Derivation of equation for velocity of P wave & S wave

P-wave propagates with the help of expansion and compression of the medium, so the motion of the particles is in the direction of propagation of the wave whereas the S-wave propagates with the relative perpendicular motion of the particles. P-waves are also called longitudinal wave/compressional wave and S-waves are also called transversal wave/shear wave. The derivation of equation for velocity of P-wave/S-wave is quite complicated and is very difficult to understand so here, we would only deal with the 1 dimension accoustic wave equation.
You can derive the velocity of S-wave or P-wave if you know 3 equations. First is the Newton's ssecond law of motion (F=ma), Hooke's law for elasticity and equation of sound waves. Easy, isn't it ?
Let us consider that P is the pressure that is exerted on the subsurface layer by the wave incident on it and x is the distance travelled by the wave in order to do so. þ is the density of the formation on which the given wave is incident. ø and µ are the modulus of elasticity and modulus of rigidity.
As we know that the Newton's 2nd law is given by:

dP/dx = þ* dx²/dt² (1)
As we know that as per Hooke's law, the ratio of stress and strain is equal to the modulus of elasticity; 
so,
P = ø* dy/dx (2)

differentiating equation (2) w.r.t. x, we get

dP/dx = ø* dy²/dx²

putting equation (1) in above equation, we get

þ* dx²/dt² = ø* dy²/dx²
dx²/dt² = (ø/þ)* dy²/dx²
comparing above equation with equation of sound wave we can directly obtain

V² = (ø/þ)
V = (ø/þ)^½ (3)

Now, for s-wave we can write modulus of elasticity(ø) = modulus of rigidity(µ) 
so, by replacing the above value ø in equation (3) by the above relation, we can write velocity of S-wave which is given by,

V = (µ/þ)^½
Similarly you can derive the equation of velocity of p-wave in which the modulus of elasticity is the function of modulus of rigidity and bulk modulus(K).
i.e. ø = K + (4/3µ).so, by replacing the above value ø in equation (3) by the above relation, we can write the velocity of P-wave which is given by,

V = [{K + (4/3µ)}/þ]^½

That's it !!!