The shape travels. The stuff does not

The picture that fixes this: pick a single point of the medium and watch only that.

1. A cork on the water bobs up and down in place while crests pass under it. It does not ride the wave to shore.
2. An air molecule near a speaker oscillates back and forth about a fixed spot. It does not fly across the room into your ear.

So the air in a room does not drift toward the listener when someone speaks, and a wave pool does not empty itself at one end. Energy moves along the medium; matter oscillates and stays.

The temptation is real, because the shape genuinely does travel and it is the thing your eye follows. What is worth checking every time is whether the sentence you are writing describes the pattern moving or the material moving.

Transverse means the disturbance is perpendicular to the direction of travel. Longitudinal means it is parallel. So classifying a wave requires knowing which way the particles move, and nothing else.

Sound is the case that catches people. It is routinely drawn as a smooth sine curve, and that curve is read as up-and-down motion, so sound gets called transverse. But look at what the vertical axis plots: pressure, or displacement measured along the travel direction. The air molecules move back and forth in the same line the sound travels, and the compressions and rarefactions line up with propagation. Sound is longitudinal.

The habit to build: ask what the vertical axis of the drawing is plotting before reading motion off its shape. A graph is a graph. The shape of the plot decides nothing about the classification.

For a mechanical wave, the speed is fixed by the medium alone. On a string,

$$v = \sqrt{\frac{F_T}{m/L}},$$

the tension over the mass per unit length. Nothing about how the wave was made appears anywhere in that expression.

So the two things a person at the end of the rope can do are both irrelevant to $v$:

1. Shaking harder raises the amplitude. The pulse is taller and arrives at the same time.
2. Shaking faster raises the frequency. More pulses per second, each still travelling at $v$.

To change $v$ you have to change the medium: tighten the string, swap in a lighter one, or warm the air for sound. Shouting louder does not make sound arrive sooner, which is worth noticing as an everyday check.

A single pulse is one disturbance travelling through the medium; a periodic wave is a train of them, produced by a source that keeps oscillating. Everything above applies to both.

What arrives at the far end is energy, and how much depends on the amplitude and the frequency together rather than on either alone. That partly explains why amplitude and frequency feel like one quantity, and they are still two independent properties of the wave, read off two different axes.

One more distinction worth having early: the speed of the wave is not the speed of a particle of the medium. The wave crosses the room at $v$; the particle it passes moves a few millimetres, slowly, and returns. Confusing those two is a compact version of the whole topic.