Question

deduce the expression for the speed of a compressional wave through an extended solid

Answer 1

Compression wave is not other wave it is just a name of longitudinal wave.
            we know, longitudinal wave travel in all types of matters. here question is this wave travel in solid. so we have to find expression of speed of longitudinal wave in solid.
   for finding speed of longitudinal wave in solid,first of all we find speed of it in liquid (fluid)
         a pulse (compression) travel in fluid filled tube. we observed that, after elementary time (Δt)  net force =foward force - backward force
                                F = pA -(p + Δp)A = -pA---------(1)

now, we know,
            mass = volume × density
                m = V × ρ
for elementary mass Δm = ΔV × ρ
                                       = A.Δx × ρ
                                       = Av.Δt × ρ
here, the air occupies a volume V = AvΔt outside the pulse is compressed by an amount of ΔV = AΔv.Δt as it enters in the pulse.
   so, ΔV/V = Δv/v---------------(2)

now,  F = Δm.a = Av.Δt × ρ .Δv/Δt = Aρv.Δv--------(3)

from (1) (2) and (3)
   pv² = -Δp/[ΔV/V] = bulk modulus = B
   v= √(B/ρ)
    
now when longitudinal wave propagates in solid bar.the situation is somewhat different from that of fluid filled in a tube of constant cross section, since the bar expands slightly Sidewise when it is compressed longitudinally.
by the same reasoning as that just given, that  the velocity of a longitudinal pulse in the bar( solid) is given by
               v = √{Y/ρ}
here Y is young's modulus of solid. and ρ is density of the solid.
           hence, speed of compressional wave through extended solid is v = √{Y/ρ}