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"THE USE OF ACOUSTIC WAVEGUIDES IN DETECTING SYSTEMS"

A Thesis submitted for the Degree of

DOCTOR OF PHILOSOPHY

in the

UNIVERSITY OF LONDON

by

BRYAN WOODWARD, M.Sc., D.I.C., A.Inst.P.

Department of Physics Imperial College of Science and Technology South Kensington LONDON, S.W.7. NOVEMBER, 1968. 2

ABSTRACT

The study of the mechanisms by which a reverberant tank of water induces stress in semi-immersed, cylindrical, elastic waveguides has been carried out to provide information for the design of a safety device for the incidence of boiling in a nuclear reactor. At each tank there is a distinct distribution of acoustic throughout the volume of water, and it has been shown that - these pressure fields can actuate longitudinal vibrations in semi- immersed rods (and tubes). The vibration amplitude depends directly on the rod length, and is a maximum at the resonant lengths of the rod.

The accelerations of the upper ends of these rods were measured • by means of a calibrated accelerometer and an attempt has been made to relate this to the acoustic pressure at the lower immersed end.

However, there is also a contribution to the vibrations created by the spatially-varying acoustic pressure developed along the curved side of the rods, although it appears that the end-face contribution is predominant. In order to investigate more explicitly this "end- contribution" the curved surface of a guide rod was isolated from the surrounding water by an enclosing concentric tube.

Investigations were also made on the influence of the form and of the physical dimensions upon the response of the waveguides. 3

ACKNOWLEDGEMENTS

It is a pleasure to record my gratitude to Dr. R. W. B. Stephens of the Physics Department, Imperial College, for providing me with the opportunity to work for a Ph.D. degree, for the interest he has taken in the supervision of this research and particularly for his patient reading and constructive criticism of the thesis.

Many other people in the Physics Department have given me consid- erable assistance, notably Mr. A. Sutherland, Mr. M. Oakey ( Group), and Mr. F. Martin (Student's Workshop) in the design and construction of the waveguide support mechanism; the Main Workshop staff under the direction of Mr. W. Shand, in build- ing the support and traverse framework and Mr. M. Jackson in photographic services. To all these people I convey my thanks.

I wish also to express my thanks to the United Kingdom 'Atomic Energy Authority, the sponsors of this research, and to Mr. A. J. Walton, Mr. I. D. MacLeod and Mr. F. Latham for their advice and suggestions during the initial stages of the project at U.K.A.E.A. Risley, Lancashire.

The final presentation of this work would have been impossible without the incalculable efforts of Joy Dunning who typed the thesis in its entirety. To her I am extremely grateful. I extend this gratitude to the members of the Acoustics Group who have enlightened the past two years by their geniality- and with whom I have had the good fortune to work.

Finally I wish to extend my thanks to my family for their unfailing help and encouragement throughout the years.

B.W.

4

CONTENTS

Page No

ABSTRACT 2

ACKNOWLEDGEMENTS 3

CONTENTS 4

INTRODUCTION 9

CHAPTER 1 REVIEW OF RESEARCH RELEVANT TO THE PRESENT 15 PROBLEM

1.1. Introductory Review 15

1.2. Reverberant Sound Fields 18

1.3. Sound Radiation from Cylinders Immersed in 29 Fluids

1.4. Coupling of Plane Waves to a Semi-Immersed 31 Waveguide

1.5. Coupling of Spherical Waves to an Immersed 34 Waveguide

CHAPTER 2 PRESSURE FIELDS IN RESONANT ENCLOSURES 39

2.1. Pressure Field Configuration in Rectangular 39 Enclosures 2.2. Pressure Field Configuration in Cylindrical 45 Tanks

2.3. Description of the Co-ordinate System used 50 in the Experiments 5

Page No.

CHAPTER 3 EXPERIMENTAL SYSTEM 52

3.1. Plan of Research 52 3.2. Description of Apparatus 54 3.3. Method of Supporting Waveguides 59 3.4. Pressure Transducer Mountings 64 3.5. Vibration Isolation and Noise Reduction 65 3.6. Mounting of Accelerometers 66 3.7. Choice of Waveguides 69 3.8. Calibration of Apparatus 70 (a) Level Recorder 70 (b) Microphone Amplifier 71 (c) Accelerometers 71 (d) Pressure Probes 71 3.9. Experimental Errors 75 (a) Readings 75 (b) Positioning of Waveguides and Probes 75 (c) Calibration Procedure 76 (d) Reading Signal Levels on Level Recorder 76 Paper

CHAPTER 4 PRELIMINARY INVESTIGATION OF ACOUSTIC 80 FIELDS IN THE EXPERIMENTAL TANK

4.1. Justification of Experiments 80

4.2. Finding the Natural Resonance of 82 the Tank

4.3. Variation of Acoustic Pressure with Position 85 of Probe: Source Transducer in Fixed Position

(a) Variation in the z-direction 85 (b) Radial Variation 86 (c) Angular (0) Variation at a Constant 88 Radius 6

Page No.

CHAPTER 4 CONTINUED

4.4. Variation of Acoustic Pressure with Position 92 of Driving Transducer: Probe in Fixed Posi- tion (a) Variation in the z-direction 92 (b) Radial Variation 93 4.5. Concept of "Effective Vessel Radius" 98

CHAPTER 5 USE OF SOLID ROD WAVEGUIDES AS SOUND 101 DETECTORS IN THE EXPERIMENTAL TANK

5.1. 'Introduction to Experiments with Solid Rods 101

5.2. Direct Measurements of Longitudinal Reson- 104 ance Frequencies of Solid Cylindrical Wave- guides 5.3. Identification of Peaks on Spectra for 110 Semi-Immersed Waveguides 5.4. Variation of Waveguide Response with Position 112 of Waveguide: Transducer in Fixed Position (a) Variation in the z-direction 112 (b) Radial Variation 113 5.5. Variation of Waveguide Response with Position 119 of Source Transducer: Waveguide in Fixed Position (a) Variation in the z-direction 119 (b) Radial Variation 120 5.6. Significance of Rod Dimensions 127 (a) Rods of the same Length but Different 127 Diameters (b) Dependence on Rod Length 130 5.7. Investigation of Energy Transfer Mechanism 139 between Acoustic Medium and Waveguide (a) Contribution of Plane End-Face of a 139 Rod (b) Contribution of Cylindrical Boundary 141 of rod Page No.

CHAPTER 5 CONTINUED

5.8. Effect of Bending a Waveguide 148 5.9. Comparison of Responses for Similar Waveguides 151 off Stainless Steel, Copper, Brass and Dural- umin

5.10. Dependence of Probe and Waveguide Response on 153 Power of Source

5.11. Effect of Variation of the Angle of Immersion 155

CHAPTER 6 FURTHER QUALITATIVE EXPERIMENTS 159

6.1. Introduction 159

6.2. Comparison of Responses for Rods and Tubes 160 of the same length and material

6.3. Response of Finned Tube Waveguides 164 6.4. Use of an Accelerometer to Investigate Tank 168 Wall Vibration 6.5. Spiral Waveguide Characteristics 171 (a) Experimental Procedure 171 (b)Discussion 172

CHAPTER 7 INTERPRETATION OF RESULTS AND GENERAL 180 CONCLUSIONS

7.1. Forced Vibrations in a Semi-Immersed Wave- 180 guide (a) Discussion of the Problem 180 (b) Mechanism of Driving Force 181 (c) Damping 184 (d) Equation of Motion for Forced 184 Vibrations (e) Rate of Energy Supply to the Waveguide 186

8

Page No.

CHAPTER 7 CONTINUED

7.2. Correlation of Acoustic Pressure and Wave- 188 guide Acceleration 7.3. Proposals for Further Study 191 1. Wire Waveguides 191 2. Phase Measurements 192 3. Horn Adaptor 192 4. Further Investigation of Energy 192 Transfer Mechanism 5. Pulse Propagation in Waveguides 193 6. Practical Design 193

APPENDIX I CALCULATION OF TANK RESONANCE FREQUENCIES 194

APPENDIX II PROGRAM TO COMPUTE "EFFECTIVE VESSEL RADIUS" 206

APPENDIX III PROGRAM TO COMPUTE YOUNG'S MODULUS AND 'Qs' 209 FOR WAVEGUIDES

APPENDIX IV ACCELEROMETERS AND TRANSDUCERS 219

GLOSSARY 230

LIST OF SYMBOLS 231

REFERENCES 235

BIBLIOGRAPHY 239 9

INTRODUCTION

The objective of this research was to provide information for designing an acoustical safety device to be used in the Proto- type Fast Reactor (PFR) under construction at Dounreay, in Caithness, Scotland.

The PFR is a 250 Megawatt reactor due to be operational by 1971 and will act as a forerunner to reactors of perhaps 1000 MW or more which will be commissioned in the 1980s.

In such a reactor the cooling agency is liquid sodium which has a normal working temperature between 400°C (at the inlet) and 600°C (at the outlet) but localized boiling may occur if there is a loss of coolant flow or if there is a power surge due to some perturbation of the nuclear reactivity. In order to avert a poss- ible fuel melt-down and subsequent release of fission products, various safety features must be incorporated, and it has been proposed that among these should be an acoustic waveguide acting as a detector of the signals produced by liquid boiling. The signals could then activate a Boiling Noise Detection System (BONDS) to shut down the reactor or operate a warning system.

Unfortunately there is likely to be a significantly high level of background noise generated by sodium pumps and by cavitation nucleation and collapse. The low frequency pump noise may be fil- tered out but there remains the difficulty of distinguishing boiling noise from cavitation noise. Since both have essentially broad-band "" spectra it is desirable to eliminate the cavitation noise which would otherwise occur all the and activate the detection system. If this is possible the boiling noise alone can be detected 10

when the reactor becomes overheated. Wood (1963) has invest- igated this cavitation problem using carefully designed pumps with transparent casing so that the bubbles were visible.

Experiments at Oak Ridge in the United States by Huntley et al (1966) and by MacPherson (1967) revealed that owing to its high tensile strength and surface tension, sodium does not boil without superheating it well above its, saturation temper- ature. They used an electromagnetic pump to demonstrate that these properties could suppress the onset of cavitation noise, but only in the absence of entrained microbubbles of gas since otherwise the gas would lower the tensile strength.

Holt and Singer (1968) have reported that the pressure- temperature history definitely influences the degree of super- heating required for incipient boiling in sodium. They detected the initiation of boiling both with thermocouples and with an accelerometer, since the vaporization of the superheated sodium resulted in a temperature drop and caused a pressure pulse. The acoustic vibrations were found to be "the most sensitive means to detect the instant of nucleation".

The boiling and cavitation process result in a two-phase medium, and Hammitt (1968) has suggested that the "void phase" volume would expand rapidly causing a modulation of nuclear reactivity and heat flux, thereby delaying overheating. (Liquid sodium has a thermal conductivity about one hundred that for water).

Even if pump noise and cavitation noise can be eliminated there still remain problems in the detection of the boiling noise. For instance, little is known about the attenuation of 11

sound in liquid sodium as a function of frequency and distance, nor of the spectrum of boiling noise which will actually occur in the PFR.

A study is at present being carried out by Fahy (1968) to attempt to determine acoustic stress levels in various parts of reactor heat exchangers. This may lead to a knowledge of sound transmission and attenuation characteristics of the various components.

An initial approach to the problem was to use an acoustic waveguide to detect the sound produced by a whistle mounted at the centre of each of the fuel "pins" in the reactor core. Fig. 1.2 illustrates a fuel pin, in which the top half contains the uranium fuel and the lower half is void so that the fission gases can be collected; the pressure of these gases increases with time. It was considered most probable that if a pin became overheated and cracked it would crack in the upper half, and hence the fission gases could escape through the whistle and consequently "blow" it. However, it was found in preliminary experiments at Risley that the whistle pulses produced had a very short duration and further, the background noise made it impractical to detect the signals.

As a sequel to this research it was decided to study the problem of detecting boiling noise by means of an — guide. The waveguide for use in the PFR has to take the form of a rod or tube of stainless steel, being the most suitable mat— erial to withstand corrosion and high temperatures. The guide will most likely be placed in the reactor so that its lower end is in the vicinity of the reactor core. (See Fig. 1.1). 12

It has been established by MacLeod (1966) and others that subcooled boiling noise is spread over an enormous freq- uency range (from less than 1 Hz up into the MHz range) and that no particular frequency in the spectrum is prominent. Hence almost any bandwidth within this wide range may be used by the detection system.

Boiling noise is caused by the growth, oscillation and collapse of vapour bubbles and can be sufficiently intense to excite the natural of a containing vessel. Because of a 6" air-gap between the PFR vessel and the surrounding earth (See Fig. 1.1), it seems feasible that the vessel will be capable of resonating in its natural modes of vibration. It is at these modes that distinct fields are established in the liquid medium, and if this acoustic pressure can excite vibrations in a waveguide that is semi-immersed in the liquid, then it may be possible to detect the boiling noise. 13

Sodium Level 6 Heat 3 Sodium Exchangers Rotatable Shield Pumps with Control Tubes Thermal

1 ---Insulation , • • • • •

771

Fuel Core Neutron Shield

Structural Concrete 6" Air Gap

Multilayered Sodium Vessel

FIG.1.1 SCHEMATIC DIAGRAM OF PROTOTYPE FAST REACTOR. 14

Stainless steel 'can' (8 feet long,1" dia.)

Fuel

/

Whistle IF

Void

FIG.1.2 PFR Fuel Pin. 15

CHAPTER 1 REVIEW OF RESEARCH RELEVANT TO THE PRESENT PROBLEM

1.1. Introductory Review

So far as is known the problem of extracting sound energy from a liquid contained in a reverberant vessel by means of a semi- immersed waveguide, has not before been attempted. Because this is a new type of problem, the results of this work could not be directly compared with the results of other authors. In view of this an attempt has been made to relate the results to the available theory of resonant• enclosures and finite waveguides.

There have been a vast number of publications on both theor- etical and experimental aspects of acoustic waveguides, since the appearance of the notable papers by Pochhammer (1876) and Chree (1889), who independently investigated longitudinal and transverse wave prop- agation in infinitely long cylindrical rods.

Bancroft (1941) has published numerical solutions of the dispersion equations for propagation, whilst Hudson (1943) and Abramson (1957) have dealt with dispersion for transverse wave propagation.

Baltrukonis et al (1961) studied axial-shear vibrations of infinite concentric cylinders in relation to the design of solid propellant rocket motors. This was extended by Kumar (1, 1966) who obtained frequency equations for torsional vibrations of finite com- posite cylinders, either concentric or bonded end to end. He also produced vibration patterns showing surface displacements of copper and steel cylinders placed end to end, taking the case when one cylinder 16

has a real propagation constant and the other has an imaginary constant.

Much of the foregoing theoretical work on acoustic wave- guides has been outlined by Redwood (1960), the chapter on con- tinuous waves in solid cylinders being of particular interest. A dissertation by Woodward (1966) summarizes papers having mostly an experimental bias.

