Body and surface waves

Jascha Roedel, winter semester 2015/16

The term wave describes the spatial propagation of an oscillation or disturbance through a medium such as solid bodies, liquids or gases. Thereby, waves do appear in various forms and differ in their characteristics. A general differentiation between types of elastic waves can be made by classifying those into two basic types of elastic wave motion: body and surface waves.

Body Waves

Body waves occur in the homogeneous half space. In case of a homogeneous and isotropic matter their propagation is spherical-symmetrical. There exist two main types of body waves regarding aspects of velocity and the oscillation of particles caused by a propagating wave, which can be further differentiated between longitudinal and transverse waves. The intensity of the wave decreases with increasing distance due to the growth of the spherical wave front, which is referred to as ‘geometric spreading’. The amplitude decreases with:

$//<![CDATA[ \begin{array}{l}\frac{1}{r}\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}r\end{array} //]]>$ = distance to wave source [m]

In general, the correlation between wavelength, the wave’s frequency and its velocity is important:

$//<![CDATA[ \begin{array}{l}f = \frac{\nu}{\lambda}\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}f\end{array} //]]>$ = frequency [Hz]

$//<![CDATA[ \begin{array}{l}\nu\end{array} //]]>$ = propagation velocity [m/s]

$//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ = wavelength [m]

Longitudinal Wave

Characteristics

The oscillation or motion of disturbance within the longitudinal wave follows its direction of propagation. The longitudinal wave transmits tensile and compressive stress. Therefore, it is also called compression wave. Furthermore, it is named P-wave (Primary wave) and dilatational wave. ^[2] The term Primary wave implies that the longitudinal wave has the highest velocity of all waves and is always registered first. ^[3] The compression wave leads to a change in volume and form of the particles in a matter, hence, the expression dilatational wave. ^[4]

Propagation

The propagation velocity of the wave depends on several factors for example the elastic parameters of the medium. The most influential factors are density and Young’s modulus. The restoring force of the shifted particle determines the propagation velocity of the wave. There exist several solutions to describe the propagation velocity of the P-wave. By assuming the matter to be homogeneous, isotropic, and indefinitely expanded, the propagation velocity can be determined as:

$//<![CDATA[ \begin{array}{l}\nu_p = \sqrt{\frac{E (1 - \sigma)}{\rho(1 + \sigma)(1 - 2 \sigma;)}}\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}\nu_p\end{array} //]]>$ = propagation velocity of the P-wave [m/s]

$//<![CDATA[ \begin{array}{l}E\end{array} //]]>$ = Young's modulus [N/m²]

$//<![CDATA[ \begin{array}{l}\sigma\end{array} //]]>$ = Poisson's ratio

$//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ = density [kg/m³]

The following table illustrates some examples for the propagation velocity of longitudinal waves in different mediums: ^[1]

Transverse Wave

Characteristics

The displacement of particles due to a transverse wave is perpendicular to the direction of propagation and the wave transmits shear stress. Therefore, it can only be found in matters that are able to carry shear forces. It does not occur in liquids or gases. ^[2] In contrast to P-waves the transverse wave deformation does not lead to a change in volume or density. ^[4] Other names that refer to transverse waves are shear wave, S-wave (Secondary wave), rotational wave, distortional wave or equivoluminal wave. ^[2] S-waves can be further classified into horizontal polarized shear waves (SH) and vertical polarizing shear waves (SV). Both of these terms describe the same wave type but a differentiation is useful in order to evaluate the unequal characteristics during reflection of the wave. ^[3]

Propagation

For most matters the speed of the S-wave is always smaller than the speed of the P-wave. The propagation velocity of shear waves in the homogeneous, isotropic and indefinitely expanded matter is determined as:

$//<![CDATA[ \begin{array}{l}\nu_s = \sqrt{\frac{E}{2 \rho (1 + \sigma)}}\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}\nu_s\end{array} //]]>$ = propagation velocity of the S-wave [m/s]

$//<![CDATA[ \begin{array}{l}E\end{array} //]]>$ = Young's modulus [N/m²]

$//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ = density [kg/m³]

$//<![CDATA[ \begin{array}{l}\sigma\end{array} //]]>$ = Poisson's ratio

For most elastic matters the following coherence can be assumed:

$//<![CDATA[ \begin{array}{l}\nu_s \asymp \frac{\nu_p}{\sqrt{3}}\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}\nu_s\end{array} //]]>$ = propagation velocity of the S-wave [m/s]

$//<![CDATA[ \begin{array}{l}\nu_p\end{array} //]]>$ = propagation velocity of the P-wave [m/s]

