time. This progressive property will become obvious. Try to figure out the velocity with which your function advances! (we will study this later) The negative sign in front of the time term causes the wave to propagate in the direction defined as positive (if that seems confusing, try plotting more functions over time, and examine the results). If you replace the negative with a positive (or instead consider a negative value of omega), the wave will propagate in the negative direction.

A very special and important case of a wave is the mathematical function $f({\bar {k}}{\bar {r}}-\omega t)=\sin({\bar {k}}{\bar {r}}-\omega t)$ , or in one dimension, $f(kx-\omega t)=\sin(kx-\omega t)$ . This is a sinusoidal wave – it oscillates up and down infinitely in both directions, and moves as time progresses. I mentioned that waves have the quality of repeating over and over, the quality of periodicity. However, many functions of the form mentioned above do not seem to repeat. However, you will find that ALL waves can be decomposed into a sum of many of these simple, infinitely repeating waves when you learn about Fourier transformations.

More than any other concept, physicists are finding that waves characterize the structure of the universe at every scale imaginable. As you learn about the physics of waves in everyday life, keep an open mind towards finding waves and wave behavior everywhere you turn.

Let's consider a very well-known case of a wave phenomenon: water waves. Waves in water consist of moving peaks and troughs. A peak is a place where the water rises higher than when the water is still and a trough is a place where the water sinks lower than when the water is still.

So waves have peaks and troughs. This could be our first property for waves. The following diagram shows the peaks and troughs on a wave.

In physics we try to be as quantitative as possible. If we look very carefully we notice that the height of the peaks above the level of the still water is the same as the depth of the troughs below the level of the still water.

Waves are repetitions of physical quantity in a periodic manner, carrying energy in the process. The water waves, for example, can be visualized to repeat any of the physical quantities like "peaks", "troughs", "potential energy" or "kinetic energy". Even, we can visualize water waves as the motion of disturbance (energy). It is the energy aspect of waves that is central to the understanding of different types of waves, many of which are not visible.

Looking closely at the water wave, we can recognize that crests and troughs basically represent of extreme potential and kinetic energies in addition to representing rise and fall of water from the still level. At the peak, energy is only potential, whereas energy is only kinetic at the trough. Similarly, propagation of electromagnetic wave is associated with repetitions of magnetic and electric field in space with certain periodicity. As existence of electrical or magnetic fields does not require any medium, electromagnetic waves can move even in the absence of any medium.

Characteristics of Waves: Amplitude

We use symbols agreed upon by convention to label the characteristic quantities of the waves. The characteristic height of a peak and depth of a trough is called the amplitude of the wave. The vertical distance between the bottom of the trough and the top of the peak is twice the amplitude. To put it simply, the amplitude is the distance of the wave from the medium, to the crest or trough

Worked Example 1

Question: (NOTE TO SELF: Make this a more exciting question) The height of the wave from the medium is 2m. What is the distance from the peak to the trough. What

Answer

The amplitude is 2m. (Read above paragraph to know why). The distance from the peak to trough is 4m.

Characteristics of Waves : Wavelength

Look a little closer at the peaks and the troughs. The distance between two adjacent (next to each other) peaks is the same no matter which two adjacent peaks you choose. So there is a fixed distance between the peaks.

Similarly, you'll notice that the distance between two adjacent troughs is the same no matter which two troughs you look at. But, more importantly, its is the same as the distance between the peaks. This distance which is a characteristic of the wave is called the wavelength.

Waves have a characteristic wavelength. The symbol for the wavelength is the Greek letter lambda, $\lambda$.

The wavelength is the distance between any two adjacent points which are in phase. Two points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They don't have to be peaks or trough but they must be separated by a complete number of waves.

Characteristics of Waves : Period

Now imagine you are sitting next to a pond and you watch the waves going past you. First one peak, then a trough and then another peak. If you measure the time between two adjacent peaks you'll find that it is the same. Now if you measure the time between two adjacent troughs you'll find that its always the same, no matter which two adjacent troughs you pick. The time you have been measuring is the time for one wavelength to pass by. We call this time the period and it is a characteristic of the wave.

The period of the wave is denoted with the symbol $T$.

Characteristics of Waves : Frequency

There is another way of characterising the time interval of a wave. We timed how long it takes for one wavelength to go past. We could also turn this around and say how many waves go by in 1 second.

We can easily determine this number, which we call the frequency and denote f. To determine the frequency, how many waves go by in 1s, we work out what fraction of a waves goes by in 1 second by dividing 1 second by the time it takes T. If a wave takes 1/2 a second to go by then in 1 second two waves must go by. ${\frac {1}{\frac {1}{2}}}=2$. The unit of frequency is the Hz or s⁻¹.

Waves have a characteristic frequency.

$f={\frac {1}{T}}$     f : frequency (Hz or s−1) T : period (s)

generally, the frequency of a wave is the number of crests that pass by per unit time.

Characteristics of Waves : Speed

Now if you are watching a wave go by you will notice that they move at a constant velocity. Thinking back to rectilinear motion you will be able to remember that we know how to work out how fast something moves. The speed is the distance you travel divided by the time you take to travel that distance. This is excellent because we know that the waves travel a distance $\lambda$ in a time T. This means that we can determine the speed.

$v={\frac {\lambda }{T}}$     v : speed (m.s−1) $\lambda$ : wavelength (m) T : period (s)

There are a number of relationships involving the various characteristic quantities of waves. A simple example of how this would be useful is how to determine the velocity when you have the frequency and the wavelength. We can take the above equation and substitute the relationship between frequency and period to produce an equation for speed of the form

$v=f\lambda$     v : speed (m.s−1) $\lambda$ : wavelength (m) f : frequency (Hz or s−1)

Is this correct? Remember a simple first check is to check the units! On the right hand side we have speed which has units ms⁻¹. On the left hand side we have frequency which is measured in s⁻¹ multiplied by wavelength which is measure in m. On the left hand side we have ms⁻¹ which is exactly what we want.

Speed of a wave through strings

The speed of a wave traveling along a vibrating string (v) is directly proportional to the square root of the tension (T) over the linear density (μ):

$v={\sqrt {\frac {T}{\mu }}}\,$

μ is equal to the mass of the string divided by the length of the string.

$\mu ={\frac {M}{L}}$