1. Let’s talk about waves. A good formal definition of a wave is: A wave is a perturbation that is periodic in space and in time. However, the term “perturbation” may sound somewhat weird to people who have just started learning the wave theory. Let us then replace this sophisticated term by a more familiar and intuitively clearer notion of “displacement”.

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

3. A wave propagating in one direction can be fully characterized by three parameters: the wavelength, the frequency f, and the amplitude A (i.e., the maximum displacement value). We can then calculate the value of the displacement (call it y) at any point x and at any time instant t using the equation:

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

5. In other words: Also, we can introduce the speed of the wave. Note that during a single oscillation period each “wave crest” Travels a distance equal to a single wavelength – hence we get the speed V :

6. Also, we can introduce the speed of the wave. Note that during a single oscillation period each “wave crest” travels a distance equal to a single wavelength – hence we get the speed of the wave V : Combining the two, we can readily obtain a new formulafor the frequency:

7. And by putting all these things in into the initial wave equation form, we get: Which all are equivalent equation forms.

8. More about waves. You will see now a few slides explaining the 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 right on 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.

9. But the progressive motion of the beads is only an illusion! They cannot move in the horizontal direction because they are 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 is vertical oscillation! (and the same is true for all other beads, of course).

10. Now, let’s consider two parallel waves – their amplitudes may 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.

11. 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.

12. 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 then of the other equation, you would not be able to see any difference! Then, what is the phase good for? Well, in the case of a single wave it is not particularly relevant. However, if we have two or more waves, their phases become really important!

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

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

15. is called the phase difference, and more often the phase shift. So, here the phase shift between the two waves is ½π (or 90 degrees, if you don’t like radians! – however, I don’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 such phenomena we will start talking somewhat later.

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

17. Again, an animation: About such ways, we say shortly that they are out of phase.