More relevant to the current problem, Kumar (2, 1966) also drew surface displacement diagrams for axially symmetric vibrations in finite cylinders with different length/diameter ratios.

Pollard (1962) in a review of the Pochammer-Chree theory for solid cylinders, discussed the solutions for longitudinal, torsional and flexural waves then went on to consider wave propa- gation in a finie rod, introducing an analogy between a rod and an electrical transmission line. He considered the work of Kolsky (1953) on of pulses at the free end of the rod and also discussed mechanical systems, in which the response near resonant peaks is of most interest. The resonances of finite rods is an important feature of the present work and will be discussed at length later.

It will be shown in Chapter 5 that the responses of the semi-immersed waveguides are forced vibrations initiated by the reverberant pressure fields in the containing vessel. These vib- rations are predominantly longitudinal in nature and the ratio a/A (radius to wavelength) for all waveguides used did not exceed that for the lower limit of dispersion (that is, the region in which the velocity of sound becomes a function of the wavelength). 17

A number of topics under the sub-headings to follow were considered to be pertinent to the problem under investigation. 18

1.2. Reverberant Sound Fields

The science of reverberant fields can be conveniently intro- duced by reference to the work of Waterhouse (1955) who showed that in such a sound field the sound energy is distributed into interfer- ence patterns at the reflecting boundaries. This was explained by the fact that although the mean energy flow at all points is the same in all directions, the phases of the wave trains near the reflecting surfaces are not random because of reflection of the waves at these surfaces. It is this non-randomness of phase that gives rise to the interference patterns.

In his physical explanation of interference patterns Waterhouse — considered plane waves striking normally a plane rigid reflector. The sinusoidal pattern produced (Fig. 1.3(a)) was due to the addition of incident and reflected waves, with a difference of phase between them depending on the distance from the reflector. At the reflecting surface the two waves are in phase and the mean squared pressure (which is proportional to the potential energy of the system) is doubled. At a distance X/4 away, the waves are n out of phase and the mean squared pressure is zero.

For sound waves incident at oblique angles, the interference pattern produced is also sinusoidal but the wavelength increases with the angle of incidence as shown in Figs. 1.3(a),(d),(g).

But in a reverberant field, waves are incident on the bounding surfaces at all angles, and for each angle a separate pattern can be considered to form. Rayleigh (1945) showed that the mean squared are additive so that a resultant interference pattern as shown in Fig. 1.3(j) is produced.

19

e=o0

2kx 2kx 2kx (a) (b) (c) 2 2

P2 v2 E 1 1

2kx 2kx 2kx (d) (e) (f)

E

2kx 2kx 2kx (g) (h) (i) 2 2 2

E r r 1 1 1

0 0 0 2kx 2kx 2kx (j) - (k) (1)

FIG.1.3 INTERFERENCE PATTERNS FOR SOUND INCIDENT ON A PLANE REFLECTOR. 20

This pressure, or potential energy distribution, is given by:

< pr> = 1 + sin 2kx/2kx where the function (sin 2kx/2kx) represents the instantaneous radial pressure distribution of spherical waves emanating from a simple source.

Waterhouse went on to explain that in any acoustic field, energy exists in two forms, potential (pressure) and kinetic (part- icle velocity) and that the energy is equally divided between these forms. For the case of sound waves incident normally on a plane reflector the P.E. and K.E. are equal and opposite so that on addition they give an energy density that is equal at all points. See Fig. 1.3(c).

When the angle of incidence is not 0°, the energy density varied from point to point as in Figs. 1.3(f),(i). The kinetic energy distribution, averaged for waves incident from all directions over a hemisphere, is given by:

sin 2kx sin 2kx < v2 > 1 + ( - cos 2kx) /k2x2 (1.2) r 2kx 2kx as is represented by Fig. 1.3(k).

The reason why the curves for P.E. and K.E. differ is because pressure is a scalar quantity whereas is a vector.

By adding the two expressions together, the energy density distribution (E) is obtained: 21

< E > = 1 + (sin 2kx/2kx - cos 2kx) / k2x2 (1.3)

Clearly, the energy density varies according to the value of x and hence the sound field can be described as reverberant. (See Glossary).

In addition to this theory, the effect of bandwidth on interference patterns was considered, and some of this work was experimentally verified. Fig. 1.4 gives an indication of the var- iation of < p2 > with distance x from a reflecting wall in a rever- berant sound field. For a continuous band of frequencies extending from vi to v2, equation (1.1) becomes:

2

1 sin 2kx < p2 1+ dx (1.4) r k -k 2kx 2 1 k 1 where k = 27rv/c.

We now have an explanation of the mechanism of the constructive interference of sound waves in the most general case. But what happens in an enclosure of a particular shape ?

Morse (1948) has provided answers to this question by describing how normal modes of vibration are established in rectangular rooms, and given formulae for the allowed frequencies.

He stated that a sound source in a room will generate two types of vibrations: 22

2.0 2.0

Single Frequency 1 Octave Bandwidth Pr Pr

r X/4 0.6 ititill1110.6 0 2 4 6 8 10 2 4 6 8 10 2kx 2kx 2.0 2.0

+ 10% Bandwidth 2 5 Octave Bandwidth Pr Pr

0.6 i i i i i i 1 1 1 1 0.6 ttIIIIIIII 0 2 4 6 8 10 0 2 4 6 8 10 2kx 2kx

FIG.1.4 EFFECT OF BANDWIDTH ON PRESSURE DISTRIBUTION IN A SOUND FIELD.

23

(i)steady-state vibrations at the source frequency, (ii)transient free vibrations at the frequencies of the room modes, which decay with time.

Type (i) consists of many standing waves whose amplitudes depend on the source frequency and position, and on the "wave impedance". After the transients have decayed only these steady- state vibrations remain, and when the source is turned off the standing waves remaining damp out exponentially. (This is the rev- erberation phenomenon).

Though the boundary conditions for a rectangular tank and a rectangular room are different, the expression giving the frequencies of the normal modes is the same. It is given by:

2 ( n Fiz) 2 (en 2 (1.5) . 2.rr • 2 ixx )

where lx, 1 , I are the lengths of the sides, and n , n , n are y z x y z integers

The corresponding frequency equation for a cylindrical room or a cylindrical tank is:

2 4/ [nz]2 [amni v= (1.6) L z a where a is the nth solution of the Bessel function J (x) divided mA m by Tr (to comply with the boundary conditions); 1 is the tank length z (or height); "a" is the tank radius: and "c" is the sound velocity (1) in water.

24'

Morse (1932) was the first to derive formulae for the number of normal modes (N) in rectangular rooms and for the aver- age number of modes with frequencies in given bandwidth (dN). His formulae were:

N = (4V/3c3)v3 (1.7)

dN = (4V/c3) v2 dv (1.8) where V is the volume of the room.

The validity of these formulae was questioned by both Bolt (1939) and Maa (1939) who recognised independently that they are correct only when the limiting wavelength is negligibly small com- pared to the room dimensions.

Bolt's corrected formula for the number of normal modes of vibrations was given as:

3 47Vv3 2vV + CR1 (1.9) 3c3 2vV + (c/2)111 where R = (1 2 + 2 + (1 1 2 and 1 , 1 are the room x 1y ) (1y 1z ) z x ) x y' 1z dimensions.

From an alternative derivation, Maa obtained the following result.

N - 4Vv3 3Lc2 1 + 3Sc (1.10) 3c3 16Vv 8rrVv2 25

+ 1 + 1 where S is the room surface area and L = 1x y z

Although equations (1.9) and (1.10) appear quite different they were found to be in good agreement.

Roe (1941) extended equation (1.10) to apply to a cylinder (and also to other shapes) and his equation for the number of normal modes with frequencies less than v was given by:

47Vv3 TrSv2 3c3 4c2

Morse (1948) pointed out that the allowed frequencies in this case are not so evenly spaced as for rectangular rooms. This is due to the symmetry about the cylindrical axis, so that several standing waves, with different directions but nearly the same fre- quency, can be generated.

This preceding theory pertains directly to but the same reasoning can be applied to our case of the water-filled tank. The significant difference is that for the hard-walled room the pressure is a maximum at the boundaries, whereas for a water-filled tank it is a minimum.

The Fortran program in Appendix I computes all the possible tank resonance frequencies up to 10 KHz, and the results clearly show the uneven spacing as observed for by Morse. It must be emphasised that by no means all of these were identifiable in practice.

(2) Compared with the great amount of research published on architectural acoustics, the converse case of tank acoustics has 26

received little attention.

However, Smith and Schultz (1961), with reference to work by various other authors, discussed analogies between the two dis- ciplines in a report: "On Measuring Transducer Characteristics in a Water Tank". They considered expressions given by Waterhouse (1958) for the sound power output of simple monopole and dipole sources as a function of source position in various reflecting environments. In general the power output of a source differs significantly from the free-field value if the source-reflection distance is less than a wavelength. But Smith and Schultz suggested that for a monopole source in a water tank, the wall effects are relatively unimportant for distances greater than X/4, where A is the wavelength at the mid- frequency of the sound output.

It is interesting to apply this result to our cylindrical tank which was on average 45.5 cm radius and was mostly used with water at a depth of 120 cm. With the (spherical) source transducer at the geometrical centre (as for most experiments), then it was situated well outside the A/4 limit for the fundamental tank resonance, since at this frequency, (approx. 1400 Hz), X/4 was about 25 cm. Obviously, at higher order resonances X/4 was smaller.

Also, Smith and Schultz related the Sabine A to the rate of decay of pressure in a tank after the source is turned off, by:

K = cA/8V neper/time unit (1.12) where K = decay rate in neper/unit time (pressure E exp (-Kt)); V is the tank volume and c is the sound velocity in the containing liquid. However, this is of little importance from our standpoint

27

since continuous waves and not impulses were used, and primarily the steady state of normal modes were of interest.

In contrast to the equations (1.7) to (1.11) Stroh (1959) derived an expression for modal density by introducing the concept of "mean free path", 1 , which is the average path length between c "collisions of sound rays" (i.e. the average distance travelled between reflections). This mean free path was given by:

1 = 4V/S (1.13) c

where S is the tank (room) surface area.

In this wave theory, dimensions were measured in units of 1/3 . a characteristic length, L = V c - 1.5 1c.

The corresponding characteristic frequency was then: v = c/L and the frequency ratio was given by: c c

= v/v = L (1.14) c

The final expression for modal density was in terms of these parameters:

dN n S = 47412 + — (1.15) dp '4 2

and the average number of modes/bandwidth A was

4Kv2 [ 1 SA N - 1+-- . (1.16) v3 8 V • 28

Further water-tank experiments for the purpose of transducer calibrations have been attempted by Batchelder (1963), who investig- ated the reverberant sound field in a concrete test tank at audio frequencies. He used a tank measuring 30 x 12 x 7 feet deep and found that his experimental results differed from theoretical predictions; the number of peaks in the frequency-response charact- eristic corresponded to only the normal modes tangent to the .end planes, as if at least one end wall were completely absorbing. The system was found to be unusually complicated and it was suggested that this might arise from sound transmission through the concrete walls and floor into the surrounding earth, which was not necessarily homogeneous. Hence the tank was not sufficiently reverberant nor sufficiently absorptive for calibration purposes.

Unlike this "loaded" tank, the present cylindrical tank was mounted on anti-vibration mounts to suppress structure-born vibrations, and was free to resonate at its normal modes. It will be shown in Chapter 4 that distinct interference patterns, as elaborated by Morse for hard-walled enclosures, were established at these natural frequencies of the tank. 29

1.3. Sound Radiation from Cylinders immersed in Fluids

The possibility of the modification of the sound field in a water tank by the immersion of a rod or tube, also has to be con- sidered.

Theory of scattering of sound by cylinders has been expounded by Rayleigh (1904), Lamb (1924), Koyama (1933), Morse (1936) and Lax and Feshback (1948), to name but a few.

Experiments for plane waves incident on wooden cylinders situated in an anechoic room have been carried out by Wiener (1947), whilst Cook and Chrzanowski (1946) have performed similar experiments using cylinders of various materials in a room. In the latter work impedance and absorption measurements were made using firstly fibreglass cylinders, then cylinders whose shells were con- structed from perforated sheet steel and whose cores consisted of perforated paper and layers of fibreglass and cloth. The authors concluded that the absorbent cylinders could have absorption coeff- icients greater than unity. More recent work for plane waves incident on obstacles in anechoic rooms has been carried out by Harbold and Steinberg (1963).

However, the cylinders used in our present research were metal rods and tubes, with low absorption coefficients compared with these specially-designed absorbers, so few conclusions can be reached for this case using the results of analogous cases.

Bauer et al (1948) experimented on the scattering of ultra sonic waves in water by cylindrical rods and tubes whose radii were much larger than the wavelength of the beam. The beam frequency was 1145 KHz corresponding to a wavelength of 1.3 mm and the obstacles 30 were a 1" (6.35 mm) steel rod, a in (12.7 mm) steel rod, and a 1" (15.875 mm) hollow polystyrene tube. At this frequency the dimensions of the tank used were large enough to prevent the form- ation of resonances. The conclusions reached were that trans- mission through and reflection by the obstacles was small compared with the around them.

This differs considerably from our simulated reactor vessel in which, for the audible frequency range used, the wavelength of sound was much larger than the lateral dimensions of the largest waveguide used (!", 19.05 mm). Most of the detailed analysis was undertaken at frequencies below 5 KHz; at this frequency the wave- length of sound in water was approximately 30 cms, leading to well- defined modes.

Hence at these low frequencies it was considered that the presence of a waveguide in the water would not appreciably alter the field configuration. 31

1.4. Coupling of Plane Waves to a Semi-Immersed Waveguide

So far we know something about the resonant tank but the fundamental problem of greatest interest is that of the relation- ship between pressure in the water and vibration in an immersed waveguide. Though it appears that this specific problem has not been investigated, experiments involving the coupling of plane waves in a liquid to a semi-immersed rod have been attempted. The work of Blitz (1963) is so closely related that it will be briefly presented here.

Unlike our present case where the waveguide is placed in a reverberant sound field, Blitz used a tapered metal rod (unspecified) immersed in a liquid such that sound waves were normally incident to the lower end surface as depicted in Fig. 1.5. The rod acted simply as an acoustic transformer between the sound field and the transducer elements, and the system was found to be most sensitive when the rod had a natural frequency the same as the frequency of the sound source.

Blitz measured relative intensities in water in which sta- tionary waves were generated at 40 KHz. Signals from the acceler- ometer were fed to a valve-voltmeter and r.m.s. readings of 3 volts were obtained when the water was excited to just above the cavitation threshold. This is approximately two orders of magnitude greater than for the signals obtained in the present experiment at the same frequency. The main reason for this is most likely to be due to the use of high-amplitude plane waves, though the amplitude of the received signals also depends on the response characteristic of the accelerometer. (In the present case the accelerometer had a meas- ured first resonance at approximately 54 KHz). 32

Barium Titanate Element Metal Blocks I Accelerometer

Tapered brass Rod

Liquid(Water) 1 I Sound Waves

FIG.1.5 TAPERED ROD IN A LIQUID.

