Mechanical disturbances do not propagate through a solid instantaneously. When one part of an elastic material is displaced, neighboring material responds through elastic forces, and the disturbance travels through the material as a wave.

For small longitudinal disturbances in an ideal slender elastic rod, the propagation speed is

$$\boxed{c=\sqrt{\frac{E}{\rho}}},$$

where

• $E$ is Young's modulus, which measures the material's resistance to elastic stretching or compression;
• $\rho$ is mass density, which measures the inertia that must be accelerated as the disturbance passes.

The formula expresses a general competition between restoring stiffness and inertia.

Why greater stiffness makes waves faster

If two materials have the same density but one has a larger Young's modulus, the stiffer material develops a larger restoring stress for the same strain. Neighboring regions are therefore accelerated more strongly, so the disturbance propagates faster.

Because

$$c\propto\sqrt E,$$

multiplying $E$ by four doubles the wave speed when density is unchanged.

Why greater density makes waves slower

If two materials have the same stiffness but one has a larger density, more mass must be accelerated for a comparable local deformation. The disturbance therefore propagates more slowly.

Because

$$c\propto\frac1{\sqrt\rho},$$

multiplying the density by four halves the wave speed when $E$ is unchanged.

Dimensional check

Young's modulus has units of pressure:

$$[E]=\mathrm{Pa}=\frac{\mathrm{kg}}{\mathrm{m,s^2}}.$$

Density has units

$$[\rho]=\frac{\mathrm{kg}}{\mathrm{m^3}}.$$

Therefore

$$\left[\frac{E}{\rho}\right] =\frac{\mathrm{kg/(m,s^2)}}{\mathrm{kg/m^3}} =\frac{\mathrm{m^2}}{\mathrm{s^2}},$$

so

$$\left[\sqrt{\frac{E}{\rho}}\right]=\mathrm{m/s},$$

as required for a speed.

Worked example

For a steel-like material, take

$$E=200\times10^9,\mathrm{Pa},$$

$$\rho=7850,\mathrm{kg/m^3}.$$

Then

$$c=\sqrt{\frac{200\times10^9}{7850}} \approx5.05\times10^3,\mathrm{m/s}.$$

Thus

$$\boxed{c\approx5.1,\mathrm{km/s}}.$$

If a rod made from this material is $2.0,\mathrm m$ long, the earliest a disturbance introduced at one end can reach the other in this model is approximately

$$t=\frac{L}{c} =\frac{2.0}{5.05\times10^3} \approx4.0\times10^{-4},\mathrm s.$$

So the opposite end begins responding only after about

$$\boxed{0.40,\mathrm{ms}}.$$

Scope of the relation

The expression

$$c=\sqrt{E/\rho}$$

is the ideal result for small longitudinal waves in a slender rod with linear elastic behavior. Bulk waves in three-dimensional solids can depend on other elastic properties, including shear modulus and Poisson's ratio, and anisotropic materials can have direction-dependent wave speeds.

The broader physical principle remains: mechanical wave speed is set by how strongly the medium restores a disturbance compared with how much inertia must respond.