Reflection of Body Waves

Layer boundaries, surface boundaries, and inhomogeneity in elastic mediums are places where reflection and refraction processes occur. These areas cause a change in material properties. ^[4] When falling upon such surfaces, a part of the wave energy is reflected back into the first medium and another part is transmitted or refracted into the second medium. ^[5] While spreading and propagating through a medium, the waves are reflected and scattered. Due to those processes, a conversion of longitudinal waves to transverse waves (and vice versa) takes place. ^[6] That is why in elastic bodies P-waves and S-waves always appear coupled. The transition from one wave form into another wave form is called mode splitting. The amount of reflected wave parts depends on the impedance which describes the wave resistance:

$//<![CDATA[ \begin{array}{l}Z = \rho * \nu\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}Z\end{array} //]]>$ = impedance [kg/m²s]

$//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ = density [kg/m³]

$//<![CDATA[ \begin{array}{l}\nu\end{array} //]]>$ = propagation velocity of the wave [m/s]

Factors like the incidence angel, impedance characteristics, and Poissons ratio of the material, affect the amplitude of the reflected wave. In order to determine the ratio of amplitudes of the reflected wave and the inclined wave, the complex reflection coefficient needs to be calculated. In the case of normal incidence on a material interface, it is given by:

$//<![CDATA[ \begin{array}{l}R_{1,2} = \frac{Z_2 - Z_1}{Z_2 + Z_1} = \frac{\rho_2 \nu_2 - \rho_1 \nu_1}{\rho_2 \nu_2 + \rho_1 \nu;_1}\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}Z(1,2)\end{array} //]]>$ = impedance of the medium [kg/m²s]

$//<![CDATA[ \begin{array}{l}\rho(1,2)\end{array} //]]>$ = density of the medium [kg/m³]

$//<![CDATA[ \begin{array}{l}\nu(1,2)\end{array} //]]>$ = propagation velocity of the wave in the medium [m/s]

The amount of reflected energy is determined as $//<![CDATA[ \begin{array}{l}R^2\end{array} //]]>$. ^[1]

Surface Waves

Surface waves are created on free surfaces in the half space as well as on boundary layers in the layered half space.^[2] Whenever a body wave undergoes reflection and refraction, new wave types can be generated. The result of these processes is a complicated wave pattern consisting of a mixture of longitudinal and transverse waves. Surface waves can be understood as a superposition of longitudinal waves and transverse waves, hence, surface waves are rotational and dilatational. ^[1]

The propagation of surface waves is parallel to the surface of a medium or the interface between differing matters. The wave energy is mainly transported on the surface. Surface waves spread in only two dimensions. The velocity of propagation is again determined by the medium’s density and elasticity. In comparison to body waves, they fall off more slowly with increasing distance from the source. Since the wave front of the surface wave is cylindrical and not spherical, the effects of geometric spreading have a lower impact. A decay can be described with the following. The amplitudes decrease with:

$//<![CDATA[ \begin{array}{l}\frac{1}{\sqrt{r}}\end{array} //]]>$ ^[4]

$//<![CDATA[ \begin{array}{l}r\end{array} //]]>$ = distance to wave source [m]

There exist many types and combinations of surface waves. The most important ones regarding science fields like seismology, engineering and non-destructive testing shall be named in the following.

Rayleigh Wave

Characteristics and Propagation

The Rayleigh wave forms on the surface of a homogeneous half space as a result of the superposition of a P-wave and a S-wave. ^[6] The displacement of particles caused by the Rayleigh Wave follows an upright ellipse. ^[4] The motion of disturbance is in the plane perpendicular to the free surface as well as parallel to the direction of propagation, meaning it has vertical and horizontal components. ^[2] The characteristics of Rayleigh Waves are complex and determined by several factors such as the relative amplitude, the direction of propagation, the plane of polarization and the phase relationship between the waves themselves. It is also affected by the ratio between the longitudinal and transverse wave speed. One of the striking characteristics of the Rayleigh Wave is its decreasing amplitude with increasing depth away from the surface. ^[7] The penetration depth is assumed to be similar to the wavelength. ^[8]

Rayleigh wave propagation is along the free surface. The velocity of surface Rayleigh waves is given by:

$//<![CDATA[ \begin{array}{l}\nu_r = \frac{0.87 + 1.12 \sigma}{1 + \sigma}\sqrt{\frac{E}{2 \rho(1 + \sigma)}}\end{array} //]]>$ ^[8]

