Interference of two circular waves

What is it about?

Two sources $E_1$ and $E_2$ at a distance $d$ dip into the water of a ripple tank in step and send out circular waves of wavelength $\lambda = \frac{c}{f}$ . Where they meet, the displacements add up (superposition). Whether they reinforce or cancel at a point $P$ depends on the path difference $\Delta s = s_1 - s_2$ :

maximum (constructive): $\Delta s = k \cdot \lambda$ , amplitude $2\hat y$ ,
minimum (destructive): $\Delta s = \left(k + \tfrac{1}{2}\right) \cdot \lambda$ , the water stays at rest.

In general $A = 2\hat y \cdot \left|\cos\frac{\delta}{2}\right|$ with $\delta = 2\pi \cdot \frac{\Delta s}{\lambda} + \Delta\varphi$ ( $\Delta\varphi$ : phase difference of the sources). All points with the same $\Delta s$ lie on a hyperbola with foci $E_1$ and $E_2$ . Since $|\Delta s| \le d$ , for sources in phase there are only the orders $|k| \le \frac{d}{\lambda}$ – for $d = 3\lambda$ the $7$ antinodal lines $k = -3, \dots, 3$ .

Far away the hyperbolas approach straight lines through the centre: $\sin\alpha_k = k \cdot \frac{\lambda}{d}$ – the same formula as for the double slit.

The snapshot shows crests light and troughs dark, the amplitude pattern shows how strongly the water oscillates on average. The waves are shown greatly slowed down; damping is neglected.

Try it

Press “Restart the waves” and watch the pattern build up.
Drag the cork $P$ onto a calm line and read off $\Delta s$ . Repeat on another calm line.
Switch to the amplitude pattern with all lines. Load example
Drag $E_2$ outwards. How many antinodal lines are there for $d = 2\lambda$ , $3\lambda$ , $4\lambda$ ?
Set $\Delta\varphi = 180°$ . What happens on the perpendicular bisector? Antiphase
Show the far field. Far field

Tasks

Task 1: Maximum or minimum?

Load task

$d = 6\,\text{cm}$ , $\lambda = 2\,\text{cm}$ , in phase. $P$ is $8\,\text{cm}$ straight above $E_2$ . Find $s_1$ , $s_2$ , $\Delta s$ and decide.

Show solution

$s_2 = 8\,\text{cm}$ , $s_1 = 10\,\text{cm}$ , $\Delta s = 2\,\text{cm} = 1 \cdot \lambda$ : antinodal line of order 1, amplitude $2\hat y$ . Show in the simulation

Task 2: Counting lines

Load task

How many antinodal and nodal lines are there for $d = 7\,\text{cm}$ and $\lambda = 2\,\text{cm}$ ?

Show solution

Antinodal: $|k| \le 3.5$ , so $7$ lines. Nodal: $|\Delta s| = 1, 3, 5, 7\,\text{cm}$ , so $8$ lines; the two with $|\Delta s| = d$ lie on the line through the sources, outside them. Show

Task 3: Far field

For $d = 4\,\text{cm}$ and $\lambda = 1.2\,\text{cm}$ , find the directions of the antinodal lines far away.

Show solution

$\sin\alpha_k = 0.3 \cdot k$ : $\alpha_1 \approx 17.5°$ , $\alpha_2 \approx 36.9°$ , $\alpha_3 \approx 64.2°$ ; $7$ antinodal lines in total. Show

Ideas for the classroom

Use: upper secondary, interference of mechanical waves, as a bridge to the double slit; a good companion to a real ripple tank, since here everything can be paused, slowed down and measured. With val=0 the values in the picture are hidden for measuring tasks.
Model: equal amplitudes, no damping, ideal point sources.
Common stumbling blocks:
- “There are no waves on the nodal lines.” – Both waves pass through; they just cancel.
- “The path difference is the distance between the sources.” – It is the difference of the paths to $P$ ; only on the axis outside the sources is $|\Delta s| = d$ .
- “A stable pattern needs any two waves.” – It needs equal frequency and a fixed phase difference (coherence).