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with a description of what is happening in a water tank when two waves meet, mix and interfere with each other. To do this, we must find a convenient way of representing the positions of the peaks and troughs of each wave. In the technical jargon, these are known as phases. Colloquially things are described as ‘in phase’ if they reinforce one another in some way, or ‘out of phase’ if they cancel each other out. The word is also used to describe the Moon: over the course of around twenty-eight days, the Moon passes from new to full and back again in a continuous waxing and waning cycle. The etymology of the word ‘phase’ stems from the Greek phasis, which means the appearance and disappearance of an astronomical phenomenon, and the regular appearance and disappearance of the bright lunar surface seems to have led to its twentieth-century usage, particularly in science, as a description of something cyclical. And this is a clue as to how we might find a pictorial representation of the positions of the peaks and troughs of water waves.

Figure 3.2. The phases of the Moon.

Have a look at Figure 3.2. One way to represent a phase is as a clock face with a single hand rotating around. This gives us the freedom to represent visually a full 360 degrees worth of possibilities: the clock hand can point to 12 o’clock, 3 o’clock, 9 o’clock and all points in between. In the case of the Moon, you could imagine a new Moon represented by a clock hand pointing to 12 o’clock, a waxing crescent at 1:30, the first quarter at 3, the waxing gibbous at 4:30, the full Moon at 6 and so on. What we are doing here is using something abstract to describe something concrete; a clock face to describe the phases of the Moon. In this way we could draw a clock with its hand pointing to 12 o’clock and you’d immediately know that the clock represented a new Moon. And even though we haven’t actually said it, you’d know that a clock with the hand pointing to 5 o’clock would mean that we are approaching a full Moon. The use of abstract pictures or symbols to represent real things is absolutely fundamental in physics – this is essentially what physicists use mathematics for. The power of the approach comes when the abstract pictures can be manipulated using simple rules to make firm predictions about the real world. As we’ll see in a moment, the clock faces will allow us to do just this because they are able to keep track of the relative positions of the peaks and troughs of waves. This in turn will allow us to calculate whether they will cancel or reinforce one another when they meet.

Figure 3.3 shows a sketch of two water waves at an instant in time. Let’s represent the peaks of the waves by clocks reading 12 o’clock and the troughs by clocks reading 6 o’clock. We can also represent places on the waves intermediate between peaks and troughs with clocks reading intermediate times, just as we did for the phases of the Moon between new and full. The distance between the successive peaks and troughs of the wave is an important number; it is known as the wavelength of the wave.

The two waves in Figure 3.3 are out of phase with each other, which means that the peaks of the top wave are aligned with the troughs of the bottom wave, and vice versa. As a result it is pretty clear that they will entirely cancel each other out when we add them together. This is illustrated at the bottom of the figure, where the ‘wave’ is flat-lining. In terms of clocks, all of the 12 o’clock clocks for the top wave, representing its peaks, are aligned with the 6 o’clock clocks for the bottom wave, representing its troughs. In fact, everywhere you look, the clocks for the top wave are pointing in the opposite direction to the clocks for the bottom wave.

Using clocks to describe waves does, at this stage, seem like we are over-complicating matters. Surely if we want to add together two water waves, then all we need to do is add the heights of each of the waves and we don’t need clocks at all. This is certainly true for water waves, but we are not being perverse and we have introduced