# Why is the formula for the speed of sound in solids different from that in liquids and gases?

We know that sound waves are longitudinal waves and that the propagation of longitudinal waves depends on the bulk modulus $$B$$ (volumetric elasticity). Why, then, does the speed of sound in a solid depend on the Young’s modulus $$Y$$ of the medium and the density, $$v = \sqrt{(Y/\text{density of the medium})}?$$ The form should be the same as for liquids and gases: $$v = \sqrt{(B/\text{density of the medium})}.$$

There is more than one speed of sound in a solid. In an isotropic solid the speed depend on the two Lame coeffcients $$\lambda$$ and $$\mu$$. The speed of longitudinal "p" waves in a bulk solid is $$c_{p}= \sqrt{\frac{E(1-\nu)}{\rho(1+\nu)(1-2\nu)}}$$ where $$E=\mu\left(\frac{3\lambda+2\mu}{\lambda+\mu}\right)$$ is Young's modulus and $$\nu=\frac 12 \frac{\lambda}{\lambda+\mu}$$ is Poisson's ratio. The speed of transverse "s" waves is $$c_s= \sqrt{\frac{\mu}{\rho}}.$$ The simple formula $$c_{rod}= \sqrt{\frac {E}{\rho}}$$ applies only to longitudinal waves in a long thin rod.
• Is the short answer that the Poisson's ratio of a fluid $\nu = 0$ which makes the first expression $c_p = c_{rod}$. Jan 22, 2021 at 1:06
• No. It is the shear modulus $\mu$ of a fluid that is zero. The fluid cannot sustain shear waves. Then Poisson's ratio is $\nu=1/2$ and Young's modulus is zero. This is because you can stretch a fluid with no effort as long as you allow it to shrink transversely so that its volume stays fixed. In a fluid it is the bulk modulus $\kappa$ that counts, and with $\mu=0$ we have $\kappa=\lambda$ and $c_{fluid}= \sqrt{\kappa/\rho}$. Jan 22, 2021 at 18:54