Sound Absorbent 17,-) Material - Pick-up Coil

Plastic Cover

Polarizing Coil Nickel Rod

Sound Waves

FIG.1.6 MAGNETOSTRICTIVE PROBE MICROPHONE. 33

Blitz also described a design for a magnetostrictive probe microphone for liquids using a similar arrangement, shown inf Fig. 1.6. Again the lower end face of the waveguide was normal to the direction of plane wave propagation.

The rod diameter was only 0.5 mm so that there was no modification of the wave configuration, and Blitz claimed that reasonable results were possible when such a rod was used in water at frequencies up to 600 KHz.

In the design shown a nickel rod was held vertically in a fluid (unspecified), and its curved surfaces were covered by a plastic tube in an attempt to expose only the free end to the sound waves. The sound waves travelled up the rod and caused a current to be induced in a pick-up coil by the inverse magneto- strictive effect. The rod was polarized by a D.C. polarizing coil, and it was asserted that standing waves in the rod could be eliminated by means of absorbent material placed at the upper end.

Reasonable criticism can be levelled at this experiment, and some of this has been substantiated by experimental verification (Chapter 5). Firstly, the plastic tube would be almost trans- parent to sound since its characteristic impedance is approximately the same as that for water. Secondly, the sound waves in the water would initiate forced vibrations in the rod and these would have a maximum amplitude each time the frequency of the sound source coincided with a natural (longitudinal) frequency of the rod. If it were considered desirable to use the device "off resonance", either a length of rod which was not a resonant length could be chosen, or the source frequency could be changed. Lastly no mention was made of the possibility of spreading of the plane waves or of the resonances of the tank containing the liquid. 34

1.5. Coupling of Spherical Waves to an Immersed Waveguide

In a theoretical analysis White (1958)has derived an expression for the disturbance in a one-dimensional waveguide resulting from a spreading spherical sound wave which is loosely coupled to the guide at all points along its length.

This is an idealized derivation since the waveguide was assumed to be in an infinite volume of liquid, i.e. no modes of a containing vessel were considered to be present.

The waveguide, capable of transmitting waves without dis- persion, is completely surrounded by a medium in which the spherical waves are generated.

Fig. 1.7 illustrates the geometry of the problem in which the source is at a perpendicular distance x from the origin of the guide co-ordinates z.

Z/x = 0 Z/x = 2 ROD

V x FLUID

z

Fig. 1.7.

White asserted that the coupled disturbance to an elementary length of the guide would be directly dependent on the following parameters:

(i) a function of the spherical wave, f(z) Fo[ t (z2 x2)1/c] where c is the speed of the spherical wave in the liquid;

35

(ii) the length of the element dz;

(iii)a coupling factor K which was the same for all points on the guide.

If this disturbance travels with speed v from point z = z at time t to z = Z at time T, then T = t + 1(Z - z)/v1 , and the total coupled disturbance for an elementary length dz is then:

S(Z,T) = K.f(z).F0 [ T - 1(Z - z)/v1 - (z2 + x2)1/c] dz -00 (1.17)

The possibility of a contribution directly proportional to the spreading wave at the point Z was also mentioned.

White then went on to calculate the waveform of the coupled disturbance for a "unit impulse" spreading wave so that the response of any other type of wave could be calculated from this. The coupled signal was, in this instance, given by:

OD S (Z,T) = i K.f(z).F [r - 1(Z - z)/v1 - 1 1 (z2 + x2)1/c ] dz (1.18) where the function F [T, v, Z, z, x, c] is zero except when the 1 argument is zero, that is , when:

I (Z - z)/v I + (z2 + x2)1 /c (1.19)

Differentiation of equation (1.18) gives two expressions:

cdT dz = for z > Z (1.20) [z/(z2 + x2)i + c/v]

36

cdT and dz - for z < Z [zi(z2 4. x2)1 - civi (1.21)

Substitution into equation (1.17) yields a useful expression for the total coupled wave due to a single impulse of sound energy radiating spherically:

K.c. f(zi) S (Z,T) - 1. [c/v - zi/(z1 x2)1] (1.22) K.c. f(z )

[c/v + z /(z2 4. x2)] g g where z denotes values of z less than Z; z 1 gdenotes values of z greater than Z; the plus sign applies when contributions are additive regardless of arrival direction; and the negative sign applies if the direction of propagation alters the sign of the contribution of the signal.

For the case of a solid rod immersed in a fluid, and cap- able of transmitting longitudinal waves, White's expression for an expanding pressure pulse was given as:

p = P.R.F [ t - (z2 + x2) / (z2 + x2) (1.23) where p, P are instantaneous and peak pressure respectively, and R is a reference distance.

By Hooke's law, the effect of a pressure p acting on an elementary length dz is to create an axial strain c = op/E. The strain produces a displacement of (ap/E)dz which means that each 37

face is displaced from the midpoint of the elementary section by an amount (ap/2E)dz. The pressure variation with time pro- duces a corresponding change of displacement, or particle velocity:

du = n/at = (adzi2E)4/3t (1.24)

The total particle velocity is the sum of all these elementary contributions with the correct time delays.

White enlarged upon this by giving an expression for the particle velocity in the rod due to a step function in pressure:

(x2 + x2)-1 (z2 x2)-1 aPRc u(Z,T) = 2E [ [c/v - zi/(q. + x2)I] [c/v + x /(x2 + x2)1] gg (1.25)

Using the relation, Stress = -pvu, and the relation, Axial Strain (e) = Stress/pv2, together with a strain contribution proportional to local pressure and given by:

crPR.F CT - (Z2+ x2)1 /c1 /E (Z2 + x2) I (1.26) 2 the total axial strain resulting from a step-rise in pressure is then:

r (zi x2)-1 (z2 + x2)-1 aPRc e(Z,T) - v [civ - zi(x2 [c/v + z g/(z2 x2)1] 1 1 x2) Z] g

2v.F - (Z2 + x2)i/c] 2 c(z2 + x2)1

(1.27)

In the last two sections we have dealt with work that is within the realm of the present research. Although this is of 38

relevance, neither the experimental work of Blitz, nor the theor— etical treatment by White is directly useful in this study of the properties of a waveguide semi—immersed in a reverberant vessel. However, the relations between axial strain and the pressure producing it will be of interest in a proposed sequel to this present work in which pulsed will be used.

FOOTNOTES

1. For a complete explanation of the Bessel function solutions used in the evaluation of the tank resonance frequencies, see Appendix I.

2. A comprehensive treatise, covering most aspects of architect— ural acoustics, has been published by Morse and Bolt (1944), Rev. Mod. Phys. 16(2), 69.

39

CHAPTER 2 PRESSURE FIELDS IN RESONANT ENCLOSURES

2.1. Pressure Field Configuration in Rectangular Enclosures

A vessel containing a liquid vibrating in one of its natural modes (eigenfunctions) is said to "resonate", and at each of the "resonance frequencies" there is a definite pressure (1) field established within the liquid medium . In analysing the pressure configuration consideration will be given first to the case of a rectangular room, which however normally has boun- dary conditions converse to those for a vessel containing a liquid.

The general wave equation applicable to this problem, in terms of rectangular co-ordinates is:

. alp alp a2p1 alp (2.1)

8x2 ay2 az2 c2 at2

where p is the function (pressure) and c is the wave velocity.

The solution of this equation is:

cos cos = co. (wx x/c) . (w y/c) . (w sin sin y sin z z/c) exp(-iwt) (2.2)

where w = Aw2 w2 w2).

Although the boundary conditions for a rectangular vessel and a rectangular room are different, the expression giving the 40

frequencies (eigenvalues) of the normal modes is the same:

[ (nx12 )2 ln1121- 2 v = p (2.3) 2r 1 x z where 1 , 1 , I are the lengths of the sides, and n , n x y z x y , nz are integers.

A for a given combination of n , n , n is set x y z up when plane waves, generated in the direction given by the direction cosines w /w, w /w, w x z/w, on multiple reflection from the walls give rise to a standing wave.

Morse (1936) has listed seven classes of standing waves:

x-axial, parallel to the x-axis, (n,n = 0, n y z x A 0) y-axial, parallel to the y-axis, (n , n = 0, n 0) x z y # z-axial, parallel to the z-axis, (nx, ny = 0, nz 0 0) y,z tangential waves, parallel to the (y,z) plane, (n = 0, ny0 0, n x z0 0) x,z tangential waves, parallel to the (x,y) plane, (n = 0, nzA 0, n.0 0) x,y tangential waves, parallel to the (x,y) plane, (n z = 0, n 0, n 0 0) x A y oblique waves (n , n , n x y z 0)

Each class of wave has a different reverberation time depending on boundary absorption, e.g. oblique waves are absorbed more readily than tangential waves.

For a tank which is lightly supported, that is, not externally loaded in any way, the walls are pressure-release surfaces. There is usually some residual pressure at the liquid/metal interfaces but at the water-air interface, where almost lossless reflection occurs, this is very small.

The variation of pressure throughout the tank volume depends 41 on the particular mode, or resonance. The fundamental mode will first be considered.

If the pressure is to be zero at all the boundaries, then the lowest possible frequency, given by equation (2.3), is obtained when each of the integers n , n , n is unity. Thus the fundamental x y z resonance frequency of a rectangular tank is the (111) mode.

The modes represented by one or two of the integer constants being zero, i.e. (100), (010), (001), (110), (101), (011), are impossible in practice since the boundary conditions are not satis- fied. Plane wave modes cannot be propagated since across the wavefront the pressure must be uniform and if at the boundaries this is zero, as for pressure-reslease boundaries, then it must be zero over the whole wavefront.

The (111) mode is represented pictorially in Fig. 2.1. The pressure at the boundaries is everywhere zero, whereas it is a max- imum at the geometrical centre. Thus, a probe measuring the pressure along r AB would measure a maximum at C and minima at A and B. Similarly, along DE it would measure a maximum also at C and minima at D and E.

Now suppose that a line which does not pass through the centre C is chosen, say PQ, in this case the maximum at R will be less than that at C, depending on the distribution along the line XY. So for this first mode, the pressure is a maximum at C and decreases along any line from this point.

The next mode will be represented by one of the following combinations of n , n x y , nz : (112), (121), (211). The mode with the lowest frequency will depend on the lengths of the sides of the 42 tank, i.e. on the magnitudes of 1 , 1 1 in equation (2.3). y, z

For the boundary conditions to be satisfied for this mode, the volume will effectively consist of two separate (111) config- urations as shown in Fig. 2.2. For this mode there are two "loops" of pressure along one direction, whereas along each of the other two directions there is only one.

Higher order modes can be represented by combinations of the (111) configuration as shown in Fig. 2.3.

This, then is the relevant theory for the modal pressure distribution in a rectangular tank. The equivalent theory for cylindrical tanks can now be discussed. 43

FIG.2.1 FUNDAMENTAL (111) MODE FOR A RECTANGULAR TANK.

(211)

Dividing Plane

(121)

(112)

1 x------_, / y FIG.2.2 (112),(121),(211) MODES FOR A RECTANGULAR TANK. 44

(234)

(312)

(455)

FIG.2.3 HIGHER ORDER MODES FOR A RECTANGULAR TANK. 45

2.2. Pressure Field Configurations in Cylindrical Tanks

It has been explained that for a rectangular tank a pressure field (other than for the fundamental mode) can be thought of as a combination of rectangular volumes, all with the same pressure configuration, and all the same size. Each volume, or "unit of pressure", must satisfy the boundary con- ditions, that is, at the boundaries of each unit there must be pressure minima.

Similarly a cylindrical tank can also be subdivided, but the volumes are not necessarily all the same size or shape, since the pressure configuration is more complicated. Subdivision of a cylindrical volume produces concentric cylinders, shorter cylinders end-to-end, segments, or combinations of these.

The form of the radial and z-direction distributions of pressure (see Fig. 2.6) can be obtained from the solution of the general wave equation in cylindrical co-ordinates:

cos p = . (mg)). (w z/c) • J (w r/c) . exp(-iwt) (2.4) sin sin z m r where w = Aw2 w2).

It is apparent from this equation that at a given depth z, the radial variation is a Bessel function, and at a given radial distance r, the z-variation is a sine function.

It has been verified experimentally with the present apparatus that at the fundamental mode of vibration of the tank, the acoustic pressure is a maximum at the geometrical centre, and it falls to a minimum (theoretically zero) at the boundary surfaces 46

(a) (101) fundamental

(201) (b)

(c) (302)

Each 3-dimensional figure shows the pressure configuration in the planes ABCD only. (b) shows division of the original cylinder into 2 cylinders end-to-end. (c) shows also concentric division.

FIG.2.4 J o- TYPE MODES OF A CYLINDRICAL TANK. 47

(a) £ less than 1.71a n = 1 z n = 1

(b) n = 2 z n = 1

(c) n z = 3 n = 2

Boundary conditions satisfied by functions above Jo, i.e. m greater than zero. FIG.2.5 DIVISION OF A CYLINDER FOR HIGHER-ORDER MODES. 48 as represented pictorially in Fig. 2.4(a). This means that a probe moved from the surface of the water to the base of the tank down the central axis detects a pressure varying from zero to a maximum then back to zero again. Likewise, measurements along any diameter at a constant depth also reveal a central maximum. In the present notation this mode is referred to as the (101) mode. These integers refer to constants in the equation from which the tank resonance frequencies (v) are obtained:

2 nz 2 1 1 (2.5) 2- [ ( ( where a is the nth solution of a Bessel function J , divided mn m by 'a' is the tank radius; 'nz' is an integer 1, 2, 3, etc. 't' is the water depth; and 'c' is the velocity of sound in water.

In the (101) mode n = 1, that is, there is one "loop" of z pressure in the z-direction of the tank, and the first zero 0 ) 0,1 of the Jo Bessel function satisfies the boundary conditions. Hence the (102) mode corresponds to nz = 1 and the second zero of the Bessel function J o: (202) corresponds to nz = 2 (two "loops" of pressure in the z-direction) and the second zero of the Bessel function Jo: (212) corresponds to n = 2 and the second zero of the function J and so on. 1

By substituting the appropriate constants in equation (2.5) a whole series of resonance frequencies can be computed (see Appendix I). In the present study using a tank with £ = 120 cm and a = 45.5 cm, the total number of computed modes at 20°C up to 10 KHz is 524, but the total number of modes possible in theory is 1048, because each mode is is doubly degenerate corresponding to a duality of cos WO in the characteristic function. This means that in the solution of the wave 49

equation (2.4), m can be be a positive integer, where +m corresponds to sin (mcb) and -m corresponds to cos (m4). Hence, numerically, a = a and there are correspondingly -m,n m,n twice as many modes satisfying the boundary conditions. The figure in Appendix I illustrates only positive solutions.

