Physics 9702 · Paper 2

Interference

Connect two speakers to the same signal and walk in front of them. Some places are loud. Some places are almost silent.
The pattern never moves. That fixed pattern of loud and quiet places is .

What interference is

Wire two loudspeakers to the same signal generator. Now they make sound waves of the same wavelength, in step.
Where the two sets of waves overlap, they add by superposition (from the last page). Some points always add up to loud. Some always cancel to quiet. This steady map of loud and quiet is the interference pattern.
Learn this line for the exam: interference is the superposition of waves from two coherent sources.
Light does the same. Two overlapping light sources make bright and dark bands, not one flat glow.
The same experiment works for every wave: two dippers in a ripple tank (water), two speakers (sound), two slits (light), or two gaps in a metal plate (microwaves). The exam can name any of them.

Two loudspeakers on one signal generator, their sound waves overlapping in front — Two speakers on one signal generator: where the sound waves overlap, they interfere.

Adding up and cancelling out

Where the two waves arrive in phase (crest on crest), they add: . Two equal waves make double the height — loud, or bright.
Where they arrive in antiphase (crest on trough), they cancel: . Two equal waves make zero — quiet, or dark.
When they are only half in step, they only partly add. The point is not the loudest, but it is not silent either.

phase φ

φ = 90°·resultant amplitude = 2A cos(φ/2) = 1.41·A·in between

Two coherent sources overlap and make a fixed pattern. In phase, they add: loud or bright. Antiphase, they cancel: quiet or dark.

Your turn— tap to reveal the worked answer (9702/23/M/J/24 Q4)

Two sources send ripples across a tank. At one point a line of crests from one source keeps meeting a line of troughs from the other. What do you see there, and why?

Answer: the water there stays almost flat — this is destructive interference. Crest meeting trough means the two waves arrive in antiphase, so by superposition they cancel and the resultant is about zero. (9702/23/M/J/24 Q4)

Which one happens: the conditions

You now know in phase means add, antiphase means cancel. To tell which one happens at a point, count the path difference (the extra distance one wave travels) in wavelengths.
A whole number of wavelengths of extra distance: the waves line up crest on crest, in phase. They add. This is constructive — loud and bright.
A whole number and a half of wavelengths: a crest meets a trough, antiphase. They cancel. This is destructive — quiet and dark.

\begin{gathered} \text{loud (constructive):}\quad \text{path difference} = n\lambda \\[4pt] \text{quiet (destructive):}\quad \text{path difference} = \left(n + \tfrac{1}{2}\right)\lambda \end{gathered}

Symbols

$\text{path difference}$ = extra distance one wave travels (m)
$\lambda$ = wavelength (m)
$n$ = any whole number: 0, 1, 2, 3, … (none)

The gold line is the blue wave plus the green wave. Top: a whole number of wavelengths of path difference, so the crests line up and the waves add. Bottom: a whole number and a half, so crest meets trough and they cancel to nothing.

Using the conditions

Work out the path difference in wavelengths, then match it to one of the two lines.

Worked example

Deciding whether waves add or cancel

Two sources make waves of wavelength 6 cm. At a point, one wave has travelled 9 cm further than the other. Add or cancel?

Path difference in wavelengths: $\frac{9}{6} = 1.5$ wavelengths.
1.5 is a whole number and a half, so it fits $\left(n + \tfrac{1}{2}\right)\lambda$ with $n = 1$ .
That is the destructive condition, so the waves cancel: this point is quiet (or dark).
Now change one number. Make the extra distance 12 cm. Then $\frac{12}{6} = 2$ whole wavelengths. That is the constructive condition with $n = 2$ , so the point is now loud (or bright).

Answer

\text{destructive: } 1.5\lambda

Path difference in whole wavelengths: add, so loud and bright. Whole number and a half: cancel, so quiet and dark.

Your turn— tap to reveal the worked answer (9702-style)

Waves of wavelength 4 cm meet at a point after one has travelled 10 cm further than the other. Add or cancel?

Answer: $\frac{10}{4} = 2.5$ wavelengths. That is a whole number and a half, so it is destructive — the waves cancel and the point is quiet or dark. (9702-style)

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