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Fundamentals of Acoustics: Wave Propagation

By Jeff Dobson 20 January 2021

In this Fundamentals of Acoustics blog, we will investigate some of the basic principles of how sound waves propagate within a material. 

Wave Propagation Modes

Acoustic waves can propagate in different modes which are characterized by the particle vibrations. The most common wave modes are described below:

Longitudinal Waves

The oscillation direction of particles within a longitudinal wave is the same as the direction of the wave propagation. This wave mode can also be referred to as compressional waves due to the compressional and dilation forces active on the wave. Longitudinal waves can be generated in solids, liquids and gases. 

Shear Waves

The particles within a shear wave oscillate perpendicularly to the wave direction of motion. These waves are also referred to as transverse waves due to the particle’s motion being transverse to wave direction. Shear waves require an acoustically solid material for propagation and so are not supported within liquids or gases.

Surface (or Rayleigh) Waves

Surface waves travel along the surface of a relatively (with respect to wavelength) thick solid material. These waves penetrate to a depth of one wavelength and combines both longitudinal and transverse particle motion to create an elliptic orbit motion. 

Plate Waves

Plate waves are like surface waves except can only be generated in thin solid materials which are a few wavelengths thick. Plate waves propagate parallel to the surface and through the full material thickness. Lamb waves is a type of plate waves and the particle motion of this wave is complex and can exist in several different modes. Common modes are symmetrical and asymmetrical which is determined by the particle motion either side of the median plane of the material.


As sound travels through a medium, its amplitude reduces with distance due to attenuation. This attenuation is made up of scattering and absorption of the sound energy within a material. Attenuation is dependent upon frequency with higher frequencies having a higher attenuation amplitude due to the smaller wavelength. 

Reflection and Transmission

When an acoustic wave encounters a boundary between different acoustic impedance materials, the wave will be reflected. The difference in acoustic impedance (impedance mismatch) will determine the percentage of sound energy which will be reflected. This percentage can be calculated using the below equation where Z1 and Z2 are the acoustic impedances of the materials either side of the boundary.

R= (Z2-Z1Z2+Z1 )² x 100%


As acoustic waves travel through a material interface at an oblique angle, the wave will be reflected and refracted if the materials have different refraction indices. This is the same behavior that can be observed with light. 

The reflected and refracted wave propagation angles can be calculated using Snell’s Law. Snell’s Law describes the relationship between the wave velocities (VL) and propagation angles (θ), as shown in the diagram and equation below.

Wave Propagation

sin sinØ1VL1  = sin sinØ2VL2 

This blog has provided an overview of the fundamentals of wave propagation for acoustic waves. Next in this series is a look into acoustic energy!

Jeff Dobson
Jeff Dobson

Jeff Dobson is a senior application engineer at OnScale and has recently completed an Engineering Doctorate at the University of Strathclyde in Glasgow, focused on NDT simulation. Jeff is an experienced FEA engineer with an expertise in advanced NDT simulations, such as modeling ultrasonic inspections of composite components. Jeff has over 7 years experience with OnScale and supports activity across current and future applications, whilst also managing consultancy projects and providing technical support.