An Investigation Of Surface Acoustic Waves

An Investigation of Surface Acoustic Waves By David.R.Gilson

Abstract This experiment investigates the surface ultrasonic properties of five different materials, primarily Aluminium. Using theory, it is determined which mode of bulk acoustic wave (longitudinal or shear) gives rise the surface (Rayleigh) acoustic waves. For aluminium, the wavelength of the surface acoustic wave (SAW) is determined by experiment. From this and the measured SAW velocity, the frequency of the SAW is calculated.

Introduction Surface Acoustic Waves (SAWs) have applications in a wide range of technologies. SAWs are used can be used to detect flaws, faults and breaks surfaces of different materials. This can be useful detecting faults in materials before more serious faults occur. SAWs are also applied in televisions. SAWs are used as a signal filter (in the form of a quartz crystal) in the intermediate frequency amplifier, which prepares the RF signal for demodulation into baseband video and audio. When an oscillatory force is applied to the surface of a solid, mechanical waves arise. These are known as acoustic waves. Such bulk waves occur in two modes, longitudinal and shear waves. Longitudinal waves are where the axis of vibration of the particles is parallel to the direction of propagation of the wave front. This causes alternating compressions within the solid. In shear waves, the axis of vibration is perpendicular to the direction of propagation of the wave. With this, the bonding between the particles within the solid experiences shearing forces. Acoustic waves extend to the surface of the solid as well as throughout the bulk. Both modes of oscillation can occur in the surface of the solid. Of course, surface shear waves are polarised in the plane of the surface because that is the only axis of vibration available (at surface depth). Another surface wave which can exist is called the Rayleigh wave. This wave takes the form of an elliptical wave. The Rayleigh wave is a mixture of shear and longitudinal waves. It is sustained from reflections of acoustic waves on and beneath the surface. This system is analogous to fibre optics, where light is guided along a glass fibre by reflections with an angle less than the critical angle of the material. In general the system which arises when an acoustic wave enters a solid is shown in figure 1.

Figure One. In figure 1 the incident wave is longitudinal (as in the ensuing experiment). As would be expected, there is a reflected and refracted (transmitted) longitudinal wave. Also, through some mode conversion process, reflected and refracted shear waves exist too. It is the refracted shear and longitudinal waves that generate the Rayleigh wave at the surface. The velocity of the Rayleigh wave can be represented by the theoretical expression,

 { 2

  }

CR C 6R 8C 4R C 2S 16 2 24 ⋅ 6 − 4 C R 2 − 2 −16 1− 2 =0 CS CS C S CS C L CL



(1)

Where CR is the Rayleigh wave velocity, CL is the longitudinal wave velocity and CS is the shear wave velocity. Usually, depending on the type of material, shear waves or longitudinal waves are favoured at the surface. It can be determined which wave mode, is the major constituent of the Rayleigh wave using Snell's law. If the angle of incidence and the speed in both materials are known, then the angle of refraction can be calculated. Since the Rayleigh wave is a surface phenomena and does not penetrate far into the material, it can be assumed that the wave mode with the greatest refraction angle (i.e. the wave closest to the surface) is the mode that gives rise to the Rayleigh wave. Hence, the refraction angle can be calculated thus,

ArcSin





C 1 sin i =r C2

(2)

(where r is the angle of refraction.) The Rayleigh wave does not penetrate far into the material (because of it's very nature). The energy of the Rayleigh wave falls exponentially along the depth of the material. It is assumed for a SAW that the wave energy drops to 1/e of the surface value at a depth of one wavelength.

Figure Two. In fact, from this statement the following expression can be conjectured,

E=E 0 e

−

x 

(3)

Where E is the amplitude (E0 being the amplitude at zero depth). This provides a basis for experimental determination of the Rayleigh wavelength. First of all, if we take equation 3 and square it we can see how a linearly varying distance, x, varies with the square of the amplitude. This gives,

E 2= E 20 e

−2 x 

(4)

2

If we then divide throughout by E 0 , this gives normalised values for the amplitude. This is useful when the amplitude of the wave is measured in the form of a voltage from a Cathode Ray Oscilloscope (CRO). Equation 4 can be linearised by taking logarithms of both sides, so that statistical (least square fit, (LSF)) methods can be applied.

