superposition of waves
A wave carries crests and troughs through space, and when two waves meet at the same place the displacement at that instant simply adds. That is superposition. The speed is fixed by v = fλ, the two waves reinforce or cancel each other, and a small difference in frequency makes beats. What this lesson follows is not a wave standing still. It is a traveling wave that actually flows. The height at the meeting place is the sum.
A traveling wave flows sideways while keeping the shape of its crests and troughs. The wavelength λ is the distance from one crest to the next, and the frequency f is the number of crests that pass one point in 1 second. Their product v = fλ is the speed at which the wave advances. In a real medium the speed is fixed almost entirely by the medium, so raising the frequency shortens the wavelength by the same amount. Move λ and f and the way the speed is fixed shows itself. The spacing of the crests is the wavelength.
Overlap two waves of the same frequency and the real wave at each point is the sum of the two displacements. If the phase difference Δφ is 0, crest meets crest and the wave grows much larger. That is reinforcement. If the phase difference is π, a crest meets the other trough and the wave cancels flat. Headphones that erase noise use exactly this cancellation. They make a wave whose phase is off by π from the incoming sound and wipe the sound at the ear. Move the phase difference and the summed curve changes with it. Crest on crest grows, and crest on trough vanishes.
Overlap two waves whose frequencies differ only a little, and they fall in step and out of step over and over, so the size of the sum slowly grows and shrinks. That slow sway is a beat. The frequency of the beat equals the difference of the two frequencies, |f₁ − f₂|. When you tune an instrument, you sound the two notes together and adjust until the beat disappears, and then the two frequencies have become exactly the same. Move f₂ to bring the two frequencies close and the beat slows down so you can see it. The smaller the difference, the slower the sway.
Superposition can be checked right at one point. Slide the vertical probe left and right, and at that place the height of each wave and the value of their sum come out together. At every position the sum is always just the two heights added. Because the sum is exactly the sum of the two heights at every point, the two waves pass through each other and then go on with their original shapes, as if the other had never been there. Move the probe and the sum of the two values lands on the actual sum curve. The shape after the crossing matches the shape before the meeting, and that is another face of superposition.
Superposition holds the same way no matter how the wave shakes. In a transverse wave the medium shakes perpendicular to the direction of travel, and that is the curve we usually see. In a longitudinal wave the medium shakes back and forth along the direction of travel, and places that are crowded alternate with places that are sparse. The two pictures are the same wave drawn in two ways. Light is a transverse wave and sound in air is a longitudinal wave, yet superposition applies to both in the same way. Press the toggle and the same shaking appears in the two shapes. Only the drawing differs. The rule of adding is the same.