Snapshot versus history
There are two very different graphs of a wave, and beginners mix them up constantly. A snapshot plots displacement y versus position x at one frozen instant — like a photo of the whole rope. On it, the repeat distance is the wavelength. A history plots displacement y versus time t at one fixed point — like watching a single cork bob. On it, the repeat time is the period.
The wave function
A single formula captures both graphs at once. A sinusoidal wave traveling in the +x direction is written as the wave function below, where A is the amplitude, k is the wave number, \omega is the angular frequency, and \varphi sets the starting phase.
The traveling sine wave. The combination (kx − ωt) moves in +x; flip to (kx + ωt) for a wave moving in −x.
The wave number counts radians of phase per metre; the angular frequency counts radians per second.
Divide the two definitions and the master relation reappears: v = \omega / k = f\lambda. And notice what each little piece of the medium does — as time runs, every fixed x oscillates as a sine of t. Each particle performs simple harmonic motion; the wave is just those oscillations, staggered in phase from point to point.
What sets the speed?
If the source doesn't set the speed, what does? The medium — through a tug-of-war between two properties. A restoring property (stiffness) yanks each bit back toward rest and hurries the wave along; an inertial property (mass) resists the motion and slows it. For a wave on a stretched string this is exact: the speed depends on the tension T and the linear mass density \mu (mass per unit length).
Wave speed on a string: more tension → faster; heavier string → slower.
This is exactly why tuning a guitar works. Tightening a string raises T, so v rises, so its resonant frequencies rise, and the pitch goes up. The thick low strings have large \mu, giving a slower wave and a lower pitch. (Idealization: this formula treats the string as perfectly flexible and ignores its own bending stiffness and any damping — good enough for real strings, not perfect.)
Longitudinal waves and sound
Sound is a longitudinal wave in air. As it passes, air molecules jiggle back and forth along the travel direction, packing into compressions and spreading into rarefactions. Your eardrum feels the resulting pressure swings. The speed of sound in air is about 343\text{ m/s} at 20\,^{\circ}\text{C} — but roughly 1480\text{ m/s} in water and 5000\text{ m/s} in steel. Speed depends on the medium and its temperature, never on how loud or high-pitched the sound is.