Elastic waves

Jonas Holtmann, summer semester 2016

“A wave […] is a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points.” ^[1]

Waves can be divided in two main categories. Firstly mechanical or elastic waves which propagate through a medium and secondly the electromagnetic wave which does not require a propagation medium and can therefore exist in vacuum. This article deals with the elastic/ mechanical waves and their use in non-destructive testing. For electromagnetic waves see Electromagnetic waves.

Definition

The principle of elastic waves is based on the restoring force on particles in matter. When a particle is moved from its original position (e.g. by an excitation) a restoring force is acting on it. This force can be calculated by the Hooke’s law and acts in the direction of the original direction. Because the particles in matter are linked to each other the displacement of one particle leads to a displacement of its surrounding particles. This leads to a propagation of energy through the medium ^[3].

Elastic waves can be divided into transversal (shear / S-) and longitudinal (compressional / P-) waves. Transversal waves consist of oscillations perpendicular to the direction of propagation. Longitudinal waves consist of oscillations parallel to the direction of propagation. ^[2] Figure 1 and 2 illustrates a transversal and longitudinal wave. Both types of elastic waves need a propagation medium.

Another way of categorising elastic waves is by body and surface waves (Figure 3).

Body waves propagate spherical symmetrical in a body while surface waves only propagate in 2D on the surface. In addition to transversal and longitudinal waves there are further types of surface waves, such as Rayleigh- and Love waves . Rayleigh waves are a superposition of transversal and longitudinal waves. They propagate by an elliptical particle movement and only exist up onto a depth of approximately one time the wavelength. ^[4] Love waves can only propagate in layered objects. It propagates in the form of a horizontally oscillating transversal wave.

In objects with small geometrical dimensions compared to the wavelength there exist Lamb- and torsional waves. Both waves result of a superposition of transversal and longitudinal waves. ^[5] The Lamb wave can occur in two different wave modes. In the symmetrical mode the corresponding particles on the surfaces move symmetrical to the horizontal plane through the middle of the plate. In the asymmetrical mode the corresponding particles on the surface move in the same direction. ^[4]

All presented waves can be used in NDT to determine properties of the test object.

Mathematical description

A plane wave can be fully described by the following wave equation:

$//<![CDATA[ \begin{array}{l}\frac {\partial^2 \xi}{\partial z^2} = \frac {1}{v^2} \frac{\partial^2 \xi}{\partial t^2}\end{array} //]]>$ (Equation 1)

With:

$//<![CDATA[ \begin{array}{l}\xi (z,t)\end{array} //]]>$: wave function

$//<![CDATA[ \begin{array}{l}z\end{array} //]]>$: direction of propagation

$//<![CDATA[ \begin{array}{l}v\end{array} //]]>$: phase velocity

$//<![CDATA[ \begin{array}{l}t\end{array} //]]>$: time

There are infinite solutions to this differential equation which all describe a wave. By setting specific boundary condition a concrete wave can be described. ^[3]

Important quantities of waves

Wave speed: The speed of the wave $//<![CDATA[ \begin{array}{l}v\end{array} //]]>$ depends on the material properties of the object it is travelling in. It links the frequency $//<![CDATA[ \begin{array}{l}f\end{array} //]]>$ with the wavelength $//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ by Equation 2.

$//<![CDATA[ \begin{array}{l}v = f * \lambda\end{array} //]]>$ (Equation 2) ^[7]

Wavelength: The distance between two neighbouring wave maxima (Figure 4). ^[7]

Frequency: The inverse of the time $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ for one complete cycle of an oscillating wave. The frequency and the wavelength are linked by the wave speed by Equation 2. ^[7]

Refraction: When a wave crosses the interface into another medium the direction of propagation changes depending on the wave speed of the medium (Figure 5). The angle of refraction can be calculated by the formula of Snellius: $//<![CDATA[ \begin{array}{l}\frac {sin (\alpha_1)}{sin (\alpha_2)} = \frac{c_1}{c_2}\end{array} //]]>$ (Equation 3)

Furthermore the mode of the wave can change at the interface e.g. a transversal wave can change into a longitudinal wave. ^[4]

Reflexion: The wave changes its direction in such a way that it returns into the medium it originated from. The angle of incidence equals the angle of reflection (Figure 5). ^[6]

Absorption: The wave interacts with matter in such a way that its energy is transformed into another energy form, e.g. into heat. ^[8]

Use in NDT applications

Ultrasonic inspection: Ultrasonic waves are transmitted into the test object to detected defects or to characterize the material properties. ^[4] The ultrasonic waves have a frequency greater than 20 kHz (Figure 6.1).

Impact echo: Similar to the ultrasonic inspection sound waves are transmitted or generated in the test object. In contrast to the ultrasonic inspection the impact echo testing uses waves in a broader frequency spectrum e.g. in the audible bandwidth (Figure 6.2). ^[9]

Acoustic emission analysis: The acoustic emission analysis uses sound waves that are generated by cracks or other activities in the tested material. By recording the sound waves with multiple sensors information about the location and size of the crack can be obtained (Figure 6.3). ^[10]

Local Acoustic Resonance Spectroscopy/Tapping test: Elastic waves are generated inside a test object by exciting the surface with an impact hammer. The waves propagate through the material and interact at possible interfaces e.g. cracks or flaws. By recording the wave acoustic with a microphone, information about cracks and flaws in the object can be obtained (Figure 6.4).

Literature

1. Freegarde, T.: Introduction to the physics of waves. 2013.
2. Hering et al.: Physik für Ingenieure. 2012.
3. Demtroeder, W.: Experimentalphysik 1. Mechanik und Wärme. 2015.
4. Schiebold, K.: Zerstörungsfreie Werkstoffprüfung - Sichtprüfung. 2015.
5. Große, C.U.: Einführung in die Zerstörungsfreie Prüfung im Ingenieurwesen. Grundlagen und Anwendungsbesipiele. 2015.
6. Schröder et al.: Technische Optik. Grundlagen und Anwendungen. 2014.
7. Tipler et al.: Physik für Wissenschaftler und Ingenieure. 2015.
8. Beyerer et al.: Automatische Sichtprüfung. Grundlagen, Methoden und Praxis der Bildgewinnung und Bildauswertung. 2012.
9. Alampalli, S.: Structural materials technology IV. 2000.
10. Große et al.: Acoustic Emission Testing. Basics for Research - Applications in Civil Engineering. 2008.
11. DGZFP-IE (2010): Merkblatt über die Anwendung des Impakt-Echo-Verfahrens zur zerstörungsfreien Prüfung von Betonbauteilen. DGZfP, 2011.