Morse (1948), in a study of the converse of this case, for cylindrical rooms, asserted that for 2, < 1.71a the funda- mental mode is as represented in Fig. 2.5(a) in which the air oscillates across a diameter. If this is so for an omnidirec- tional source the room would have a preferential direction of vibration. This type of mode would be very weak or unstable in a tank since it has a low-pressure plane through the tank centre, where for the predominant Jo-type modes the pressure is a maximum. For this mode to exist there would have to be pressure maxima at the two points A and B in Fig. 2.5(a) falling to zero at the boundaries of the demi-cylinders. The predominant low- frequency pressure variations in our tank (for which 2, = 2.66a) are likely, therefore, to be as in Fig. 2.4, but angular variations may occur in the higher modes, characterized by a division of the cylinder by one or more diametrical planes. or n eiative 50

2.3. Description of the Co-ordinate System used in the Experiments

All positions in the tank are referred to by the cylindrical co-ordinates (r, e, z) and considered as points, though for large diameter waveguides these positions are only approximately correct. An estimation of these positional errors is discussed in Section 3.9(b). The radial entity r varied from 0 to 45.5 cm (see program in Appendix II on computation of "Effective Vessel Radius"). The angle 0 was measured relative to an arbitrary vertical line painted down the inside of the tank, and varied from 0° to 360°; and the axial entity z varied from 0 at the water surface down to plus 122 cm at the base of the tank i.e. vertical axis is positive downwards. This depth of 122 cm represented the maximum possible depth of the water, which was the overflow depth at the rim, but for most exper- iments the depth was slightly less i.e. 120 cm.

Each waveguide was directly calibrated so that the depth (+z) of its end-face below the water surface was known.

Fig. 2.6 illustrates the co-ordinate system.

FOOTNOTES

1. When referring to the tank,. "resonance" and "mode" are synon- omous, whereas for a finite waveguide only the term 'resonance' is used. It is usual, in this context to talk about a reson- ance of a certain mode of propagation, such as the longitudinal (symmetric), flexural (or transverse, or antisymmetric) or torsional (shear). In this case the longitudinal mode would be most applicable. 51

0 0=0 (arbitrary)

r positive

FIG.2.6 CYLINDRICAL CO-ORDINATE SYSTEM. 52

CHAPTER 3 EXPERIMENTAL SYSTEM

3.1. Plan of Research

The general objective of this work was to investigate the nature of acoustic energy transfer from a liquid to a solid wave- guide semi-immersed in this liquid.

The justification for using water to replace sodium in the model laboratory experiment is because the characteristic impedance (pc) of water at room temperature is approximately equal to that o (1) of molten sodium in the temperature range 4000-600 C . Since water and liquid sodium are fairly similar acoustically, effects in sodium can be expected to be similar to those observed in water. This similarity is fortunate as the cost of building and maintaining a sodium rig, which must be carefully temperature-controlled and operated under the most stringent safety precautions, would be pro- hibitive.

In view of these considerations it was decided to use a cylindrical, galvanized-iron tank (approximately 4 feet deep by 3 feet in diameter), filled with ordinary tap water as the acoustic medium, in order to simulate the PFR, which has a cylindrical vessel with a hemispherical base.

First thoughts suggested that a waveguide would only transmit vibrations at its natural resonance frequencies in the longitudinal mode, but in practice this assumption was found to be erroneous.

The tank chosen for this work had a fundamental resonance 53 frequency of 1.4 KHz when filled with water, which was of the same order of magnitude. as that of the waveguide in longitudinal resonance.

It is readily possible to simulate conditions in the reactor by producing bubbles in the water tank, say by local boiling around an immersed heater element. As an alternative to using a real bubble source, a simulating electrical source could be utilized by using a transducer driven by a white noise generator. The power of the generator is spread over a wide frequency range, so the energy per unit bandwidth is small and a power amplifier is necessary. It would thus be possible to study narrow bands of frequency by means of a highly selective frequency analyser, but it was found that more meaningful results could be obtained by using pure tones. These were produced by a Frequency Oscillator driving a transducer directly. 54

3.2. Description of Apparatus

3.2(a) Sound Generator

Fig. 3.1 shows the general arrangement of the apparatus used.

The Beat Frequency Oscillator (B.F.O.) was a Brilel and Kjaer type 1012 producing continuous discrete waves over the frequency range 2 Hz to 20 KHz. This instrument had the facility that it could be matched to loads of impedances 6n, 600, 6008, and 6 KZ. The load in this sytem was a spherical transducer of Barium Tit- anate (Bo Ti 0) which was 2.54 cm (1") in diameter (type P3283 man- 3 ufactured by United Insulators Company Limited). It had a resonance frequency of approximately 81 KHz. Since most of the measurements were made at the lower end of the frequency range in which the water- immersed transducer was a high-impedance source, the 6 Ka matching position on the B.F.O. was used. (See Fig. 3.2). Over the oper- ating range 1-10 KHz the voltage remained constant to within ± 0.5 dB. For instance, a setting of 30 volts r.m.s. at 1 KHz would be found to fall to 28 volts r.m.s. at 10 KHz which is equivalent to a voltage decrease of 0.5 dB.

3.2(b) Sound Detection

Three methods of sound detection have been used: (i) miniature Lead Zirconate Titanate (PZT) cylindrical trans- ducers acting as pressure probes in the water; (ii) waveguides of various materials in the form of rods and tubes, semi-immersed in the water, with a piezoelectric accelerometer (Acos ID1003 manufactured by Cosmocord Ltd) mounted on the upper end of each; (iii) a piezoelectric accelerometer mounted on the outside wall of the tank. 55

CATHODE. RAY FREQUENCY SIGNAL IN OSCILLOSCOPE COUNTER

calibration signal

MICROPHONE LEVEL RECORDER BEAT FREQUENCY AMPLIFIER 2304 OSCILLATOR 2603

flexible drive cable

ACCELEROMETER

driving transducer (source)

E: ACCELEROMETER cylindrical water tank cylindrical pressure probe waveguide

anti-vibration mounts

FIG.3.1 DIAGRAMMATIC PLAN OF APPARATUS.

in 0 0 In 0 Ln 0 n 0 (volts) • • • • • 0 In co co 01 (KHz) 80 0 .

70 0

60

50

Slope = Impedance in Kilohms 40

Source: 1' diam. Barium Titanate sphere Voltages measured across output of BFO on a Standard AVO meter (2.25% accuracy for A.C.) Current measured in series with output 20 Source Frequencyat resonance: 81 KHz aesistanceat resonance: 440 ohms Capacitance at resonance: 8000 pf Bandwidth at resonance: 10 KHz 10 Q: 8 Power Efficiency: 80%

FIG.3.2 VARIATION OF SOURCE TRANSDUCER IMPEDANCE. I I 1 .• I 1 0 10 15 20 25 30 35 40 45 50 (mA) 57

The pressure transducer depended for its operation on the piezoelectric effect which is briefly summarized in Appendix IV.

The waveguide was essentially a simple acoustic transformer which transformed acoustic energy in the water medium to mechanical energy in the waveguide and this was transformed to electrical energy by the accelerometer.

Signals detected by these devices were amplified by 20, 40 or 60 dB by a Brilel and Kjaer Microphone Amplifier (type 2603), having a linear frequency characteristic in the range 2 Hz to 40 KHz. The amplified signals were then fed to a high-speed Level Recorder (BrUel and Kjaer type 2304) so that permanent recordingscould be obtained.

The type 2304 comprised an interchangeable potentiometer to which the input was fed, and the potentiometer slider was mechan- ically coupled with the driving coil and the recording stylus. The time constant and thus the lower limiting frequency could be changed by switching in different capacitors with the "Lower Limiting Frequency", which regulated the D.C. voltage. It was found to be convenient to use a 50 dB potentiometer, hence this switch was set to 50 dB in order to use the maximum dynamic range of the potentiometer.

The writing speed was variable in 9 steps from 50 mm/sec to 1000 mm/sec, and this was set to 300 mm/sec so that there was neither "overshoot" nor "undershoot" of the stylus. A gearbox between the motor and the paper drive gave 10 speeds: 100, 30, 10, 3, 1, 0.3, 0.1, 0.03, 0.01, 0.003 mm/sec (the last four correspond- ing respectively to: 36, 10, 8, 3, 6, 1.08 cm/hour). Since the paper was calibrated for particular speeds only, a convenient setting was chosen so that it was slow enough for a signal to reach its peak 58

value but fast enough for a recording to be made in a matter of seconds. This setting was lmm/sec.

The variable oscillator and the level recorder were conn- ected by a flexible drive cable so that as the frequency range was scanned automatically, a trace of the D.C. values of the amp- lified signals was obtained on calibrated paper. This frequency- calibrated paper was specifically made for use with a 50 dB poten- tiometer in the frequency range 0-20 KHz. The recording appeared as a transparent line on a red wax background thus permitting photographs to be obtained.

The Tektronix Oscilloscope (type 545B) was used to calibrate the level recorder (procedure described in section 3.8) and also to monitor the resonance peaks observed (described in section 4.2).

The frequency counter was mainly used for measuring the exact frequencies of the natural modes of the tank, dealt with fully in section 4.2. 59

3.3. Method of Supporting Waveguides

Figs. 3.3 and 3.4 show the waveguide support which fac- ilitates placing the lower end of the waveguide in any desired position in the tank.

Because of the cylindrical symmetry of the tank it was relevant to use the cylindrical co-ordinates (r, e, z) as explained in Chapter 2, but for reasons of stability and ease of construction it was decided to use a specially-designed support with three degrees of freedom described in rectangular co-ordinates (x, y, x).

The framework for the waveguide support was constructed form 1" angle iron (401" x 471" x 6") which rested on foam rubber pads in order to vibration-isolate it from the rigid dexion frame- work surrounding the tank. Four l" diameter stainless steel rods, used as runners as shown in Figs. 3.3 and 3.4 allowed movement in the x- and y- directions. The x- and y- "trolleys" were designed to slide freely on pulling or pushing by hand, thereby rendering a complicated automatic system unnecessary.

The tall paxolin tube was anchored to the small "trolley" and movement in the z-direction was brought about by use of a system of pulleys attached to this tube.

Waveguides in the form of rods were inserted into rubber bungs which rested in a bakelite collar supported by fine steel wires of the pulley system (Fig. 3.4). Preliminary tests showed that the position of the bung on the waveguides made no detectable difference to their propagation properties, since the bungs were very compliant. 60

FIG.3.3. VIEW OF APPARATUS 61

upper pulleys (---) (- -) (two also on opposite side)

accelerometer cable

accelerometer

rubber bung

bakelite collar waveguide

lower pulleys (six in each set)

paxalin support tube

clamp for support wire

windlass 5" square brass mounting plate

clamp steel "runner" waveguide

FIG.3.4 WAVEGUIDE SUPPORT AND PULLEY SYSTEM. 62

In an experiment to investigate the mode of energy transfer between the water and a semi-immersed rod (section 5.6), a rod was surrounded by a dural tube in such a way that the curved sides of the rod were isolated from the water. The support for this arran- gement is shown in Fig. 3.6.

With these arrangements a waveguide could be raised or lowered by winding a handle; the maximum range of travel was app- roximately equal to the depth of the water (120 cms). Provision was made for two waveguides to be used simultaneously by having two sets of pulleys but this was not used.

Some of the waveguides used were not heavy enough to exert sufficient tension on the steel wires so it was necessary to use a simple rubber-covered clamp to secure these guides in a vertical position. This made no detectable change to the received signals. (A heavy steel support collar would have been preferable).

The whole support system was remarkably noise-free when left undisturbed.

Rubber moulding paste

Waterproof cable Lead Zirconate Phosphor Bronze Titanate cylinder Earthing strip braid

FIG.3.5 PRESSURE PROBE 63

accelerometer support ..--- wires

securing screws

bakelite collar

dural tube waveguide

end of waveguide tight-fitting touching water rubber insert

FIG.3.6 ROD SUPPORT USING SURROUNDING TUBE TO EXCLUDE WATER. 64

3.4. Pressure Transducer Mountings

The pressure probe used for most of the field plotting was a Lead Zirconate Titanate (PZT) cylinder (0.95 cm length, 0.5 cm outside diameter, 0.05 cm wall thickness). Like the driving trans- ducer mentioned in section 3.2, this transducer was piezoelectric.

If a piezoelectric crystal is coated with conducting elect- rodes on certain faces and it is subjected to a mechanical strain, an electric charge will occur across these electrodes. In this case the strain was introduced by pressure between the inner and outer walls of the cylindrical element, the resulting charge produced being directly proportional to the induced strain(2).

As tap water is conducting, it was necessary to isolate the conducting surfaces of the transducer from each other. Fig. 3.5 illustrates how this was achieved using Escorubber (SR300) silicone rubber moulding paste. The inner surface of the cylinder was in contact with a phosphor-bronze "spring" which was connected to the central wire of a co-axial waterproof cable. In this way it was not neccessary to attempt the difficult task of soldering directly to the inner face. Soldering the earth braid of the cable to the outer face of the cylinder was done with extreme care to avoid thermal shock.

A second probe was made in exactly the same way except that the sealing compound used was 'Araldite'. This probe (a Brush Clevite tube, No. 8-4020 PZT-5A) which was calibrated in pV/pbar so that its response and the responses of waveguides could be directly com- pared, had the dimensions: length, 1.27 cm; outside diameter, 0.635 cm; wall thickness, 0.051 cm. 65

3.5. Vibration Isolation and Noise Reduction

In order to damp out extraneous vibrations the tank was supported on four anti-vibration "Barrymounts" which had approx- imately the correct loading when the tank was full of water. The mounts had a natural frequency of 12 Hz and isolated vibrations above 20 Hz. With 600 lbs (272.4 kgm) loading on each there should be a 98% isolation at 50 Hz, and a corresponding decrease in percentage efficiency of vibration isolation when the mounts are not correctly loaded.

However, the tank, together with the water (120 cms deep), weighed approximately 860 kgm, and very little structure-born vib- ration occurred through the rigid floor of the sub-basement of the Physics building where the tank was situated.

Earthing was always a major problem and the best method was to use the mains earth. There was no standard way in which earth leads were connected between the components of the apparatus and the best arrangement was found by trial and error. 66

3.6. Mounting of Accelerometers

The ID1003 accelerometer was supplied with a 4BA grub screw with which connection was made to the waveguides. In the solid cylindrical rods, a 4BA tapped hole was made in the upper end and the accelerometer was screwed on tightly with the aid of a spanner. In a few cases the accelerometer was stuck to the rod with "Durofix" adhesive and the variations in response using the two methods of fixing were mostly within the experimental error.

When performing experiments with tubes, the accelerometer was screwed to a small end piece, of the same material as the tube, which was cut out on a lathe and stuck rigidly into the upper end of the tube with "Durofix". In this way any vibrations in the tube were transmitted to the rigidly attached endpiece and hence to the detector. FIG.3.7. WAVEGUIDE COLLAR, SUPPORT BUNG AND ACCELEROMETER Coin of the Realm 10 New Pence ACCELE 'Acos' Accelerometer RO METERS Phosphor-Bronze Strip , P PZT Cylinder ROBES AN Probe P1 D TRANS NW"

D Mounted Probe P2 UCER

Barium Titanate Source Trans- ducer .41 69

3.7. Choice of Waveguides

Most of this work utilized stainless steel waveguides since this is the material most suitable for use in the reactor. In addition it does not corrode or rust when left in water for a long time, and is readily available. The only minor disadvantage found with stainless steel was difficulty with machining.

