Why do solids support transverse waves but fluids generally do not?

Transverse waves require a restoring shear stress; solids resist shear and so transmit transverse (shear) waves, whereas ordinary fluids cannot sustain static shear and carry only longitudinal pressure (sound) waves.

What sets the speed of a mechanical wave?

It is set by the medium's stiffness relative to its inertia: roughly the square root of an elastic modulus or compressibility divided by the density, so stiffer or lighter media carry faster waves.

Waves in Continuous Media

Disturbances in elastic solids and fluids propagate as mechanical waves governed by the wave equation, including sound in fluids and longitudinal and transverse waves in solids.

Definition

Waves in continuous media are propagating disturbances of a fluid or elastic solid, governed by the wave equation that follows from applying the continuum equations of motion to small deformations, with a propagation speed set by the medium's stiffness and density.

Scope

This topic covers mechanical wave propagation in continuous media: the derivation of the wave equation from continuum dynamics, the speed of sound in fluids and the longitudinal and transverse elastic waves in solids, dispersion relations, energy and momentum transport by waves, and basic wave phenomena such as reflection and the Doppler effect. It connects continuum mechanics to acoustics and seismology.

Core questions

How does the wave equation arise from the dynamics of a continuous medium?
What determines the speed of sound in fluids and elastic waves in solids?
How do longitudinal and transverse waves differ in solids and fluids?

Key concepts

Wave equation
Phase and group velocity
Longitudinal and transverse waves
Speed of sound
Dispersion relation
Reflection and the Doppler effect

Key theories

Wave equation for elastic and acoustic waves: Small disturbances in an elastic or fluid medium obey the wave equation, with propagation speed determined by the ratio of an elastic modulus or compressibility to the medium's density.
Sound waves in fluids: Acoustic waves are small adiabatic compressions and rarefactions whose speed is set by the fluid's compressibility and density, propagating as longitudinal pressure disturbances.

Clinical relevance

The theory of mechanical waves underlies acoustics and noise control, ultrasound and nondestructive testing, the seismology used to study earthquakes and the Earth's interior, and sonar and underwater acoustics, anywhere disturbances travel through solids or fluids.

History

Newton first estimated the speed of sound from the elasticity of air, and d'Alembert derived the one-dimensional wave equation for the vibrating string in 1747. The full theory of elastic and acoustic waves was developed in the nineteenth century, culminating in Lord Rayleigh's comprehensive Theory of Sound and the analysis of surface waves that bear his name.

Key figures

Jean le Rond d'Alembert
Isaac Newton
Lord Rayleigh

Seminal works

french1971
landaufluid1987

Frequently asked questions

Why do solids support transverse waves but fluids generally do not?: Transverse waves require a restoring shear stress; solids resist shear and so transmit transverse (shear) waves, whereas ordinary fluids cannot sustain static shear and carry only longitudinal pressure (sound) waves.
What sets the speed of a mechanical wave?: It is set by the medium's stiffness relative to its inertia: roughly the square root of an elastic modulus or compressibility divided by the density, so stiffer or lighter media carry faster waves.