E 2 −2 x =e E 20

(5)

2 ln

 

(6)

ln

 

(7)

E −2 x =  E0 E −x =  E0

If we rewrite equation 7, the linear form can easily be seen,

ln  E=ln E 0 − Where the negative of the reciprocal wavelength is the gradient.

x 

(8)

Experimental procedure. A pair of ultrasonic wedge transducers were used in this experiment to create an acoustic source and to detect acoustic signals. These devices are composed of a ceramic element that oscillates when a current is passed through it and also causes a current to flow when it is oscillated by an external force. There is an analogy to audio microphone/speaker devices, with these transducers. To use these transducers, a coupling gel had to be used. This gel allows the ultrasonic waves to transit between source and sample without being absorbed by the surrounding air. Air absorbs such frequencies that these ceramic devices use very strongly. The coupling gel also acts as a lubricant to slide the transducers along the surface.

SAW Speed Measurement To determine the SAW speed, the transducer pair were placed at incrementing distances from each other, then the corresponding temporal increment for the pulse to reach the receiving transducer was measured on the CRO. The distances were measured by Vernier callipers, which are accurate to a high degree. The transducers were connected to a CRO and a signal generator as shown below.

Figure Three. The above configuration allowed the time taken for the acoustic pulse to travel from one transducer to the other to be measured on the CRO. The corresponding distances and times could then be used directly to calculate the SAW speed. This process was performed on samples of Aluminium, Brass, Mild Steel, Crown Glass and Perspex. A test performed on each sample before SAW was one to determine whether or not there were actually any SAWs present. By applying pressure on the surface of the sample between the transducers, the trace displayed on the CRO changed. This was because the SAWs were being prevented from reaching the receiving transducer, and the SAW component of the pulse was removed from the signal. If there was no change in the signal, this meant there was no SAW present. This latter effect was observed for the perspex sample. However, the signal for perspex was very weak. The interface material in the transducers was perspex too. So as to be expected there was no acoustic bulk wave refraction for the perspex sample, and hence very little energy available to sustain a SAW.

SAW Wavelength and Frequency determination The wavelength of the SAW in aluminium was measured by exploiting equation 3. An aluminium sample with a slit that increased in depth along the length of the block was used to impede the SAW. The block was 24cm±0.1cm long. The slit went from zero depth at one end of the block to 0.25cm±0.01cm depth at the other end of the block (see figure 4). Since the SAW had a very limited depth penetration, once the slit was at a certain depth the SAW would not be able to overcome the barrier that the slit provided. The transducer pair were placed at either side of the slit. The transducers were incrementally moved along the block and the depth of the slit and the signal magnitude (from the CRO) were recorded.

Figure Four. As the slit became deeper, the signal decreased (as to be expected). Care had to taken to keep the transducers at a constant distance from the slit. Because varying distance would affect the signal strength (to some extent), and the signal strength would not have been exclusively controlled by slit depth.

Results Acoustic Refraction Angles Below is a table of text book values for longitudinal, shear and Rayleigh (SAW) wave velocities (see "Kaye and Laby" in the references section). Longitudinal, CL (m/s)

Shear, CS (m/s)

Rayleigh, CR (m/s)

Aluminium

6374

5102

2906

Brass

4372

3451

1964

Perspex

2700

2177

1242

Glass

5660

5342

3127

Mild Steel

5960

5196

2996

Table One. Text book values of velocities of various acoustic wave modes in different materials. Using equation 2 the refraction angles for a bulk longitudinal wave from perspex into these materials (given the velocities and fixed incidence angle of 63°) can be calculated. Then as discussed as earlier, the main cause of the SAW can be determined. Below is a table of results from these calculations. Longitudinal Refraction Angle

Shear Refraction Angle

Principal Wave Mode

Aluminium

22.17°

50.65°

Shear

Brass

33.38°

Undefined

Longitudinal

Perspex

63.00°

Undefined

Longitudinal

Glass

25.15°

44.70°

Shear

Mild Steel

23.81°

48.04°

Shear

Table 2. Table of results, showing dominant component of Rayleigh waves in particular materials. The entries denoted undefined relate to mathematically non-existent refraction angles. The physical meaning of this is that there were no refracted waves of a particular type. From the above results, we can interpret that there was no acoustic wave mode conversion for Perspex and Brass (this would have been expected for perspex for, as discussed below). Given that the interface of the transducers is perspex no mode conversion would be expected, and none is predicted by theory too.

