waves
For a string, the formula for wave speed is $v=\sqrt{\frac{T}{\mu}}$, where $\mu=\frac{m}{L}$. The greater the linear density, the more massive the string is per unit length, the more inertia it has, and the slower the wave propagates.
However, for a sound wave, wave speed is fastest in densest media. Why is that the case? Doesn’t inertia come into play as well, slowing down the wave?
Consider a short string of length $dx\;.$ We will use small-scale approximation so that the magnitude of tension $T$ is not changed by the transverse displacement of the string.
Potential energy can be calculated by finding how much the string becomes long when it is displaced transversely. say, it has a length of $ds$ when it is deformed. therefore the work done on the deformation is the required potential energy as $$\text{PE}= U= T(ds-dx)\;.$$ Now, \begin{align}ds&= \sqrt{(dx^2 + dy^2)}\\&= dx\sqrt{(1+(\delta_x y)^2)}\;.\end{align} Using Taylor series & neglecting higher-order terms of $\delta_x y^2$ (since, by small-scale approximation, $\delta_x y \ll 1$), we get $$ds - dx\approx \frac{1}{2} {(\delta_x y)}^2 \;dx. $$ Therefore, potential energy $$\text{PE}\approx \frac{1}{2} T {(\delta_x y)}^2\; dx\;.$$ Therefore, potential energy density \begin{align}\frac{dU}{dx}&\approx \frac{1}{2} T {(\delta_x y)}^2\\&= \frac{1}{2} \rho v^2 \left(\frac{\partial y}{\partial x}\right)^2\end{align} using the definition of velocity $v= \left(\frac{T}{\rho}\right)^{1/2}\;.$
Best Answer
For a solid, the speed of sound is given by
$$c = \sqrt\frac{B}{\rho}$$
where $B$ is the bulk modulus, and $\rho$ is the density.
So contrary to your assertion that "wave speed is fastest in denser media", the same type of relationship holds as for string: speed goes up if the force constant (restoring force for displacement) is greater, and it goes down as the inertia increases.
An example - speed of sound in diamond (density ~ 3.5 g/cc) is about 12000 m/s; in lead (density ~11.3 g/cc) it is about 1200 m/s. While lead is denser, the speed of sound is 1/10th of the value in diamond. This is of course because diamond has lower density, and much greater bulk modulus (large force for small deformation).