Why is displacement zero for a particle when pressure is maximum and i want a detailed perfect explanation for displacement - pressure wave ? i want a very accurate answer

Well, your question is not very clear as the conditions that you have mentioned are very vague.

But where propagation of pressure waves are concerned, they are longitudinal waves. At every node, there is max pressure and at every anti node, the pressure is minimum or sometimes zero. Any solution to the classical wave equation can represent a pressure wave. Refer to the classical wave equation in any physics text book. If you do not know how to solve it, then fix a solution and check for the values that you want. You can randomly try out trigonometric circular functions which will mostly work in simple cases.

If you can also mention where from you are measuring the displacement, I can give you a more precise answer.

the pressure fluctuation depends on the difference between the displacement at neighbouring points in the medium. Quantitatively, the change in

volume( /_\ V ) = S(y₂-y₁) = S[y(x+/_\x,t) - y(x,t)]

in the limit /_\x----- 0, the fractional change in volume dV/V(volume change divided by original volume) is dy(x,t)/dx

the fractional volume change is related to the pressure fluctuation by the bulk modulus B, which definition is B=-p(x,t)/(dV/V)

p(x,t) = -Bdy(x,t)/dx the negative sign arises because when dy(x,t)/dx is positive , the displacement is greater at x+/_\x than at x

at t=0  ,  y(x,t) and p(x,t) describe the same wave, these two functions are one-quarter cucle out of phase: at any time, the displacement is greatest

where the pressure fluctuation is zero, and vice versa. In particular, note that the compressions( points of great pressure and density) and rare fractions

9points of lowest pressure and density) are points of ZERO displacements...

we have already know that pressure is directly proportional to density and inversely proportional to the volume

hence, displacement is zero when pressure is maximum...