For example, in solids, the velocities of transverse and the longitudinal waves depend on the shear modulus and compressive modulus, respectively, and shear modulus is less than compression modulus (the velocities also depend on the density, which is the same for both types of waves).
No, it is not true that sound travels faster in denser media. In fact it travels slower.
In the adiabatic approximation we assume that the portions of the gas vibrate so fast that it is not able to exchange heat with the surroundings. The the longitudinal displacements can be shown to satisfy the wave equation
$$\frac{\partial^2 u(x,t)}{\partial t^2}=\frac{\gamma P_0}{\rho_0}\frac{\partial^2 u(x,t)}{\partial x^2}.$$
As you can read, the velocity of this wave is
$$v=\sqrt{\frac{\gamma P_0}{\rho_0}},$$
where $\gamma$ is the adiabatic constant, $P_0$ is the gas pressure and $\rho_0$ is the gas density.
If you speak right after inhaling Helium you will sound at higher pitch. The reason is that the speed of sound in Helium is higher than in air. Air is denser than Helium.
For liquids the speed of sound is given by
$$v=\sqrt{\frac{B}{\rho_0}},$$
where $B$ is the volumetric elasticity, defined as
$$B=-\frac{\Delta P}{\Delta V/V},$$
i.e. it gives the relative change in volume when we change pressure. Again we see the speed of sound decreases when we increase density.
Best Answer
Imagine a metal rod and you hit one end of it with a hammer.
A compression pulse travels down the rod and back again after reflection at the other end and the cycle of reflections is repeated - this is a longitudinal wave motion.
However as the pulse travels down the rod I would imagine that the walls of rod bulge out and then return, so this is equivalent to a transverse wave.
That bulging out then compresses and rarefies the air in the vicinity of the rod and so you get longitudinal waves in air produced.