Rods of copper and brass, and rods and tubes of duralumin were also used in subsidiary comparative experiments.

Experiments were conducted with waveguides of different lengths and diameters and since at the outset of this work very little was known about the propagation of waves in semi-immersed rods, the length of the paxalin supporting tube was chosen so that it allowed a wide range of lengths to be accommodated. The support collar could take waveguides with diameters up to approximately 1" and the fine support wires could easily be changed to take the weight of very heavy rods.

To begin the experimental work the lengths of rods were chosen such that their fundamental longitudinal resonance frequencies were of the order of the fundamental resonance frequency of the tank. For example, a 2.6 metre stainless steel rod has its fundamental at approximately 1 KHz, just below the tank fundamental of 1.4 KHz. This did not prove an important consideration and eventually a range of rod lengths between 60 cms and 260 cms was used.

A series of rods all of the same length, with standard diameters: 11t 1" 3t1 In VI 3n 8 4 , g, , 8 , a , was used to investigate any dependence on cross-sectional area (and/or on area of curved surface). This is explained fulley in section 5.5(a). 70

3.8. Calibration of Apparatus

3.8(a) Level Recorder

When calibrated the 2304 Level Recorder rectified A.C. input signals and recorded D.C. levels in the 50 dB range between 10 mV and 3.16 volts. Thus any signal from the probe or accel- erometer required amplification (a 2603 Microphone Amplifier was used) to fall within this range. The 10 mV calibration output from the 2304 was found to be giving 12.5 mV which was arithmet- ically an inconvenient calibration voltage, so another calibration method was devised. This procedure utilized the "Calibration Output" of the 545B Tektronix Oscilloscope. The output was a continuous 10 KHz square wave for a wide range of discrete voltages.(3)

This square wave output on rectification became a steady D.C. (on r.m.s.) level equal to half the "square-wave" value. Thus a 2 volt calibrating square-wave was rectified to 1 volt by the level recorder.

Since the recording paper was specifically for use with a 50 dB potentiometer and the lower limiting input voltage was to be 10 mV, then the other 10 dB levels were as depicted in Fig. 3.10, according to the relationship:

dB (volts) = 20 log V /V 10 1 2

To check that the "0 dB" level was 10 mV, a 20 mV square- wave signal was led to the recorder with the paper drive operating. If the resulting stylus trace was not on the lowest line of the paper it was adjusted to be so with the "Zero Level Adjustment" control. The accuracy of this setting was checked at the 20 dB 71

level (200 mV calibration output giving 100 mV on the paper), and at the 40 dB level (2 volts output giving 1 volt on paper).

The 2603 amplifier, when used with a short-circuited input, superimposed a noise level of approximately 30 pV, i.e. 10 dB, on the trace, hence this was the minimum signal that could be recorded. Fig. 3.11 shows the actual signal levels for different amplifier gains.

3.8(b) Microphone Amplifier

This amplifier was adjusted to conform with the graduated gain scale, (i.e. calibrated by applying a square wave signal from the Tektronix calibration output, and adjusting the input sensitivity of the amplifier when viewing the output on the oscilloscope screen).

3.8(c) Accelerometers

The ID 1003 accelerometers were precalibrated by the manufacturers and those used had sensitivities of 8.5 mV r.m.../ and 9.0 mV gpeak r.m.s./gpeak.

3.8(d) Pressure Probes

The sensitivity (S) of a probe (below resonance) can be shown to be a function of both the "static" capacitance (C ) o of the crystal and the capacitance of the connecting cable (Cc).

If V is the signal voltage measured and s = o (constant).p, then the sensitivity can be expressed as V /E. o In terms of the capacitances this becomes: V V C o o o S - (pV/pbar) kp C +C o c 72

Sensitivity measurements for probes with exactly the same specification as P3 (see Appendix IV) have been made at the National Physical Laboratory, using cable of capacitance (C1) such that (C + C') was 2000 pF. Since the sensitivity o is proportional to 1/(Cc + Cc), the sensitivities of two sim- ilar probes can be compared. Thus

C + C" 1 o c 3200 S C + C' 2000 2 o c

Therefore the sensitivity of P3 was:

S = 5.51/8 2

On the voltage scale, the sensitivity at -120 dB re 1 volt was pV/pbar.

The level at which this 'calibrated' probe had a sen- sitivity of 1pV/pbar was therefore -124 dB re 1 volt. The other probes P1 and P2 could therefore be directly calibrated from corresponding pressure field plots, such as Fig. 4.3.

73

V = 2 volts PP

(a) Full sine-wave

V = 1 volt

(b) Half sine-wave (Full wave rectification)

V = 0.707 volts r.m.s.

(c) Root-mean-square voltage

FIG.3.9 REPRESENTATION OF VOLTAGES.

dB Recorded Voltage (after amplification) 50 3.16 V 40 1 V. 30 316 mV 20 100 mV 10 31.6mV 0 10 mV

FIG.3.10 VOLTAGE LEVELS ON 50 dB RECORDER PAPER. 74 dB re. 10mV Signal Gain dB re. 10mV Signal Attenuation 50 3.16 V 70 31.6 V 45 1.77 V 65 17.7 V 40 1 V 60 10 V 35 562 mV 55 5.62 V 30 316 mV 50 3.16 V 25 177 mV 1 45 1.77 V 10 20 100 mV 40 1 V 15 56.2mV 35 562 mV 10 31.6mV 30 316 mV 5 17.7mV 25 177 mV 0 10 mV 20 100 mV

30 316 mV 90 316 V 25 177 mV 85 177 V 20 100 mV 80 100 V 15 56.2mV 75 56.2 V 10 31.6mV 70 31.6 V 5 17.7mV 10 65 17.7 V 100 0 10 mV 60 10 V -5 5.62mV 55 5.62 V -10 3.16mV 50 3.16 V -15 1.77mV 45 1.77 V -20 1 mV 40 1 V 10 31.6mV 5 17.7mV 0 10 mV -5 5.62mV -10 3.16mV -15 1.77mV 100 -20 1 mV -25 562 pV -30 316 pV -35 177 pV -40 100 pV

-10 3.16mV -15 1.77mV -20 1 mV -25 562 pV -30 316 pV -35 177 pV 1000 -40 100 pV -45 56.2pV -50 31.6pV -55 17.7pV -60 10 pV

FIG.3.11 SIGNAL LEVELS FOR GAIN/ATTENUATION OF 2603 AMPLIFIER. 75

3.9. Experimental Errors

3.9(a) Frequency Readings

Measurements of frequency were taken with a Digital Fre- quency Counter (Advance Electronics Timer Counter TC7) which was accurate to ± 1 Hz. Many of the stronger modes of the tank in the audible range could, after experience, be roughly checked by ear. In practice all modal frequencies were obtained by monitoring the waveform on the oscilloscope and noting the frequency when the displaced waveform-attained a maximum amplitude. It was possible to measure the frequency of most of the lower-order res- onance peaks to within ± 1 Hz but these values changed slightly (± 3 Hz) from day to day owing to changes of temperature.

The computer program in Appendix II gives some indication of how much the resonance frequencies change with temperature and depth of water.

3.9(b) Positioning of Waveguides and Probes

On all waveguides used fiducial marks were painted at 5 cm intervals starting from the lower end. Similarly, the leads of the driving transducer and the probes were marked, in each case starting from the centre of the sphere and cylinders respectively. For convenience measurements were only taken at 5 cm intervals (to an accuracy of ± 0.1 cm).

To obtain the (r, 0) positions of the waveguide the loc- ation of the central axis of the tank had to be known. Since cylindrical co-ordinates were used, a 0° (arbitrary) vertical ref- erence line was painted down the inside of the tank. Reference o o o lines were also painted at 90 , 180 and 270 . One of the trolleys 76

could traverse the 00-1800 positions, the other the 900-2700 positions.

In many experiments the driving transducer was situated at the geometrical centre of the water volume. It was placed in this position by lowering it to half the water depth and lining up the suspension lead with both the 00-1800 line and the 900-2700 line. The position of a waveguide could now be located by meas- uring from the central axis of the tank to the longitudinal axis of the waveguide with a metre rule. The accuracy of these meas- urements was ± 0.1 cm.

The measured depth of the water at the tank walls differed from measurements over the complete section owing to the warped nature of the tank base. The error would be of consequence only when measurements were taken near this base.

3.9(c) Calibration Procedure

The resolution of the potentiometer was ± 0.5 dB or 1% of the total travel but since the accuracy of the oscilloscope cal- ibrator was about ± 3% then the error involved in this calibration procedure was ± 1.5 dB.

3.9(d) Reading Signal Levels on Level Recorder Paper

The errors involved in the measurement of signal levels on the paper depended to a certain extent on the scanning speed of the B.F.O. At very high speeds the modal frequencies were scanned so quickly that there was insufficient time for the mode to "build up" fully. A test was carried out to give some idea of this error.

The level recorder gear-box was used on different ratios and the driving transducer was driven by the B.F.O. Using a 77

piezoelectric probe in an arbitrary position traces were obtained for 4 speeds of the gearbox and a comparison was made between the amplitude of the peaks at each resonance.

As an example the following table shows the variation in response for the fundamental tank mode, using the 2603 ampli- fier on two different voltage gains.

GEARBOX COG GAIN: x 1000 GAIN: x 100

FAST A 27 dB 47 dB B 28 48-49 C 30 50-51

SLOW D 30 51

The table shows that at the slower gearbox speeds, there was time for the mode to build up fully. For very accurate runs, speed C was used but this was too slow for most experiments, so B was used and 2 dB was added on to all readings. Measurements on other modes shows that this was a reasonable value to add on in order to compensate for the fast speed.

Many scans using the driving transducer and detector in the same relative places indicated a total amplitude vari- ation of about 4-5 dB (± 2.5 dB) for any given mode.

Considering the maximum possible calibration error of ± 1.5 dB, the total assessed error was ± 4 dB. However graphs of various parameters, when compared with theoretical considerations, revealed that ± 2 dB was a more realistic error, which suggests that some sources of error cancelled out. 78

FOOTNOTES

1. Parameters for liquid metals: Melting Point (°C) Boiling Point (°C) Potassium 62.3 760 Sodium 97.5 880

2. .Details of piezoelectric probes and accelerometers are given in Appendix IV.

3. For completeness, the meanings of a few pertinent terms should be explained, with reference to Fig. 3.9.

A "full sine-wave" of 2 volts (Fig. 3.9(a)), after full-wave rectification, becomes a "half sine-wave" of 1 volt (Fig. 3.9(b)). These are respectively the "peak-to-peak" (V ) and the "peak" PP (V p) values of the alternating voltage. The equivalent steady (D.C.) level of voltage is the "root-mean-square" (r.m.s.) value given by:

V r.m.s. = V x 0.707

Hence 2 volts peak-to-peak is equivalent to 0.707 volts r.m.s., or 2.8 volts peak-to-peak is equivalent to 1 volt r.m.s. 79

It I speak not this in estimation,

As what I think might be, but what I know

Is ruminated, plotted, and set down,"

William Shakespeare (Henry IV, Part I) 80

CHAPTER 4 PRELIMINARY INVESTIGATION OF ACOUSTIC FIELDS IN THE EXPERIMENTAL TANK

4.1. Justification of Experiments

An introduction was given in Chapter .1 on the interfer- ence of wave trains in air and how modes are built up between the hard-wall boundaries of a room. As a sequel to this, the theory pertaining to a vessel with pressure-release boundaries was developed in Chapter 2.

We know from the solution of the general wave equation for cylindrical symmetry that the axial pressure variation is a sine function and that the radial variation is a Bessel function. Hence it seemed a reasonable starting point to test the validity of this theory by plotting pressure fields in the cylindrical volume for different resonance frequencies of the tank.

Two ways in which this could be done are:

(1) The source transducer can be driven at a discrete res- onance frequency of the tank and readings taken as the detecting pressure probe is moved from one location to another. In this case each frequency in turn can be studied, but there are certain inherent difficulties. For instance it is necessary to "hold" onto a resonance for long periods, and to do the experiment adequately requires a noise-free travel mechanism for the probe.

(ii) The probe can be fixed in position for a complete, auto- matic scan of all the frequencies of interest. Then 81 the operation can be repeated for successive positions, both axially and radially, say every 5 cm apart. With this method all the responses of the detectable resonances are recorded in one automatic scan for each position. However, each of the peaks in the Level Recorder spectra has to be identified. It was this procedure which was adopted since it facilitated taking reproducible recordings. 82

4.2.. Finding the Natural Resonance Frequencies of the Tank

The apparatus used to find the resonance frequencies of the tank was as shown in Fig. 3.1 with the omission of the waveguide.

With the driving transducer and probe in any position even with both touching the tank walls, it was possible to obtain a frequency spectrum on the level recorder when the oscillator was driven through its frequency range (2 Hz - 20 KHz). However, larger amplitude peaks were obtained when the transducer and probe were situated near the tank centre, and a typical spectrum is shown for probe P3 in Fig. 4.1.

For a particular set of measurements in which the source transducer (driven at 30 volts) was at the geometrical centre of the tank at position (0, 0°, 60) and the probe P3 was at (5, 180°, 60) peaks representing natural resonances of the tank were obtained on the level recorder trace at the following frequencies: (Hz) 1403, 2180, 2825, 3245, 3280, 4060, 4321, 4391, 4635, 5026, 5210, 5840, 5994, 6074,

Almost all of these modes could be checked audibly, but for reliable measurements of the frequencies it was necessary to adjust the oscillator by hand until the amplified probe waveform, as displayed on the oscilloscope screen, had a maximum amplitude. The digital frequency counter would then show the resonance fre- quency to an accuracy of ± 1 Hz.

These frequencies up to 6 KHz represented by no means all the resonances in this band, since the probe would be situated in pressure minima of the reverberant fields for other resonances. 83

As the location of the probe was changed, further peaks were indeed detectable on the traces. For instance, when using the same probe (P3) at position (5, 180°, 5) and with the source transducer (at 30 volts again) at position (0, 0°, 60) as before, the following frequencies were recorded: (Hz) 1403, 1728, 2180, 2697, 2825, 3245, 3280, 3643, 3813, 4321, 4391, 4635, 5026, 5210, 5534, 5570, 5840, 5939, 5994, 6074, (See Fig. 4.2).

