Last time, we started talking about waves.A good formal definition of a wave is: A wave is a perturbationthat is periodic in space and in time. However, the term “perturbation” may sound somewhat weird to people whohave just started learning the theory of light. Let us thenreplace this sophisticated term by a more familiar and intuitively clearer notion of “displacement”.

So, a wave is a displacement that is periodic in space andin time. What this means, we can easily demonstrate using the “wave machine” that I brought to class last time.

In the case of some waves, including light waves, it doesnot make much sense to talk about a “displacement”. But this is not a big problem. Once we understand the “displa-cement waves”, we can easily extend our description to include those other “no-displacement” wave types.

We said that a wave propagating in one direction can befully characterized by three parameters: the wavelength , the frequency , and the amplitude A (i.e., the maximum displacement value). We can then calculate the value ofthe displacement (call it y) at any point x and at any timeinstant t using the equation:

However, this is not the only way – we can use a differentset of parameters, taking the advantage of relations that exist between them. First, we don’t need to use the frequ-ency. Frequency is the number of oscillations made in aunit of time (a second). So, if we divide one second by thenumber of oscillations, we get the time needed to make a single oscillation – we call it the oscillation period T.

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Also, we can introduce the speed of the wave. Note thatduring a single oscillation period each “wave crest” travelsa distance equal to a single wavelength – hence we getthe speed V :

Combining the two, we can readily obtain a new formulafor the frequency:

And by putting in into the initial wave equation form, we get:

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Which is an equivalent equation form interms of the wavelength an the wave’s speed.

More about waves. You will see now a few slides explainingthe meaning of the terms “phase” and “phase shift”.Consider a chain of beads fixed on a string:

Now, we will excite a wave traveling from the left to righton this string (e.g., by “wiggling” the left end):

The crests and the troughs move with a constant speed, and it seems that the beads move with the same speed.

But the progressive motion of the beads is only an illusion!They cannot move in the horizontal direction because theyare all fixed on the string – the can only move up and down.To show that, let’s mark a single bead with blue color:

Now one can see that the only motion of the blue bead isvertical oscillation! (and the same is true for all other beads,of course).

Now, let’s consider two parallel waves – their amplitudesmay be different, but each bead in the upper wave oscillates “in unison” with its “partner”* in the lower wave:

* By “partner”, I mean a bead with the same x coordinate in the other wave.

The situation we are taking about can be better illustrated using an animated picture:

We say about such waves that their oscillations are in phase,or, shortly, that they are in phase.

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This constant angle is what we call the phase angle, or,shortly, the phase. Does such an operation have a strong effect on the wave? No! If we made an animation of the first equation, and thenof the other equation, you would not be able to see anydifference! Then, what is the phase good for? Well, in the case of a single wave it is not particularly relevant. However, if we havetwo or more waves, their phases become really important!

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Let’s again look at the animation of two waves “in phase”:

But here is a different situation: now, the lower wave is clearly “lagging behind” the upper one by a quarter ofthe wavelength, or ¼ of the oscillation period.We can write their equations as:

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is called the phase difference, and more often the phase shift. So, here the phase shift between the two wavesis ½π (or 90 degrees, if you don’t like radians! – however, Idon’t recommend disliking radians!).In many phenomena in which two or more waves are involved,the phase shift between those waves plays a crucial role!Therefore, we pay much attention to it. However, about suchphenomena we will start talking somewhat later.

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One more interesting situation – the two waves are shiftedby one-half wavelength, so the phase shift between themis π (or 180 degrees, if you don’t follow my recommenda-Tions, and you still prefer the traditional angle measure).

Again, an animation:

About such ways, we say shortly that they are out of phase.