I have been reading the paper on wave equations, and I have some confusion in notations.

Consider the initial value problem(IVP)(Wave equation):

$\frac{\partial ^2 u } {\partial t^2}(x,t) = \Delta u (x,t), (x,t) \in \mathbb R^{d} \times \mathbb R,$ $u(x,0) =f(x), \frac{\partial }{ \partial t} u(x, 0) =g(x).$

The solution to the above Cauchy Problem is $$u(x,t)=\cos (t \sqrt{- \Delta}f) (x) + \left( \frac{\sin (t \sqrt{- \Delta})}{\sqrt{- \Delta}}g \right) (x)$$

My Question is: What does the square root sign tells us?

Is it true that to study wave equation is equivalent to study the multiplier $e^{it |\xi|}$ ($f\mapsto (e^{i t |\xi|}\hat{f})^{\vee}$)? ( I don't see why this multiplier cover both the term in the solution. Can we deduce some thing about the multiplier $\cos t |\xi|$ and $\frac{\sin t |\xi|}{|\xi|}$

Background: I am familiar with the following solution to the wave equation: $u(x,t)= f\ast \partial_t W_t(x) + g\ast W_t (x),$ where $W_t = [ \frac{ \sin 2\pi t |\xi|}{ 2\pi |\xi|}]^{\vee}.$

(But here I do not see any square root sign; I guess, there must some thing to do with square root sign ....)