It will be noted that several previously undetectable resonance frequencies have now appeared (underlined) since the probe was moved into regions of maximum pressure for these modes. • 0 84 CO

Cr)

0 tr) Ln 0 0 co rn 1--1 CO CV 0 CV 04 01 +I'

40

X30 -4

1 5 10 15 (KHz)

FIG.4.1. SPECTRUM FOR P3 (5,180°,60)

0 CO

N CO

Ctl CO 0 rs- tn 0 C'',3 CO CA r••••• r-I kf) N CV Cr) V 10p dB re

1 2 5 10 15 (KHz)

FIG.4.2. SPECTRUM FOR P3 (5,180°,5) 85

4.3. Variation of Acoustic Pressure with Position of Probe: Source transducer in Fixed Position

4.3(a) Variation in the z-direction

As predicted by theory each tank mode should have a distinct pressure pattern. To investigate this, the source transducer was sited at (0, 0°, 60) and recordings were obtained on the Level Recorder for the probe at 5 cm intervals down the line (5, 180°, 5-115). At each position of the probe the main peaks of the recorded spectrum were identified by means of the frequency counter, as described in the previous section. From these recordings the variation in peak amplitude with probe pos- ition, for any resonance frequency, could be plotted. It was clear from the 23 traces taken that only the peaks in the "unimodal" region (below about 6 KHz) showed any distinct variation. Above this frequency the response was approximately constant with depth for any particular mode, though these were so close together that the recorder registered a band of modes (the "multimodal" region), hence it was difficult to distinguish with any degree of certainty the individual modes on the recorder traces. Therefore, over most of the depth, particularly near the cylindrical axis as in this case, the acoustic pressure variation was small for any specific frequency above 6 KHz.

Fig. 4.3 shows the variation of pressure down the line (5, 180°, 5-115) for four distinct low frequency tank modes, and it is noteworthy that the nodes are not quite symmetrical. This may well be due to the difference in efficiency between the top (water - air) and the bottom (galvanized iron - air) pressure-release surfaces. However, the variations are symmetrical enough to be described as "sine", though to be more precise they are really 86

"rectified sine" variations.

Pressure - a scalar quantity - can be represented diag- rammatically as in Fig. 2.4, where the pressure at any point is a positive (or zero) entity: a "negative" pressure has no meaning in this context. The amplitude of the envelope repre- senting the pressure - which is governed by some mathematical function, in this case sine - is the parameter actually measured. This must be positive, hence the concept of "rectification".

4.3(b) Radial variation

Using the same method as for the z-variation, the radial pressure variation was determined along the line (5-45, 180°, 60). The distributions for probe P1 are shown in Fig. 4.5.

As expected, the pressure for the fundamental mode (1403 Hz) falls off from a maximum at the tank centre to a minimum at the walls. The solution of the wave equation in cylindrical co-ordinates, (equ. 2.4), shows that the radial pressure variation is a Bessel function. Hence, when the pressure falls to zero (in this case a minimum), Jm(r) = 0. When the function Jo(r) = 0, r = 2.4048, 5.5201, 8.6537, 11.7915, etc. and since a = 45.5 cm for the funda- mental (first minimum at the boundary) the solution satsifying the boundary conditions is J0(45.5) = 0.

The same radial boundary conditions apply to various other modes, including the one at 2183 Hz.

For modes with a central maximum and one minimum between the axis and the walls, the distance of this minimum from the axis is given by the ratio of the first two Bessel function solutions

87 of J : o

r _ 2.4048 = 19.8 cm 45.5 5.5201

This result agrees very closely with those plotted in the figures.

For higher order J0-type modes, where there are two min- ima between the axis and the walls, the distances of these two minima are given by:

= 45.5 2.4048 5.5201 8.6537 so that r = 12.65 and r = 29.03 an, another result which 2 2 - agrees very well with experimental determinations.

The mode at 1728 Hz (Fig. 4.3) has a minimum at the tank centre and can be described as a "weak" mode, since all the low- frequency J0-type modes are characterized by having a maximum pressure region at the centre. No radial variation was plotted for this mode since in the set of radial measurements taken the response was too close to the noise level at this depth (60 cm). It is clear from the z-variation of pressure for this mode (Fig. 4.3) that if radial measurements were taken at depths of 30 cm and 90 cm there would be central maxima. Hence, at this frequency the tank volume is effectively divided into two (101)-type modes as described in Chapter 2. Similarly, at 2180 Hz (Fig. 4.3) the cylindrical volume is effectively divided into 3 cylinders end-to-end, each with the same pressure configuration.

For these two modes (1728, 2180 Hz) the division of the cylinder is straightforward. As we have seen there are distinct 88 radial variations of pressure but for higher order modes there may also be variations with the angle 6.

4.3.(c) Angular (6) variation at a constant radius

Pressure measurements taken up to this stage have revealed that most of the low frequency tank modes are of the J -type, i.e. having pressure maxima on the cylindrical axis. o In this section we set out to show that for these modes the angular variation of pressure is zero. (See Figs. 2.4, 2.5, 2.6).

With the driving transducer at (0, 0°, 60), a cardboard disc of 10 cm radius, graduated at 10° intervals, was mounted symmetrically above it and above the water surface. The pressure probe P2, at a depth of 60 cm was moved around the periphery of the horizontal disc and a level recorder spectrum obtained at every o 10interval. Analysis of these spectra showed that the variation of response with the angle e for each of the Jo-type modes was within the experimental error of ± 2dB.

Having shown the constancy of response with 0 at one radial distance, then because of the symmetry of the system, there is every reason to believe that at other radial distances the angular pressure should also be constant.

Later measurements using waveguides and probes at different angular positions bear out this assertion. dB re. 10pV j

1 I ,

1V?-11 - 50 1----f„,,

jr7TH si• T\ 361 tv,)3

Positions of probe P1: (5,180,5-115) Position of source transducer: (0,0,60), driven at 30V Water Temp: 19.3°C Water Surface Tank Base

0 10 20 30 40 50 60 70 80 90 100 110 120 (cms Depth of Immersion)

FIG.4.3 AXIAL VARIATION OF PRESSURE IN TANK. 03 dB re 10 pV 1

Water temp: 20.7°C Source at (0,0,60), 30 V 100 110 (cm)

Water Depth FIG.4.4 COMPARISON OF RESPONSES FOR PROBES P1, P2 AND P3.

tO 0 91 dB re. 1011V 1

-50

-40

-30 I f 1403 Hz 3247 Hz -20 3280 Hz

Positions of probe Pl: (5-45,180,60) Position of source transducer: (0,0660), driven at 30 V -10 Water temp: 20.2 C

0 5 10 15 20 25 30 35 40 45 (cm.)

t -50

-40

-30 If 2183 Hz 2826 Hz -20 4064 Hz

Probe and transducer positions as above. -10 Cylindrical wall Central axis of tank of tank. (a = 45.5cm.)

I I I I I I

FIG.4.5 RADIAL VARIATION OF ACOUSTIC PRESSURE IN TANK. 92

4.4. Variation of Acoustic Pressure with Position of Driving Transducer: Probe in Fixed Position

4.4.(a) Variation in the z-direction

In section 4.3 the distribution of pressure in the tank was plotted for the source transducer conveniently situated at the tank centre (0, 0°, 60), where most of the prominent low- frequency modes have a pressure maximum. From these plots, and from considerations of the available theory (sections 2.1 and 2.2), we have a clear idea of the pressure variation for specific low- frequency (< 6 KHz) modes.

To investigate how the pressure distributions are affected with source movement the same experimental arrangement was used as for the experiment described in section 4.3(a), except that the probe (P1) was located at (0, 0°, 60) and the source transducer was moved down the line (5, 90°, 5-115). As before, Level Recorder traces were obtained for the transducer at 5 cm intervals and the amplitudes of the resonance peaks for each of the modes of interest were plotted as a function of transducer position. Typical examples are shown in Fig. 4.7.

The graphs clearly show that for each mode there is a sine pressure variation; these and other results reveal that the response for corresponding modes in the two experiments is almost identical. For instance, the distributions for modes at 1403 Hz and 2180 Hz shown for the 'probe variation' in Fig. 4.3 are very closely the same as those for the 'transducer variation' in Fig. 4.7, that is, the two responses are for practical purposes identical for reciprocal positions. (Since for these modes there is no angular pressure variation, (5, 180° 5-115) and (5, 90° 5-115) are exactly equivalent lines). 93

Consider the distribution of pressure for the two modes at 1403 Hz and 2817 Hz shown in Fig. 4.7. Within ± 30 cm from the tank centre (i.e. from z = 30 to z = 90 cm), the trans- ducer excites these modes with approximately constant pressure response, viz. approximately 40 dB and 55 dB respectively. As the transducer is moved nearer the top or the bottom pressure release surfaces, it moves into regions of diminishing pressure for these modes, hence it becomes more difficult to excite them, and consequently they have reduced maxima at the tank centre, where the probe is situated.

For modes such as the one at 2180 Hz there are two add- itional places where the "degree of difficulty" of exciting this mode is a maximum, i.e. where there are naturally pressure minima, roughly at z = 40 and z = 80 cm. When the transducer is situated at z = 20, 60 and 100 cm respectively where there are pressure maxima, a minimum of energy is required to excite the tank into this mode, so there is consequently a maximum of response as measured by the centrally placed probe.

4.4(b) Radial variation o In this experiment the probe was at (0, 0 , 60) as for the z-direction measurements but the source transducer (driven at 30 volts) was moved 5 cm at a time along the line (5-45, 270°, 60). For each position of the transducer a complete frequency spectrum was obtained when the oscillator automatically scanned the range 1-10 KHz.

Variations of amplitude of response as a function of radial distance for distinct low-frequency modes are shown in Fig. 4.8.

In section 4.3(c) it was shown that for these modes the 94

angular variation of pressure is negligible so that angular positions at the same depth and radius are reciprocal positions. Therefore these radial distributions (5-45, 2700, 60) can be directly com- pared with those in Fig. 4.5 (5-45, 180°, 60), obtained with the transducer and probe interchanged.

Once again the locations where there are minima in Fig. 4.8 are where the transducer is at a pressure minimum, thereby driving the tank into resonance with a minimum of efficiency. EVR

(cm) I .

47.0

: 46.5 cif'///////411

I I / 46.0

i (—— 45.5i_,_------II' i'''''''''''''''' '

45.0 . . 120 115 110 105 100 95 90 85 80 75 Water depth

FIG.4.6. VARIATION OF EFFECTIVE VESSEL RADIUS (EVR) WITH WATER DEPTH !dB re. 1011V I I I I I I Hz cf.\ -a""1128 12-= + t/ 'HI f z[27- T 17 Hz .1cc 17

)f\ 2180 Hz -4°1i? ,T -30 zt 1403 Hz

-20'f/ Fixed position of Probe P1: (0,0,60) Positions of source transducer: (5,90,5-115) Water Temp: (19.0 deg.C)

-10

Water Surface Tank Base 1 0 10 20 30 40 50 60 70 80 90 100 110 120 (cms Depth of Immersion) FIG.4.7 ACOUSTIC PRESSURE AT (0,0,60) FOR AXIAL VARIATION OF SOURCE.

1d13 re. 10pV 97 A

5012 Hz A ATh

A - 40 -) - A 0 2176 Hz

- 30 Water temp: 19°C 1 1400 Hz

0 0 I. 0 5 10 15 20 25 30 35 40 45 (cm. )

0 5 10 15 20 25 30 35 40 45(cm.)

- 50

- 40 0 a=45.5 cms.---

0 5 10 15 20 25 30 35 40 45(cm.) FIG.4.8 ACOUSTIC PRESSURE AT (0,0,60) FOR RADIAL VARIATION OF SOURCE POSITION. 98

4.5. Concept of "Effective Vessel Radius"

No measured value of the diameter of the water tank could be taken as accurate because of warps and irregularities made in its construction. Though the lower half of the tank was fairly free of warps the rim was slightly elliptical.

Hence, a method was devised to compute the radius which, though not measurable directly, represented the mean value and was termed the "Effective Vessel Radius" (EVR).

This method utilized equation (1.6) which was rearranged to give "a", 'the radius:

ct2 m,n a — (4.1) 4v2 n2. c2 2,2

The known values at this stage were c, the velocity of sound in water at the pervading temperature; and k, the depth of the water.

An experiment was undertaken with the apparatus shown in Fig. 3.1, using only the pressure probe (in an arbitrary position) as a means of detection. With 80 volts r.m.s. applied to the transducer (also in an Srbitrary position), the fundamental tank mode (101) produced a large amplitude deflection on the Level Recorder trace. This mode could easily be detected audibly and was monitored by the C.R.O. so that the frequency could be accur- ately measured by the digital counter.

Hence, by inserting this value of frequency into equation 99

(4.1), and using a as (2.4048/7) and n = 1 (satisfying 0,1 z boundary conditions for 1st mode), the value of 'a' could be calculated.

In section 4.3 it was shown that the next two detectable modes (1728 Hz and 2180 Hz) are also satisfied by the a solution, 0,1 though the value of nz becomes 2 and 3 respectively. By using these new constants the computed values of 'a' can be double checked.

This procedure was repeated for water depths ranging from 80 cm to 122 cm (the level of the rim) and the table Appendix II shows the resulting frequencies recorded and the computed radii.

As the tank was filled with water from the tap the temp— erature decreased slightly and the corresponding values for the velocity of sound were used in the calculations.(1)On emptying the tank the temperature remained constant with depth.

The program used to compute these radii from the values of temperature, velocity of sound and resonance frequency, was SUBROUTINE RADIUS as shown in Appendix II, and the variation of Effective Vessel Radius against water depth is shown in Fig. 4.6. The graph reveals that as the depth of water was decreased the effective radius rapidly increased, though for depths over 108 cm it was constant to ± 0.1 cm at 45.5 cm. This is considered a very reasonable value for 'a'.

At 80 cm depth the computed radius was almost 47 cm corr— esponding to a diameter of 94 cm, but no measured diameter could be more than about 93 cm.

A possible explanation to this ambiguity is that for depths 100 less than 108 am-(assuming a reasonable experimental error) the resonating volume is appreciably loaded by the walls of the tank above the water surface. In other words, when the tank is full to the rim, the resonating volume is a free cylinder, but for any lesser depth, it becomes a cylinder loaded by the tank walls above the surface, which are not now pressure-release boundaries.

Hence, for most of the experimental work, a water depth of at least 108 cm was used and the radius was taken as 45.5 cm. Using the tank almost full (conveniently at 120 cm) meant that there was a smaller error in the value '2,', since the base also was distorted.

FOOTNOTES

1. A possible source of error in the estimation of 'a' lies in the fact that the values for the velocity of sound were taken for distilled water, according to "Physical and Chemical Constants" by Kaye and Laby. Clearly these values are only approximate for the tap water used. However, the calculated values of the fundamental and several of the lower- order modes have frequencies very close to those measured. 101

CHAPTER 5 USE OF SOLID ROD WAVEGUIDES AS SOUND DETECTORS IN THE EXPERIMENTAL TANK

5.1. Introduction to Experiments with Solid Rods

Having found distinct resonances of the water tank and plotted pressure distributions throughout the volume, it was at this stage possible to substitute for the ceramic pressure probes and use various acoustic waveguides.

In this chapter we present results of experiments em- ploying solid, cylindrical rods as waveguides. The following four sections deal with the identification of waveguide responses and the manner in which these responses change with the positions of the waveguide and sound source in the tank.

