Making Waves

Michael Lotinga - April 2016


Recently-observed physical evidence for the existence of gravitational waves[1], and the development of a new class of sound wave[2], got me thinking about the many different types of waves encountered in acoustics, the history of mechanical wave discovery and the very nature of what we mean when we talk about a ‘wave’…

A wave in a very general sense might be fairly described as any collective bulk disturbance in which the change at one position is a delayed response to the change at adjacent points[3]. This broad definition could include all sorts of phenomena such as electromagnetism, gravity, oceanic movement, sounds, and possibly even abstractions such as ideas or emotions.

In acoustics we focus on a relatively narrow range of mechanical waves, but that is not to say this array isn’t varied and very intriguing! Below is a very brief potted historical timeline[4] mapping some of the main physical developments in mechanical waves:

6 BC Pythagoras investigates the origins of musical sounds and the nature of string vibrations.
1500s Vincenzo Galilei studies musical string vibrations, producing perhaps the first mathematical description of a non-linear natural phenomenon.
1630s-1640s Mersenne and Galileo Galilei investigate (independently), string vibrations, pendulums and resonances.
1678 Hooke discovers the law that bears his name, establishing the foundation for elasticity theory.
1686 Newton investigates the speed of oceanic waves and of sound in air. Publishes Principia, one of the most important scientific works in human history.
1713 Taylor completes a dynamic solution for string vibration.
1744-1751 Euler and Daniel Benoulli develop the equations for vibrations in beams, and calculate the normal modes for a range of boundary conditions.
1747 D’Alambert derives and solves the 1-dimensional wave equation for string vibration.
1787 Chladni publishes experimental results on modal plate vibrations[5].
1815 Sophie Germain develops the elasticity theory for plate vibrations, winning the Paris Academy of Sciences prize (a 1kg gold medal).
1821 Navier formulates a general theory of elasticity in solids.
1828 Poisson shows that two fundamental types of wave propagate within an elastic solid, longitudinal (pressure) and transverse (shear) waves.
1848 Stokes expounds the elastic wave descriptions, identifying specific characteristics of each wave type.
1887-1888 Strutt (Lord Rayleigh[6]) investigates surface wave propagation in solids, including the discovery of the wave type that bears his title.
1889-1917 Lamb develops plate vibration equations, investigates seismic tremor and pulse propagation and progresses the surface wave work developed by Rayleigh, also resulting in the surface wave class bearing his name (Rayleigh waves travelling in layers).
1911 Love discovers a new type of surface shear wave propagating in layered solids, explaining some seismographic anomalies.
1921 Timoshenko develops his beam vibration theory.
1924 Stoneley discovers the seismic wave bearing his name, a type of ‘leaky’ Rayleigh wave that travels along material interfaces.
1927 Sezawa identifies another surface wave type, which also bears his name.
1951 Mindlin develops his plate vibration theory.
1956 Biot shows that two sub-types of longitudinal wave propagate in fluid-loaded poroelastic media[7].
1968 Bleustein and Gulyaev predict the existence of a new surface wave in piezoelectrics, which becomes known as the Bleustein-Gulyaev wave.

The physical differences in each of the main wave types are best explained by the brilliant animations at:

2D waves:

3D P-wave:

3D S-wave:

3D Rayleigh wave:

3D Love wave:

●      P-waves[8] (Primary waves, from seismological terminology) travel fastest, with the same particle motion as airborne sound. Biot showed that two classes of P-wave, a slower and a faster version, travel in a fluid-saturated poroelastic medium.

●      S-waves[9] (Secondary waves, arriving after the P-wave) are next in the wave race, with a velocity slightly less than P-waves, and particle motion perpendicular to the propagation.

●      P-waves and S-waves form the ‘body’ or ‘bulk’ wave types that travel within the medium. Surface acoustic waves (SAWs) are combinations of longitudinal and shear wave motion, with energy ‘trapped’ near the surface.

●      In R-waves (Rayleigh waves) the particle motion is anti-clockwise elliptical relative to the propagation. R-waves travel more slowly than bulk waves, but can transmit plenty of energy; in seismology R-waves are named ‘ground roll’ and convey the most destructive component of an earthquake event, arriving a considerable time after the event is detected (from faster bulk waves).

●      Love waves are another surface wave with horizontally polarised particle motion.

●      At material boundaries waves are partially reflected, and in certain cases wave type conversions occur.

In ultrasonics, SAWs with nanometre-scale wavelengths find applications in non-destructive testing and medicine, where they can be used to manipulate tiny structures such as cells.

The new ‘surface-reflected bulk wave’ type has been used to develop a stem cell nebulisation technique aimed initially at treating lung disease. The nebulisation process is playfully described by one of the researchers as “yelling at the liquid, breaking in into vapour”. Nebulisation using piezoelectric ultrasonics in itself is not a new idea, as both medical historians and luxury domestic appliance fans will know. But the new wave allows a much higher rate of liquid transfer to be achieved than was previously possible using conventional SAWs.

What does all this have to do with gravitational waves of course? Well, not a great deal at first glance; gravitational wavelengths range from 1000s to many trillions of metres after all. But this conversation in the IOA LinkedIn group[10] raised an interesting point: the LIGO interferometers use lasers and optics to detect the tiniest distortions in space-time; to ensure there is no seismic interference this requires considerable vibration isolation of the equipment. Consequently some of the LIGO research team members are experts in this area[11].

It seems that acoustical engineering and research have promising futures at every scale of physical science!

Any comments or queries on this article can be added below (logged in members only), or emailed to:


[1] Detected at the LIGO Observatories (USA) on 11th February 2016
[3] Paraphrased from archived University of Southampton physics lecture notes
[4] The list is not intended to be exhaustive, and has been mainly compiled from the following sources:  Graff,K.F., 1991. Wave motion in elastic solids. 2nd ed (reprint). Dover Publications: New York; Ansalmet,F. and Mattei,P.-O., 2016. Acoustics, aeroacoustics and vibrations. ISTE: London; Lotinga,M.J., 2014, Investigating the accuracy of a semi-empirical model for rail-induced ground/structure-borne noise and vibration. University of Salford, MSc Dissertation; Wikipedia.
[5] Chladni’s seminal 1809 publication, Traité d’Acoustique, has recently been treated to its first full English translation
[6] Both volumes of Rayleigh’s classic Theory of Sound are freely available open-access at (Vol I: Vol II:
[7] If you are a regular JASA reader you will probably notice that both parts of Biot’s publication on this subject consistently rank on the most-cited papers list every month.
[8] Also: dilatational, irrotational, longitudinal, volume or pressure waves
[9] Also: distortional, equivoluminal, shear or rotational waves
[10] You will need to be registered, logged in and a member of the group to access the link.


Update: Rayleigh waves in space?

Having watched Rupert Thornely-Taylor's excellent Rayleigh Medal lecture ( ), I must concede that it is not Rayleigh waves that cause ground roll on planet Earth exactly, but instead a similar wave identified by Scholte that propagates at the surface of a fluid-solid interface, rather than the vacuum-solid interface required for a 'pure' Rayleigh wave ( ). Thanks Rupert!