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 = $\sqrt{\frac{Y}{\rho}}$
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}

Answer 2

A mechanical  longitudinal wave such as Sound wave which is propagated by the elastic compression of medium is called As compressional wave.

According to me there is no extended solid . I think it should be Only Solid.

Expression For Speed of Compressional wave in Solid:

When a thin rod of large length is stretched in the direction of its length, there will be increase in length with contraction in cross section and rod is set in longitudinal vibration.
Let ρ be density.
Υ be young's modulus of the material of the rod.
Let α be the cross sectional of rod.

Consider an elementary slice AB in the rod bounded by two planes A and B at x and x+dx respectively at right angles .
Let these planes be displaced by y and y+dy amount.

The distance between planes A,B :(x+dx+y+dy)-(x+y)=dx+dy

so increase in length is dy and increase in length per unit length is dy/dx

If f is total force acting on area α then

longitudinal stress=f/α
Young's modulud Y=f/α/dy/dx
f=Υαdy/dx-----------------(1)

the fore acting on second face is f+(df/dx)dx---------2

substututing value of f in equation 2

we get:

Υαdy/dx +(d/dx(Υαdy/dx))dx --------------3

resultant fore is given by difference of expression 3 and 1
Υαdy/dx +(d/dx(Υαdy/dx))dx  -Υαdy/dx
=αY( d²y/dx²) dx
but mass of element  dx is ραdx acceleration d²y/dt²
hence
d²y/dt²=Y/ρ( d²y/dx²)
d²y/dt²=c²d²y/dx²-----------(4)
where c²=Y/ρ where c is the velocity of wave propagation.from equation 4 which is a wave equation, so its solution should of the  form y=f(ct-x)+F(ct+x)

Hence we get longitudinal wave velocity as c=√Y/ρ