Since there will be certain limitations to the size of the practical waveguide for the reactor, some attention was focused on the variation of guide response with length and diam- eter. (At the time of writing provision has been made in the reactor for a 1" diameter stainless steel rod which may be about 30' in length). •

Some experiments were also conducted to investigate the coupling mechanism by which an acoustic pressure field in the water can initiate vibrations in a semi-immersed waveguide. The curved surfaces and the plane lower end-faces were separately isolated from the surrounding water by air-gaps, so that the relative contributions to the resulting vibrations could be assessed. 102

In addition the effect of bending a slender rod was investigated, and in order to obtain a more complete under- standing of this field of research, comparative experiments using rods of different materials, were undertaken. 103

FREQUENCY COUNTER

BEAT FREQUENCY OSCILLATOR

DISC TRANSDUCER WAVEGUIDE ACCELEROMETER

FLEXIBLE DRIVE CABLE

LEVEL RECORDER MICROPHONE 23 04 AMPLIFIER 2603

CATHODE RAY OSCILLOSCOPE

FIG.5.1 APPARATUS FOR DETECTING AND MEASURING WAVEGUIDE RESONANCES.

104

5.2. Direct Measurements of Longitudinal Resonance Frequencies of Solid Cylindrical Waveguides

For reasons which will be discussed later the distur- bances set up in a waveguide semi-immersed in a reverberant sound field are predominantly longitudinal in nature. Therefore it is convenient at this stage to examine the measured and cal- culated longitudinal resonance frequencies of the various wave- guides used, together with the frequencies at the '3 dB points', i.e. at those frequencies above and below the resonance value at which the r.m.s. amplitude falls by a factor 1/12.

From these measurements and from the values of the mass of the waveguide and its length and radius, the following par- ameters were calculated by the program SUBROUTINE YMOD (see Appendix III):

(±) The corrected resonance frequencies v for the system o (waveguide accelerometer). (FRQ2 in YMOD).

(ii) The corrected resonance frequencies vr for the waveguide alone. (FRQ1 in YMOD).

(iii) The corrected frequencies v and v of the 3 dB points 1 2 for the system (waveguide accelerometer). (FREQ1 and FREQ2 in YMOD).

(iv) The corrected frequencies v , v of the 3 dB points 4 5 for the waveguide alone. (FREQ4 and FREQ5 in YMOD).

(v) The 'Q' of each longitudinal resonance of the waveguide alone, from v (Ql in YMOD). r - V 5 4 105

(vi) The 'Q' of each longitudinal resonance of the system (waveguide + accelerometer) from: v o Q 2 v - v 2 1 (Q2 in YMOD).

(vii) The velocity of sound in the waveguide for each resonance and from the average velocity for a number of resonances Young's modulus, given by the "bar velocity" vo E/p. (AVBAR in YMOD).

It may at first seem curious that the parameter Q2 for the waveguide and accelerometer together should be important, but measured values of the Q of any waveguide disturbance with the waveguide in the reverberant tank, have been obtained with the use of an accelerometer.

By computing both Q1 and Q2 the effect of including the accelerometer in the vibrating system can be estimated. Both the accelerometer and the transducer load the waveguide in such a way as to decrease the resonance frequencies. A resonance frequency correction for the addition of a small mass on to a vibrating system of mass M has been given by Massey (1967):

v = v(1 + m/M) (4.1) 0 where v is the corrected frequency and v is the measured 0 frequency.

Thus, the corrected resonance frequencies v for the 0 system (waveguide + accelerometer) becomes:

v = v(1 + m /M) (4.2) 0 2 where m is the mass of the transducer used to drive the waveguide 2 106

4

KHz) 10 15

(a) Stainless Steel (W4)

I; I •11 .it

I

• , •

(b) Copper (W8)

1 A 11 ki 4 )-Prjj 1 I

(c) Dural umin (W14)

FIG.5.2 SPECTRA FOR LONGITUDINAL ROD RESONANCES 107 in the experiment.

The corrected resonance frequencies vr for the waveguide alone are similarly given by the expression:

v = v(1 + m /M + m2/M) (4.3) r 1 where m is the mass of the accelerometer. 1

For detecting the waveguide resonances the apparatus depicted in Fig. 5.1 was used. The oscillator was driven by the cable drive from the level recorder gearbox through its entire range up to 20 KHz. In this way the disc transducer (3.175 cm diam x 1.2 cm thickness; 150 KHz 1st resonance) excited all the longitudinal waveguide resonances in this range. The waveform detected by the accelerometer was amplified by the 2603 amplifier and passed to the level recorder which produced a trace (Fig. 5.2) showing the longitudinal resonances. To measure the resonance frequencies accurately the amplified waveform was displayed on the screen of the oscilloscope and the oscillator adjusted at each resonance so that the observed waveform had a maximum displacement, corresponding to a maximum acceleration of the end of the waveguide. Then the frequency was read from the counter.

Measurements of the frequencies at the '3 dB points' requires some explanation.

In Fig. 5.3 the corrected frequencies v and v from which 1 2 the 'Q' of the mode is calculated are those where the peak inten- sity level falls by a factor of two, i.e. where E /E = 2. Hence 0 1 from the relationship: 108

dB(intensity) = 10 log E /E 10 0 1 the peak intensity falls by 3 dB at v and v . 1 2

2

FIG.5.3 THE Q OF A MODE.

Since intensity (power) is proportional to the square of voltage then from the expression: dB (volts) = 20 log V /V 10 0 1 the peak voltage falls by 6 .dB at v and v . 1 2

Since E /E = V2/V2 = 2 0 1 0 1 then (V )r.m.s. = '2 (V )r.m.s. 0 1

These voltages are r.m.s. voltages as measured on the level recorder but measurements of v and v obtained for signal 1 2 voltages 6 dB below the peak value were found to be totally unreliable. This was partly due to the level recorder stylus sticking, and partly because of the recorder's limited resolution.. 109

Instead of using the recorder, the oscilloscope was used to monitor the received waveform. The relation between the peak-to-peak voltage (V ) measured on the oscilloscope PP and the r.m.s. voltage (V ) measured on the calibrated r.m.s level recorder is:

V = 2 2 . V PP r.m.s.

Since (V ) 1 r.m.s. • (1701/2)r.m.s.

- (V ) 2 r.m.s.

(V //2) r.m.s. = V /4 PP

Therefore the amplitude of the waveform on the C.R.O. at the frequencies v and v should be a quarter of that at the 1 2 resonance frequencies v.

The tables in Appendex III show the relevant parameters discussed above, for each waveguide used. 110

5.3. Identification of Peaks on Spectra for Semi-Immersed Waveguides

In the previous section we have measured the longitudinal resonances of waveguides by driving them at one end with a disc transducer and detecting the vibrations at the other end with an accelerometer. These measured values have been corrected to obtain the resonance frequencies v of the system (waveguide o + accelerometer), which was the vibrating system used in most experiments. Thus if in experiments with waveguides immersed in the water tank the pressure field excited only the disturbances at each longitudinal resonance frequency, then the peaks on a level recorder spectrum would only occur at values of vo. However, this was not so.

Fig. 5.5 shows typical frequency spectra obtained with the probe P1 and with a stainless steel waveguide W11. The position of the probes and the lower end of the waveguide was the same in each case. It is clear that the predominant peaks in the waveguide spectrum are also present in the probe spectrum.

In all experiments of this kind it was found that the disturbances activated in all the waveguides used occurred at tank resonance frequencies.. As we have demonstrated there is a reverberant field at each of these resonances (or modes). Thus we may conclude that the effect of these reverberant pressure fields is to 'force' the waveguide into vibration, irrespective of its natural resonances.

rn 111 r-. GNI 0 cry CV O 0 Ln Ln 0 N- O 00 N -4- C•4 ,--icoN100 CNI cal -1* in

40 i ! 1 1 1 I I il 1 1 1 1, 111111 1 1,1•Iiiii.', 0= 30 I, 1 III 1 1 1 iii. I 1_1_ li, li Ili 1 11111,1.1 i - .1l w 20 $.1 1 : III 1 vi 1• it 10 ,.

2 5 10 (KHz)

FIG.5.4(a) SPECTRUM FOR PROBE P1 (5,180°,60)

Ln tn N- 0 CV 0 0 r•-• CV CV 01 (ft r•-• CO

40

8 30

p 20

10

1 2 5 10 (KHz)

FIG.5.4(b) SPECTRUM FOR WAVEGUIDE W11 (5,180°,60) 112

5.4. Variation of Waveguide Response with Position of Waveguide: Driving Transducer in Fixed Position

5.4(a) Variation in the z-direction

Now that we can identify the peaks on the "waveguide spectrum" as the maxima of forced vibrations initiated by the pressure fields of the tank modes, we require to know if and how the magnitude of these forced vibrations varies with the position of the waveguide.

It would not be unreasonable to assume that the deeper the waveguide is immersed into the water, the greater should be the coupled disturbance, and hence the greater the amplitude of the received waveform, because of the increasing coupling area. In other words, as the waveguide is lowered to the bottom of the tank the recorded signal for a given tank mode should increase, since the sound energy is coupled to an increasing area. But this logical argument is entirely wrong.

In an experiment to substantiate this assertion a I" diam- eter stainless steel rod of length 136.5 cm, (W11) was immersed 5 cm at a time down the line (5, 180°, 5-115) with the source transducer at (0, 0°, 60) driven at 30 volts through the frequency range 1-10 KHz at each 5 cm interval.

The variation in peak amplitude of waveguide response with depth of immersion for three discrete tank modes is represented in Figs. 5.5, 5.6, 5.7, together with the pressure variation down the same line, obtained with probe P1. The pressure variation, as we have seen previously, is a sine variation, and despite certain perturbations, which are for the most part within the experimental error of ± 2 dB, the waveguide variation is unquestionably the same 113 function. The amplitudes are different from those of the probe response, though not necessarily smaller, as will be demonstrated - later.

It is important to realize that the co-ordinates on the plots for the waveguide represent positions of the lower end of the waveguide. Hence, the end of an immersed waveguide can be regarded as a form of probe which "sees" a variation of pressure, irrespective of the coupling areas that the curved side present to the pressure field.

Verification of this discovery has been achieved by using also waveguides of various materials and sizes. All z-direction distributions were found to be characterized by a sine function varying primarily according to the pressure magnitude at the end of the waveguide. For the purpose of simplicity and for continuity in the presentation of these results, these distributions will not be included here but in later sections.

The discontinuities for both the probe and the waveguide distributions in Figs. 5.5, 5.6, 5.7 tend very closely, though not exactly, to the theoretical pressure minima in accordance with the sine function. It has already been explained that the "squashing" of the pressure field towards the base of the tank is considered to be due to the different efficiencies of the top and bottom pressure- release surfaces.

5.4(b) Radial Variation

As would be expected from the observations on the z-variation of waveform response, the radial response also varies according to the magnitude of pressure at the end of the waveguide. It has been shown that the radial pressure distribution depends on a particular 114

Bessel function, so we can expect the radial waveguide response to vary accordingly.

Fig. 5.8(a) illustrates responses for the probe P1 and for the same waveguide as used in the z-direction study (W11). o With 30 volts acrosss the centrally-situated (0, 0 , 60) source transducer, the waveguide was moved diametrically and frequency spectra were obtained on the level recorder at each 5 cm interval. For the tank mode at 3280 Hz, the response along the line (r, 0°, 60) was found to be so similar to that along (r, 180°, 60) that it can be regarded as the same. Thus, corresponding positions along each line can be described as "reciprocal"; and for this mode the response along other radii appears to vary in exactly the same way.

The variation for a mode with 2 discontinuities is bhown in Fig. 5.8(b) for a stainless steel waveguide (W4). For this experiment the source was driven at 60 volts at (0, 0°, 60) and the end of the waveguide was traversed 5 cm at a time along the line (r, 0°, 60). dB re. lOpV

3280 Hz, Probe P1 -40

—30

-20 3280 Hz, Waveguide (W11) Accelerometer: 9.0 mV/g Water temp 19.3°C -10 Line of immersion (5,180,5-115) Source transducer position (0,0,60) driven at 30 Volts. Waveguide Stainless steel rod,.136.5 cm. long,1"(0.635cm.) diam.

I I_J I I I I I I I I I I I I I I I 1 I I I I 0 10 20 30 40 50 60 70 80 90 . 100 110 120 (ems.)

FIG.5.5 .Z-VARIATION OF PROBE.AND WAVEGUIDE RESPONSE. Ui dB re. 10pV r

-50

4060 Hz, Probe P1 -40

-30

-20 4060 Hz, Waveguide (W11)

Accelerometer: 9.0 mV/g -10 Water temp: 19.3 deg.0 Line of immersion: (5,180,5-115) Source transducer position: (0,0,60) driven at 30 Volts. Waveguide: Stainless steel rod, 136.5 cm. long,r(0.635cm.) dia. II I I I I I I lilt II it I

0 10 20 30 40 50 60 70 80 90 100 110 120 (nms.)

FIG.5.6 Z-VARIATION OF PROBE AND WAVEGUIDE RESPONSE. 1-4 rn dB re. 10pV

I I I I I I I I I I I I I I I I I I I I I I I

N\

-50( 5204 Hz, Probe _ -40 47 -30

-20 5210 Hz, Waveguide (W11)

Accelerometer: 9.0 mV/g -10 Water temp: 19.3 deg.0 Line of immersion: (5,180,5-115) Source transducer position: (0,0,60) driven at 30 Volts Waveguide: Stainless steel rod, 136.5 cm. long, in(0.635cm.) diam. I

10 20 30 40 50 60 70 80 90 100 110 120 (cms.)

FIG.5.7 Z -VARIATION OF PROBE AND WAVEGUIDE RESPONSE.

idB re. 1011V 118

sqN

3280 Hz, Probe P1 50 Accelerometer: 9.0 mV/g Waveguide: Stainless steel rod, (W11) 1 136.5 cm. long, 0.635 cm. diam. 40 Source transducer position: (0,0,60), 30 V. Water temp: 19.5°C

'N 3280 Hz, Waveguide 30

fik(} -20 (a)

o --- 0=0° 0=180 --- I I I •1 I 1 I I 1 I I 40 30 20 10 0 10 20 30 40(cm.)

-50

-40 Water temp: 20.50C Waveguide: Stainless steel rod, 136.5 cm. long, 1.27cm. dia. Accelerometer: 8.5 mV/g -30 Transducer (source) posn. as above driven at 60V

1°H (b) 4323 Hz, Waveguide (W4) _0=0° 0.180°--- IIIII I I I I 1 1 40 30 20 10 0 10 20 30 40(cm.)

FIG.5.8 RADIAL VARIATION OF PROBE AND WAVEGUIDE RESPONSE. 119

5.5. Variation of Waveguide Response with Position of Source Transducer: Waveguide in Fixed Position

5.5(a) Variation in the z-direction

This experiment was an exact parallel to the one described in the section 4.5(a), in which the pressure probe was replaced by a stainless steel waveguide. It has become apparent that the end of a waveguide can be regarded as a type of pressure probe so the results expected were similar in each case.

The apparatus used was as depicted in Fig. 3.1 with the lower end of the waveguide rod (W11) in a constant position (0, o 0 , 60). As in section 4.5(a) the source transducer was located at 5 cm intervals down the line (5, 90°, 5-115) and a recorder spectrum (1-10 KHz range) obtained at each location.

