the initial displacement and velocity
To set a wave in motion you must answer two questions, not one: where is everything right now, and how fast is it moving right now? A guitarist plucks (sets a shape, releases from rest); a piano hammer strikes (gives a velocity to a flat string). These are the two pieces of starting information the wave equation demands — the initial displacement and the initial velocity.
Because the wave equation u_tt = c^2 u_xx is second order in time, its solution needs two initial conditions at t = 0: u(x,0) = phi(x), the initial displacement (the shape of the string), and u_t(x,0) = psi(x), the initial velocity (how fast each point is moving). Together these are the Cauchy data. Compare an ordinary differential equation like m x'' = -k x for a single mass: being second order, it needs the starting position AND the starting velocity. The wave equation is the same idea spread over a whole continuum of points. Giving only phi, with no psi, leaves the problem under-determined; giving u and u_t at one instant fixes the entire past and future uniquely.
The two pieces play visibly different roles, and d'Alembert's formula shows exactly how. The initial displacement phi splits into two half-height copies that travel off rigidly, keeping their shape. The initial velocity psi contributes through an integral, which spreads its effect over an expanding interval and tends to leave a lasting plateau rather than a travelling bump. A pluck (shape, zero velocity) and a strike (zero shape, velocity) therefore sound and look quite different — which is why a plucked guitar and a struck piano have distinct tones.
Pluck (phi = a bump, psi = 0): two travelling half-bumps, shape preserved. Strike (phi = 0, psi = a localized kick): a flat raised region that widens at speed c and never quite returns to zero in the middle. Same equation, opposite-looking responses, set entirely by which of the two initial data you supply.
Displacement makes a travelling bump; velocity makes a spreading plateau.
You need exactly two initial conditions — no more, no less. Adding a third (say a prescribed initial acceleration) would over-determine the problem and generally make it unsolvable, because the equation itself already fixes u_tt from u_xx.