Waves at interfaces

At any interface (any abrupt change in acoustic impedance such as a boundary or defect), the following can occur [6, 71, 74]:

• Transmission through the boundary where refraction will occur;

• Reflection from the boundary with a change of amplitude and phase;

• Conversion of elastic energy at the boundary into a different bulk wave or a surface wave;

• Attenuation due to inhomogeneity at the boundary.

A smooth boundary causes reflection, whereas a rough boundary causes scattering [75]. If the boundary is a free surface, pure reflection will occur [71]. If the second medium cannot support the appropriate particle motion, such as at a solid/fluid boundary, the wave may totally reflect or may be partially mode-converted to a compression wave [74].

For plane progressive non-attenuating waves, if the material density isρand the wave speed in

that material isc, the acoustic impedance,z, is given by [3, 75]:

compression wave

compression wave

θ0 θ0

Figure 2.1: The reflected (θ1=θ0) and mode-converted wave (θ2) angles for an incident compres-

sion wave (θ0).

The effect of an attenuating medium (attenuation factor αat angular frequencyω) is to make

the impedance complex [3]:

z=ρc(1−jα/ωc) (2.12)

For a plane wave in a material of acoustic impedance, z1, incident on a perpendicular smooth

boundary with a material of acoustic impedancez2, the fraction of the amplitude reflected,R, and

transmitted,D, is [74, 75]: R= z2−z1 z2+z1 (2.13) D= 2z2 z2+z1 (2.14) Phase reversal (R <0) always occurs if reflected from a sonically softer material, defined as a

material with a lower acoustic impedance [75]. Air is so sonically soft that the metal-air boundary can effectively be considered a free boundary [3].

Characteristic of wave-boundary interactions for elastic waves in solids, is the occurrence of mode conversion. Although different wave modes are uncoupled in the bulk of the material, coupling occurs at boundaries through the boundary conditions, such that an incident wave is converted into two waves on reflection [71]. A plane compression wave, amplitude A0, incident

to a free boundary at angle θ0, will result in a reflected compression wave (amplitude A1, angle

θ1, speed cc) and a mode-converted shear wave (amplitude A2, angle θ2 < θ1, speed cs), with

polarisation perpendicular to the boundary (SV) [6]. For the wave not converted, the reflection is specular (angle of incidence equals the angle of reflection,θ1=θ0), and the mode-converted wave

follows Snell’s law (sinθ2/sinθ0 = c/c0) [6], which can be confirmed using the Huygens-Fresnel

principle. These angles are depicted in figure 2.1 fork=cc/cs= 2. There are critical angles past

which reflected waves can disappear [71].

The amplitudes are more complicated [6, 71, 75, 77, 78]:

A1=A0

sin(2θ0) sin(2θ2)−k2cos2(2θ2)

sin(2θ0) sin(2θ2) +k2cos2(2θ2)

(2.15)

A2=A0

2ksin(2θ0) cos(2θ2)

sin(2θ0) sin(2θ2) +k2cos2(2θ2)

(2.16) These formulae are plotted in figure 2.2a. It is also worth noting that in this case k=cc/cs

and this can be restated in terms of Poisson’s ratio ν such that k= 2(1−ν)/(1−2ν). It is then

(a) 0 10 20 30 40 50 60 70 80 90 −1 −0.5 0 0.5 1 1.5

Incident Angle (degrees)

Relative Amplitude (arbitrary)

(b) 0 5 10 15 20 25 30 −1 −0.5 0 0.5 1

Incident Angle (degrees)

Relative Amplitude (arbitrary)

Figure 2.2: The relative amplitude of a reflected (solid line) and mode-converted (dashed line) wave for an incident compression wave (a) and an incident shear wave (b).

wave amplitude can be greater than unity, but this does not violate conservation of energy, since energy transfer is a function of wave speed as well as amplitude, and shear waves are slower than compression waves [71].

For an incident shear (SV) wave (amplitude B0), a reflected shear wave (amplitudeB1, speed

cs), and a mode-converted compression wave (amplitudeB2, speedcc), the amplitudes are [6,71,78]:

B1=B0

sin(2θ0) sin(2θ2)−k2cos2(2θ0)

sin(2θ0) sin(2θ2) +k2cos2(2θ0)

(2.17)

B2=B0

−ksin(4θ0)

sin(2θ0) sin(2θ2) +k2cos2(2θ0)

(2.18) These formulae are plotted in figure 2.2b. The phase shift observed (inversion of the amplitude) is necessary for the direction of displacement to be continuous at the boundary. At 45°, an SV wave does not mode convert and is reflected as a pure SV wave. After the critical angle for an incident shear wave, the mode-converted compression wave becomes a surface-skimming compression wave [6].

A reflected compression wave has a constant phase shift for all angles, whereas a reflected shear wave has aπphase shift away from the critical angle, but a variable shift near it [6]. In addition to

the free boundary equations given here, there are also equations available for an interface between two liquids, the interface between liquids and solids, and the interface between two solids [75].

A shear wave with polarisation parallel to the boundary (SH) is reflected without mode con- version, and the amplitude does not change (there is the sameπphase shift however) [6,71,75,78].

When compression and SV waves are incident on a free boundary with very specific ratios, they can reflect as themselves [71]. However, this behaviour is not of particular interest and will not be considered further here, being more of use in the study of plates.

Diffraction occurs when a propagating wave encounters an interface that is not a reflecting plane, and is most pronounced if the wavelength is the same order of magnitude as the diffracting object. The Huygens-Fresnel principle states that each point on an advancing wavefront is the source of a new wavefront, and the summation of these wavelets gives the amplitude at any given point. Diffraction can be calculated via this principle; simple cases can be solved analytically using Fraunhofer diffraction for the far field and Fresnel diffraction for the near field, but more complicated cases must be solved numerically. Fraunhofer diffraction shows that the far-field diffraction pattern is the spatial Fourier transform of the aperture, as a consequence of the parallel-

the diffraction pattern in size and shape. Diffracted signals have less amplitude than reflected signals as the energy is spread out in all directions.