reflection and transmission
Shout toward a cliff and an echo comes back: the sound wave reflected. Look through a window and you see both the world outside and a faint ghost of yourself: light partly transmitted, partly reflected. Whenever a wave meets a boundary or a change in the medium, part of it bounces back (reflection) and part continues on (transmission). The wave equation predicts exactly how much of each.
Reflection at a fixed end is handled by a wonderfully simple trick, the method of images. Take a string fixed at x = 0 (so u(0,t) = 0 for all time). A wave moving left toward the wall must keep that point pinned at zero. The trick: pretend the string extends to the left forever, and add an imaginary mirror-image wave — the same pulse, but flipped upside down and coming from the other side. By symmetry the two always cancel exactly at x = 0, automatically satisfying the boundary condition, and on the real (right) half the image wave appears as the reflected pulse. A fixed end flips the pulse over (reflection with a sign change); a free end (where u_x = 0) reflects it right-side up (no sign change). For a half-infinite or finite string, repeated images build the full solution.
Transmission happens at an interface between two media with different wave speeds — a light string tied to a heavy one, or light passing from air into glass. There the wave splits: some energy reflects, some transmits, and the split is governed by the mismatch in impedance (roughly, how 'hard' each medium is to drive). A large mismatch reflects most of the wave (which is why you see your reflection in glass and hear an echo off a wall); a matched impedance transmits almost everything (the principle behind anti-reflection coatings and acoustic matching). Reflection and transmission are how waves interact with the structured world, and the method of images is the cleanest tool for the idealized boundary cases.
A pulse travels along a rope toward a wall where the rope is tied down (a fixed end). It reflects and comes back upside down. If instead the rope end is free to slide on a frictionless pole (a free end), the pulse reflects right-side up. The method of images reproduces both by adding a flipped or un-flipped mirror pulse.
Fixed end flips the pulse; free end does not — both via a mirror-image wave.
Reflection and transmission conserve total energy (incident energy = reflected + transmitted), but they do NOT simply add amplitudes — the split depends on impedance, and at a fixed end the reflected wave is inverted, so amplitudes can subtract rather than add.