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In case of longitudinal wave propagation, the velocity of the wave is given by $$V=\left[\frac{K+\frac{4}{3}G}{\rho}\right]^{0.5}$$ or $$Vp=\left(\frac{M}{\rho}\right)^{0.5}$$
But in the case of a solid rod, we use
$$V=\left(\frac{E}{\rho}\right)^{0.5}$$
Why do we use only $E$ (Young's modulus) instead of the constrained modulus ?
The main reason is that the rod is a one-dimensional approximation of three-dimensional motion.
Consider striking a rod on one end, which creates a longitudinal wave. The force applied and transmitted is in the direction of the rod axis, but there is nothing constraining the motion of the rod in the transverse directions. Thus, some of the motion of the bar is in the transverse direction (a demonstration of the Poisson effect).
Now consider a bulk material. As a wave applies a pressure to the material the material element has nowhere to go, since all adjacent material elements are similarly being compressed. Thus the behavior of the bulk material is inherently different than the behavior of the rod.