SAW Velocities Results were recorded for all but one of the samples. For the perspex sample the signal being received by the transducer was to weak to detect any changes, and was hence abandoned for this experiment. There was also no effect when pressure was applied to impede any SAWs. There was also no detectable change when the pressure test was performed on the brass sample, however there are still text book values for Rayleigh waves in brass and perspex. They are the slowest waves of the materials considered though. It may be that because they a so (relatively) slow that they have an insignificant contribution to the signal pulse, and are hence undetectable. The velocities were calculated using a LSF on the values of distance and time recorded. Below is a table showing the text book values, speeds and errors. Text Book Value (m/s)

Calculated Speed (m/s)

Calculated Error (m/s)

Aluminium

2906

2978

110

Brass

1964

1476

37

Perspex

1242

No Data

No Data

Glass

3127

3295

86

Mild Steel

2996

2843

63

Table 3. Experimentally determined SAW velocities compared with text book values. Below are a series of graphs depicting these results.

Figure Five.

Figure Six.

Figure Seven.

Figure Eight. As a test for these velocities, the above values can be used to see if they satisfy equation 1 (along with text book values for longitudinal and shear wave velocities). Below shows how well the equation is satisfied with the calculated values and the text book values.

Calculated Values

Text Book Values

0.33

0.00

Brass

-2.03

0.00

Glass

0.56

0.01

-0.54

0.00

Aluminium

Mild Steel

Table 4. Results for equation 1 for text book and calculated SAW velocities. The theoretical result should be zero. The results obtained using the calculated SAW speeds, show that they are quite approximate. The text book values yield much better results.

SAW Wavelength and Frequency Below is a table showing the variation of signal amplitude (voltage on CRO) with slit depth (in the aluminium sample). This data was obtained by the procedure discussed above (in the corresponding sub-section). Depth (m)

Signal (V)

Normalised Signal

0.000E+00

0.030

1.00

1.042E-04

0.020

0.67

2.083E-04

0.012

0.40

4.167E-04

0.008

0.27

6.250E-04

0.006

0.20

8.333E-04

0.004

0.13

Table 5. Variation of signal amplitude with slit depth. Normalised values are shown because these were used in calculations as the normalised wave energy for equations 4, 5, 6, & 7. Below is a graph showing the relationship between squared (normalised) amplitude and slit depth.

Figure Nine. All waves that extend in all directions in space should follow the inverse square law. That is, a linear decay of squared amplitude. We can see that this decay (fig. 9) is not linear, but exponential. This can be explained by the mechanical impedance that the slit provides to the weakly penetrating SAW.

Below is a plot of equation 7. It is the natural logarithm of the normalised CRO readings versus the slit depth. The square values are lost due to mathematical manipulation (laws of logarithms).

Figure Ten. From the data for this plot, the wavelength can be calculated statistically (LSF). Then, using the calculated SAW speed, the frequency can be calculated, as,

c= 

(9)

These are all of the values that were calculated for SAWs in aluminium, (speed, wavelength and frequency). Value

Error

Speed (m/s)

2978

±110

Wavelength (mm)

0.43

±0.05

Frequency (MHz)

6.84

±0.51

Table Six. Results for SAWs in Aluminium.

References "Tables of Physical and Chemical Constants", G.W.C. Kaye and T.H. Laby, 14th edition, Longman Group Limited 1973 "Fundamental Physics of Ultrasound", V.A. Shutliov, Gordon and Breach Science, New York, 1988 "What Video and TV", WV Publications, June, 1998, ISSN 1352-6162