Introduction

Fundamental concepts of mechanical wave propagation in solid media. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Mechanical waves in solids are disturbances that propagate transporting energy without transporting matter. Unlike fluids, solids possess rigidity both in compression and in shear, allowing the simultaneous existence of two fundamental wave types: longitudinal and transverse. This duality is exclusive to solids and forms the basis of seismology, non-destructive testing and vibration engineering.

Within the block on waves in solids, this leaf introduces the basic concepts developed in depth in subsequent leafs: phase velocity, the relationship with elastic constants, wave types and the fundamental dispersion relation.

🟢 Essential level

A wave in a solid is a vibration that travels from one point to another in the material. When you strike a steel bar at one end, particles near the impact vibrate and push their neighbors, which in turn push the next ones. The disturbance travels along the bar without the material moving from one place to another: only the vibratory motion is transferred.

There are two ways in which a solid can vibrate as a wave passes:

• Longitudinal waves: particles vibrate in the same direction as the wave advances. They are analogous to sound in air and produce alternating compressions and expansions of the material.
• Transverse waves: particles vibrate perpendicular to the direction of advance. These are exclusive to solids because they require the material to resist shear stresses, something fluids cannot do.

The speed at which the wave travels, $v$, depends on two material properties: its stiffness (how difficult it is to deform) and its density $\rho$ (its inertia). A very stiff, low-density material transmits waves very quickly; a soft, dense material transmits them slowly. Steel, which is stiff and dense, has a longitudinal velocity of about 5900 m/s. Rubber, soft and relatively dense, has velocities of barely 50 m/s.

The wave frequency $f$ is determined by the source generating the vibration and does not change when passing from one medium to another. What does change is the wavelength $\lambda$, which adapts to maintain the fundamental relationship between velocity, frequency and wavelength.

🔵 Formal level

Basic formulas

Wave propagation in solids is governed by the equations of motion of continuum mechanics. For a linearly elastic, homogeneous and isotropic solid, the wave equation decomposes into two independent modes.

For longitudinal waves in a thin bar (transverse dimension much smaller than wavelength):

Longitudinal wave speed in a thin bar

Where $E$ is the Young's modulus of the material and $\rho$ is its density. This expression shows that velocity depends exclusively on intrinsic material properties and not on geometry or frequency (in the non-dispersive regime).

For transverse waves:

Transverse wave speed

Where $G$ is the shear modulus. Since in all solids the shear modulus is less than Young's modulus (typically $G$ is between one third and one half of $E$), transverse velocity is always less than longitudinal in the same material. This difference is what allows seismologists to distinguish between P waves (longitudinal, faster) and S waves (transverse, slower).

Real student questions

• Why is the speed of sound in solids greater than in gases? Because the rigidity of a solid far exceeds that of a gas. Although solids are also denser, the effect of rigidity dominates overwhelmingly.

• Can a wave change type (longitudinal to transverse) in a solid? Yes, at every oblique interface mode conversion occurs. An incident longitudinal wave generates both longitudinal and transverse reflected and transmitted waves.

• Why are there no transverse waves in fluids? Because fluids do not resist shear stresses: when you try to deform them tangentially, they simply flow rather than restoring their original shape.

Final synthesis

Wave propagation in solids is governed by the competition between elastic stiffness and material inertia. The relevant elastic modulus ($E$ for longitudinal, $G$ for transverse) and density $\rho$ completely determine phase velocity $v$. The fundamental dispersion relation connects velocity, frequency $f$ and wavelength $\lambda$. These concepts form the foundation upon which all wave phenomena in solids are built: reflection, transmission, impedance, dispersion and attenuation.