4The velocity of each point on the string is calculated at the instant of complete overlap.∂y∂t=−2c f′(x)（t=0）……
The following sections use superposition to examine reflection, standing waves, interference, diffraction, and refraction.
5Waves reflect at an end.y(L, t)=0……
6Sinusoidal waves with the same amplitude and wavelength travel in opposite directions.y=2A sin kx cos ωt……
7The result applies to a string fixed at both ends.fn=n c2L……
8Waves arriving from 2 sources add together.|r₁ − r₂|=m λ……
9Each point on a wavefront acts as a source.Next wavefront=Common tangent surface to the secondary wavelets……
10Waves pass through an opening.Direction of the first diffraction minimum:sinθ=λd……
11Refraction is determined for regions with different wave speeds.sinθ₁sinθ₂=v₁v₂……
What this chapter has shown
For a linear wave equation, the sum of 2 solutions is also a solution. Each traveling wave retains its shape, and the displacement of the string is their sum. At the instant of complete overlap, the displacement is 0, but points corresponding to the sloping sides have nonzero velocities. Their motion causes the crest and trough to reappear and pass through each other with their original shapes.
Reducing the trough depth by half gives a displacement of +0.5 cm at the center.
In everyday lifeNoise-canceling earphones reduce noise by adding sound that is in opposite phase to the noise at the position of the ear.
Detailed summary
Because the wave equation is linear, displacements add where waves overlap. Even when a crest and a trough cancel and the string becomes straight, the points on the string retain their velocities, so the waves pass through each other with their original shapes. Boundary conditions determine how waves reflect, and superposing waves traveling in opposite directions produces a standing wave. Treating each point on a wavefront as a wave source provides a common framework for explaining interference, diffraction, and refraction.