What you can learn with this simulation, topic by topic. Follow each Try it to see it on screen.
1. The superposition principle
When two waves pass the same point of a medium, the displacement there is the sum of the displacements each wave would cause alone. After overlapping, each wave travels on unchanged (independence of waves).
y=y_1+y_2
Try it Put the probe anywhere and step through time reading y_1, y_2 and y: always y=y_1+y_2. In split view, add the two heights on the same vertical line; opposite signs subtract.
2. Phase and phase difference
Phase is the angle that tells where in its cycle a wave is. Here each wave follows the equation below, with \varphi_1, \varphi_2 the phases at t=0, x=0. For waves travelling the same way, the phase difference \Delta\varphi=\varphi_2-\varphi_1 is the same at every time and place (opposite directions: topic 8).
y_i=A_i\sin\left(\omega t-kx+\varphi_i\right),\qquad \Delta\varphi=\varphi_2-\varphi_1
Increasing \varphi by \Delta\varphi shifts the waveform by \frac{\Delta\varphi}{360^\circ}\lambda in +x. A difference of 0^\circ puts crest on crest; 180^\circ puts crest on trough.
Try it At 1\,\mathrm{Hz} (\lambda=4\,\mathrm{m}), set \varphi_2 to 0^\circ, 90^\circ, 180^\circ, 270^\circ: wave 2 moves right one grid square (1\,\mathrm{m}=\frac{\lambda}{4}) at a time.
3. Constructive and destructive interference
The amplitude A of the sum of two waves of equal frequency depends on the two amplitudes and the phase difference.
A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\Delta\varphi}
At \Delta\varphi=0^\circ, A=A_1+A_2 is largest (constructive). At 180^\circ, A=\lvert A_1-A_2\rvert is smallest (destructive), and zero if the amplitudes are equal: the string stays still. In between, A lies between these values.
Try it Toggle 0^\circ and 180^\circ with Set phase difference and read the resultant amplitude. With A_1=5\,\mathrm{cm}, A_2=3\,\mathrm{cm}: 8\,\mathrm{cm} and 2\,\mathrm{cm}; at 90^\circ, \sqrt{5^2+3^2}\approx5.8\,\mathrm{cm}.
4. The sum is a sinusoid of the same frequency and wavelength
Adding two sinusoids of equal frequency and wavelength gives a sinusoid with the same frequency, wavelength and speed; only its amplitude and phase differ. Its phase \varphi satisfies:
y=A\sin\left(\omega t-kx+\varphi\right),\qquad \tan\varphi=\frac{A_1\sin\varphi_1+A_2\sin\varphi_2}{A_1\cos\varphi_1+A_2\cos\varphi_2}
Try it Play in overlay view: all three curves move together at one speed, and crests of the sum are \lambda apart; for a phase difference between 0^\circ and 180^\circ they lie between the crests of the two waves.
5. Phase difference and shift in distance
If one wave is the other shifted by \Delta x, their phase difference is set by how many wavelengths the shift is. A shift of one wavelength is 360^\circ, back in phase.
\Delta\varphi=\frac{\Delta x}{\lambda}\times 360^\circ
So whole-wavelength shifts interfere constructively and odd half-wavelength shifts destructively. This is why two point sources make an interference pattern (Two-Source Interference simulation).
Try it With \lambda=4\,\mathrm{m} and \varphi_2=180^\circ, wave 2 is wave 1 shifted by 2\,\mathrm{m}=\frac{\lambda}{2} and the sum is smallest.
6. Oscillation at one point
At a fixed probe, y_1 and y_2 are simple harmonic motions with the same period \frac{1}{f}, and so is their sum y. With a phase difference, their peaks arrive at different times.
Try it Play and check that all three traces share one period. Time ten oscillations with the stopwatch: \frac{10}{f}. Moving the probe does not change the amplitude of y.
7. The medium changes speed and wavelength, not the interference
Both waves travel on one string, so they share v=\sqrt{\frac{T}{\mu}}. Changing tension or density changes v and \lambda=\frac{v}{f} together, but the resultant amplitude depends only on the two amplitudes and the phase difference. Interference is set by distance relative to wavelength, that is, by phase.
Try it Raise the tension from 1.6\,\mathrm{N} to 6.4\,\mathrm{N}: speed and wavelength double, while the resultant amplitude and type of interference stay the same.
8. Waves in opposite directions: standing waves
With wave 1 travelling right and wave 2 left, their phase difference changes with position. The sum no longer travels; each point oscillates in place with its own amplitude, a standing wave.
With equal amplitudes, points where \cos\left(kx+\frac{\Delta\varphi}{2}\right)=0 never move (nodes) and points with amplitude 2A_0 are antinodes. Neighbouring nodes are \frac{\lambda}{2} apart. Changing the phase difference shifts the whole pattern; unequal amplitudes leave \lvert A_1-A_2\rvert of motion at the nodes.
Try it Set wave 2 to ← and play. The string oscillates in place inside the dotted envelope \pm A(x); move the probe to a node and an antinode and compare the amplitude readout. With \lambda=4\,\mathrm{m} the nodes are 2\,\mathrm{m} apart. This is the same physics as the standing waves formed by reflection in Wave on a String.