Interference of two waves: path difference decides

Interference

Two speakers play the same note. Why is it loud in some spots and almost silent in others?

Two sources send out matching waves. Drag the point P and read its path difference: a whole number of wavelengths and the waves add up, a half number and they cancel. The loud spots line up along hyperbolas.

Drag the ringed points on the board.

High school
Superposition
Path difference
Constructive and destructive
Interference pattern

d

4.0

\lambda

1.5

x_P

1.5

y_P

2.5

t / T

0.30

\Delta r = m\lambda,\quad \left(m + \tfrac12\right)\lambda

$|PS_1|$: 4.30
$|PS_2|$: 2.55
$\Delta r = |PS_1| - |PS_2|$: 1.75
$\Delta r / \lambda$: 1.17
$A_P / A$: 1.73
$N = 2\lceil d/\lambda\rceil - 1$: 5

Constructiveno

Destructiveno

Guided tour

1Equal distances: loud
Sources $4\,\text{m}$ apart, $\lambda=1.5\,\text{m}$ . Put $P$ on the perpendicular bisector: both paths are equal, $\Delta r=0$ , crest meets crest and $P$ swings with twice the amplitude.
2Half a wavelength: silent
Move $P$ to $(0.375,\,0)$ between the sources. Now $|PS_1|=2.375$ and $|PS_2|=1.625$ , so $\Delta r=0.75\,\text{m}=\tfrac12\lambda$ : crest meets trough and $P$ does not move at all.
3One whole wavelength: loud again
At $(0.75,\,0)$ the difference is $\Delta r=1.5\,\text{m}=\lambda$ . One wave is a full cycle behind the other, which is the same as being in step, so $P$ is loud again. Along the line between the sources loud and quiet spots alternate every $\tfrac14\lambda$ .
4Counting the loud lines
The green curves are all the points with $\Delta r=m\lambda$ : hyperbolas with the sources as foci. Between the sources $|\Delta r|<d$ , so with $d=4$ and $\lambda=1.5$ there are $5$ ( $m=0,\pm1,\pm2$ ). Shorten the wavelength to $1\,\text{m}$ and there are $7$ .

Explanation

1
Waves add
Where two waves overlap, each point's displacement is the sum of what each wave alone would give it, and each wave carries on afterwards as if the other were not there. That is the superposition principle.
2
Loud or quiet: the path-difference rule
For two sources that oscillate in step with the same frequency, everything depends on $\Delta r=|PS_1|-|PS_2|$ :
$\Delta r=m\lambda\ \Rightarrow\ \text{constructive},\qquad \Delta r=\left(m+\tfrac12\right)\lambda\ \Rightarrow\ \text{destructive}$
A constructive point still oscillates, with amplitude $2A$ ; it is not stuck at a crest. A destructive point stays still when the two amplitudes are equal.
3
Why hyperbolas
A curve whose points all have the same difference of distances to two fixed points is, by definition, a hyperbola with those points as foci. So each value of $m$ traces one branch, and the pattern stays fixed in space while the waves move through it. Requiring the same frequency is what keeps it still: this is why interference needs coherent sources.

Check yourself

Q01

Two in-step sources have $\lambda=2\,\text{m}$ . A point is $7\,\text{m}$ from one and $4\,\text{m}$ from the other. It is

Q02

At a constructive point, the medium

Q03

Two sources are $3\,\text{m}$ apart with $\lambda=1\,\text{m}$ . How many constructive lines cross the segment between them?

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