Question

Is the sinusoidal pattern on the string longer or shorter with a greater propagation velocity? PLEASE EXPLAIN.

Answer 1

The velocity of propagation is a function of the tension.

For example if I tighten the tuning knob on my guitar just a tiny bit, the pitch will go up noticeably. By doing so I don't change the length of the string, so all that's changing is the mass per unit length and the tension.

So just intuitively we can see here that the velocity of propagation increases with tension and decreases with mass per unit length. It turns out to actually be

v = sqrt( T / m)

where m is mass per unit length.

Since we're only changing the tension, the length of the sinusoidal pattern (the shape the string oscillates in) cannot change. What changes is how quickly it oscillates.