$//<![CDATA[ \begin{array}{l}\nu_r\end{array} //]]>$ = propagation velocity of the Rayleigh wave [m/s]

$//<![CDATA[ \begin{array}{l}\sigma\end{array} //]]>$ = Poisson's ratio

$//<![CDATA[ \begin{array}{l}E\end{array} //]]>$ = Young's modulus [N/m²]

$//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ = density [kg/m³]

Furthermore, the following correlation exists:

$//<![CDATA[ \begin{array}{l}\nu_r = 0.9194 * \nu_s\end{array} //]]>$ ^[1]

$//<![CDATA[ \begin{array}{l}\nu_r\end{array} //]]>$ = propagation velocity of the Rayleigh wave

$//<![CDATA[ \begin{array}{l}\nu_s\end{array} //]]>$ = propagation velocity of the S-wave

Love Wave

Characteristics and Propagation

The Love wave only appears in layered matters and does not occur in the homogeneous half space. This type of wave can be described as a horizontal polarizing shear wave that emerges from interference and interreflection within layer boundaries and spreads inside the upper layer. The Love wave only occurs if the upper layer features a lower velocity than the layer below. Its velocity of propagation is lower than those of P-waves and S-waves but the Love wave propagates faster compared to the Rayleigh Wave. ^[4]

Lamb Wave

Characteristics and Propagation

Lamb waves are bending waves on a plate. They define a special type of wave motion that propagate inside a plate if the thickness of the plate is small in comparison to the wave length. It does not appear in the homogeneous half space. The Lamb wave transmits shear forces as well as bending moments. ^[6] The propagation velocity depends on the wavelength. Lamb waves are complex; not only do they occur in various forms, they also occur in either symmetrical or asymmetrical wave modes. Furthermore, the phase velocity of the lamb wave and its group velocity are not equal. All in all, lamb waves are a complicated wave form and their behavior depends on several factors. Thereby, one of the most influential is Poisson's ratio. ^[9]

Elastic Waves in Non-Destructive Testing

Wave Generation and Application

Elastic wave related techniques are able to provide a tool for non-destructive testing, as damage conditions in structures and materials influence the mechanical properties and thus the wave parameters. The examination of elastic wave propagation through a material generates useful information about the condition and characteristics of a structure. The most used waves in non-destructive testing are longitudinal body waves and the surface Rayleigh wave. ^[8]

There exist several methods to generate elastic waves for the use in non-destructive testing of materials. Transducers based on piezoelectric crystals are used to create ultra sonic impulses. Another way to create elastic waves is the production of mechanical shock. For example, a mechanical shock is generated by the impact of a ball or by hammer strokes. ^[1]

Some of the most common applied techniques in non-destructive testing shall be listed in the following: ^[1]

• Ultrasonic echo method
• Ultrasonic transmission method
• Impact echo method
• Acoustic emission analysis

Literature

1. Grosse, C. U.: Einführung in die Zerstörungsfreie Prüfung im Ingenieurwesen, Grundlagen und Anwendungsbeispiele. Munich: Lehrstuhl für Zerstörungsfreie Prüfung der Technischen Universität München, pp. 38-90. Munich, 2015.
2. Engelbrecht, J.: Questions About Elastic Waves. Berlin, Heidelberg: Springer, pp. 9-10. Berlin, 2015.
3. Studer, J. A.; Laue, J. and Koller, M. G.: Bodendynamik. Berlin, Heidelberg: Springer pp. 29-50. Berlin, 2007.
4. Haupt, W. ed.: Bodendynamik Grundlagen und Anwendungen. Braunschweig: Vieweg, pp. 54-109. Braunschweig, 1986.
5. Udias, A.: Principles of Seismology. Cambridge: Cambridge University Press, pp. 63-74. Cambridge, 1999.
6. Ingard, K. U.: Fundamentals of Waves and Oscillations. Cambridge: Cambridge University Press, pp. 523-559. Cambridge, 1988.
7. Kolsky, H.: Stress Waves in Solids. New York: Dover Publications, pp. 4-24. New York, 1963.
8. Aggelis, D.G.; Chai, H.K. and Shiotani, T.: Elastic Waves on Large Concrete Surfaces for Assessment of Deterioration and Repair Efficiency. In: Omar, M. ed.: Nondestructive Testing Methods and New Applications. Rijeka: InTech Europe, pp. 189-219. 2012.
9. Gibson, A. and Popovics, J.: Lamb Wave Basis for Impact-Echo Method Analysis. In: Journal of Engineering Mechanics 131(4), pp. 438-443. 2005.