The entire experiment was then repeated with the waveguide replaced by probe P1 for subsequent calibration purposes. (It was desirable to correlate the waveguide acceleration with the pressure initiating it; the accelerometer was calibrated at 9 mV/g and the probe P1 was calibrated from another probe in terms of pV/pbar.

Figs. 5.9, 5.10, 5.11, 5.12 show the results for several types of mode. For each, the waveguide response is lower than the probe response by (roughly) a constant amount (except for the mode at 7169 Hz, in the multimodal region). By comparison with Figs 5.5, 5.6, 5.7 for the waveguide variation the source stationary, the present figures are correspondingly identical within the exper- imental error (± 2 dB).

For all tank frequencies the z-variation of pressure is according to the sine function, with regular maxima and minima. 120

It has already been argued that when the source transducer is located in a natural pressure maximum of the standing wave pattern for a discrete tank mode, the tank is driven with the system at its least impedance (or with the maximum efficiency): when at a natural pressure minimum, the tank is driven least efficiently. Hence, as the source is moved from a natural maximum to a minimum, the pressure that the waveguide "sees" falls off (according to the sine function). A careful study of Figs. 5.9, 5.10, 5.11, 5.12 reveals that this reasoning holds for each mode.

5.5(b) Radial variation

Having shown that a waveguide could be used to measure z-variations (sine) of pressure (when calibrated), and also how such measurements were dependent on source transducer position - phenomena hiterto unknown - it remained to verify that radial waveguide responses were similarly dependent on the position of the source transducer.

Fig. 5.13 represents the variation of waveguide response at two tank modes as the source transducer was traversed along the line (5-45, 270°, 60) with the lower end of the waveguide W11 at (0, 0°, 60).

Although the radial line chosen was different from that in the experiment for the radial waveguide response with transducer stationary (in section 5.3, this was (5-45, 180°, 60) - a 'reciprocal line'), the results for corresponding modes were found to be the same. This is precisely the result predicted from z-direction observations. The results of further experiments, not recorded here, also displayed these characteristics.

It can be concluded that "reciprocal" positions of waveguide 121

and source, or probe and source, yield the same response. Simply, this means, for instance, that if we know the response for the o probe (or waveguide) at position (r , z) and with the source 1 1 1 at (r , , z ), then if the source and probe (or waveguide) are 2 2 2 exactly interchanged, the two responses will be equal.

dB re. 10pV

('1 3273 Hz, Probe P1 -50

-40

-30 C r f /r \ 7/ NN\

f\ 3273 Hz, Waveguide W11 -20 I - Source transducer line: (5,90,5-115), at 30 V. Water temp: 18.5°C. -10 Accelerometer: 9.0 mV/g.

I I I I 1 I 1 I i I I I I I I 1 I I I I 1 I I 0 10 20 30 40 50 60 70 80 90 100 110 120 (cm.)

FIG.5.9 WAVEGUIDE AND PROBE RESPONSE AT (0,0,60) FOR AXIAL VARIATION OF SOURCE POSITION.

1 dB re. 1011V

• f/ •‘j' -50

4058 Hz, Probe P1 -40

-30

-20

4058 Hz, Waveguide W11

- 10 Source transducer line: (5,90,5-115), driven at 30 V. Water temp: 18.5°C. Accelerometer: 9.0 mV/g. I I I I

10 20 30 40 50 60 70 80 90 100 110 120 (cm.)

FIG.5.10 WAVEGUIDE AND PROBE RESPONSE AT,(0,0,60) FOR AXIAL VARIATION OF SOURCE POSITION. dB re. 10iV

4620 Hz, Probe P1

-40

-30

(1\ „cr.? -20 I Vr4620 Hz, Waveguide W11

- 10 Source transducer: line (5,90,5-115), driven at 30 V. Water temp: 18.50C. Accelerometer: 9.0 mV/g.

II I I I I I I I I I I I I 1

10 20 30 40 50 60 70 80 90 100 110 120 (cm.)

FIG.5.11 WAVEGUIDE AND PROBE RESPONSE AT (0,0,60) FOR AXIAL VARIATION OF SOURCE POSITION. dB re. 10uV I I I 1 i i

It

-50 7169 Hz, Probe P1

7169 Hz, Waveguide W11 -40 MK 7. \

Source transducer at (5,90,5-115), 30 V.

0 10 20 30 40 50 60 70 80 90 100 110 120 (cm.)

FIG.5.12 WAVEGUIDE AND PROBE RESPONSE AT (0,0,60) FOR AXIAL VARIATION OF SOURCE POSITION.

126 1dB re. 10pV

- 50 „,---i 1 - ••,,i -40 3264 Hz, Probe P1

-30 f„,,,, 4 f

- 20 3264 Hz, Waveguide Wll

Source transducer: line (5-45,270,60), at 30 V. - 10 Water temp: 19°C. Accelerometer: 9.0 mV/g a=45.5 cms.

I I I I 1 10 20 25 30 35 40 45 5 15 (cm)

5012 Hz, Probe P1 N _ 4 -30

- 20 5012 Hz, Waveguide Wll -

Same details as above. _ 10

Cylindrical axis of tank Tank wall

FIG.5.13 WAVEGUIDE & PROBE RESPONSE AT (0,0,60) FOR RADIAL VARIATION OF SOURCE POSITION. 127

5.6. Significance of Rod Dimensions

5.6(a) Rods of the same length but different diameters

At this stage we know how one particular waveguide (stain- less steel rod W11 136.5 cm long x 0.635 cm diam) behaves when located at given positions, and with the source transducer also at known positions. It would be obviously of interest to know if and how this behaviour changes for waveguides of different dimen- sions, and of different materials.

For an investigation into the relative responses of wave- guides of the same material, having the same lengths but different diameters (and hence different surface areas), the following wave- guides were used: Stainless steel rods, length 136.5 cm, diameters; 81", 1", i, ll in, in,

In order to use the rods under the same conditions, each was traversed spearately along the same line (5-45, 180°, 60), and frequency spectra obtained at 5 cm intervals as in previous radial experiments. The source transducer, at (0, 0°, 60) was driven at 30 volts for one series of recordings and 60 volts for another. Though for each mode the actual pressure was 6 dB higher at all points as a result of this voltage doubling, the corresponding waveguide responses bore a more complex relation, since two different acceler- ometers were used, having calibrations 9 mV/g in the 30 volt experiment and 8.5 mV/g in the 60 volt experiment.

It was considered more convenient to take radial, rather than z-direction, measurements since few recordings were necessary, and the experiments could be completed more quickly: if we know the radial response variation for a given waveguide we can predict roughly the z-response from knowledge, gained in previous experiments, 128 of the structure of the modes of interest.

Fig. 5.14 showsthe responses of five waveguides and the pressure distribution obtained with a probe for two modes, when the source was driven at 30 volts. For the sake of clarity the ± 2 dB error lines have been omitted from the points representing the waveguide responses. It is quite clear from these preliminary results that for a given position in the tank, the responses of the five waveguides are very closely the same, and in nearly all cases within the experimental error.

Figs. 5.15, 5.16 show the results obtained when the whole experiment was repeated with 60 volts across the source transducer, i.e. an increase of 6 dB. Once again most of the response levels, taken for the six waveguides separately at any one position, are within the experimental error. There is a slight indication that the larger-diameter waveguides produce greater responses than the smaller diameter waveguides, notably at the frequencies 2180, 4070 and 4380 Hz. But this observation is by no means true at all the tank modes, in particular at the mode occurring at 3270 Hz. For all the waveguides except the one of 1" diameter the responses are remarkably similar for corresponding locations, indicating that the diameter of the waveguide is of little consequence. How- ever, for the 1" diameter rod the response is quite distinctly (about) 10 dB higher than for all the other rods; it varies accord- ing to the same Bessel function. This result was so contrary to expectation that an experiment to investigate the z-variation in response, was undertaken to check it. Fig. 5.17(b) shows that the response for the 1" guide is also about 10 dB above the axial responses for all the others. It may be argued that this is merely a freak result, but if this were so, similarly high responses would probably have been obtained at the other tank modes also. At present 129

this result cannot be fully explained. Clearly the mass of the accelerometer would have a more pronounced effect the smaller the diameter of the rod.

Figs. 5.17, 5.18 represent z-variations of response obtained using the six waveguides at 5 cm intervals down the line (0, 0°, 0-120) with the source transducer at (10, 90°, 60). Once again, as with the radial distribution, most of the 6 responses for a given location, are roughly within the limits of the exper- imental error.

Thus the waveguides measure the Bessel-distribution of pressure radially , though this function is somewhat difficult to recognise, and the sine distribution in the z-direction. The overall response of each waveguide is approximately the same despite differences in both cross-sectional area and curved-surface area. It is worthy of mention that for these low-frequency modes there is not likely to be a pressure gradient across the endface of even the largest-diameter rod. So the end-pressure, which appears to be the most significant pressure contribution acting on a semi-immersed rod, can be regarded as having the same magnitude for each of the six different diameters.

The curves in Figs. 5.15 and 5.16 display a distinct "edge effect", characterized by a significant increase in response when the rods were moved close to the tank wall. A plausable explan- ation of this effect is that it is caused by a rod being influenced by the actual vibration of the tank wall. It is noteworthy that in general the smaller-diameter rods could approach much closer than the larger-diameter rods before exhibiting an increase of response, which suggests a possible "build-up" of pressure between rod and tank wall. 130

For the tank mode at 3590 Hz which appears to be of the J type, 1 having a minimum at the tank axis, there was an increase of res- ponse as the rods were brought close to the source transducer. Since the source sphere expanded and contracted, thereby emitting spherical waves, it is reasonable to assert that it was this move- ment which influenced the rods.

From Figs. 5.17 and 5.18 it can be seen that the result of standing the rods on the bottom of the tank was to produce an abrupt increase of response: in this case they were driven directly by the vibration of the tank base.

5.6(b) Dependence on rod length

The variation of response with length for a semi-immersed waveguide was studied using a stainless steel rod of in diameter. The original length was 258.5 cm.

To begin with the lower end of the rod was placed at pos- ition (10, 180°, 60), the source at (0, 0°, 60), and a frequency spectrum in the range (1-10 KHz) was obtained in the usual way. This procedure was also repeated for the rod at (10, 180°, 20).

It was necessary to cut the rod length to 255 cm by very careful machining in a lathe, before carrying out the above pro- cedure again. The original plan was to cut the rod by 5 cm for each subsequent run, but this would have been too time-consuming. Instead, from 250 cm the length was shortened 10 cm at a time.

Figs. 5.19, 5.20 show how the response varies for several tank resonance frequencies. For certain lengths of the rod, the response can be seen to increase greatly, the maxima occurring for longitudinal resonant lengths. In other words, the pressure 131 fields of the various tank modes excite the rod into vibration but the excitation increases to a maximum if the rod's length is resonant at a tank resonance frequency. Hence in such cir— cumstance the rod and the tank resonate "in sympathy".

Though only one type of rod was used there is every reason to believe that similar results would be obtained with elastic rods of other materials: the only significant difference would be a change of resonance frequencies, owing to changes in both Young's modulus and density.

The significance of waveguide length will again be dis— cussed in the conclusions (Chapter 7).

132 1 dB re. 10101 (9.0 mV/g)

-50

3250 Hz, Probe P1 -40 0_ - 0 Measurements at (5-45,180,60) •--• -.. ...,,,....0 • 3250 Hz, Waveguides -30 •...... • N.0x A -- 1+-- - -- 1--• - -+---0 — A • -I- 20 Diam. dB x 3 35,34,32,26,27,26,25,26,26 511 + 8 35,34,31,25,25,24,26,24,24 -10 • iu 1. - ,31,28,24,- ,25,25,23,22 O 2 n 8 36,36,32,26,27,27,27,25,24 • l4 u 35,34,31,25,26,26,27,26,24 1 I I I 1 I 1 5 10 15 20 25 30 35 40(cm.)4

-50 4635 Hz, Probe P1

_ 40

4635 Hz, Waveguides (5-45,180,60) - -30 x x +-O - -x _+,- -• - + -20 Diam. dB x 3fl 29,27,- ,29,30,28,- ,26,24 4 28,27,27,29,29,24,25,- ,- + 85n • iu -" ,— ,- ,28 ,29,26,- ,- ,- -10 2 3fl 27,- ,- ,29,28,25,- ,- ,- O g In 28,- ,27,29,29,26,- ,- ,- A

FIG.5.14 RESPONSES FOR RODS OF SAME LENGTH, DIFFERENT DIAMETERS.

133 dB re 10pV (8.5 mV/1)

X xA X X X x -20 + 0 - 4 it x-.... x — • v • 1 x x x 7 11------: x x v—______e'---..._ v v ---a + + v v -10 2180 Hz

5 10 15 20 25 30 35 40 (cm)

3270 Hz -40

-30

-20

5 10 15 20 25 30 35 40 (cm)

• • • • 's+ _-x----x..., - 30 ..... +______.,x...... \. .:;..,------+.....4. -... .• sx e,f 4., X '_.5!..------a------_ 0 3590 Hz

X N--I 1 - 20 Nx 0 5 10 15 20 25 30 35 40 (cm)I A 1 1 I

FIG.5.15 RADIAL RESPONSES. FOR RODS OF SAME LENGTH, DIFFERENT DIAMETERS. 134 dB re 100 (8.5 mV/g)

4070 Hz 30 A .•\ •

-20 ,V

5 10 15 20 25 30 35 40 (cm).-1 1

4620 Hz

V X

/7 • -20 V

5 10 15 20 25 30 35 40 (cm)

FIG.5.16 RADIAL RESPONSES FOR RODS OF SAME LENGTH, DIFFERENT DIAMETERS.

2180 Hz

A 0 X o X o x A A x X X ___ A A o A A A -20 A _...... ,A X--__...... -R------___ o A------X a A X 0 0 A x ---...„...k ../— 0 0 x-,,,... ----- .5.______0 ---A-., v....-"---7-----V------V------...17...... _ v---...... v V ------**--v~ v v (a) v ------v

t I I I I I I I i I I I I I I I 1 I I 1 dB re 1011V (8.5 mV/g)

3276 Hz

v---...„ ✓ V*--..._ -v V.,._,,,v -40 VN\ V/';'----. v \v ve t + x 0 A+ I. o A I ' 1 A X 0 4 O O 1 + -30 A 1 1 X X + O A 1 0 O 0 (b) o 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 1 1 [ 1 1 1 1 10 20 30 40 50 60 70 80 90 100 110 120 (cm)

FIG.5.17 Z -VARIATION OF RESPONSE FOR RODS OF SAME LENGTH, DIFFERENT DIAMETERS.

dB re 10 diV 1(8.51 mV/) (

X A „...... ,--e----...,R A_...... -30 //t ....,, a V v ...------T X/f Z 4- A. v 1 di V /X -20 4060 Hz a

(CM) 1 I I I I 1 1 I 1 1 I 10 20 30 40 50 60 70 80 90 100 110 120 cm)

-30 •

• 6 A -20 4620 Hz

lit I

FIG.5.18 Z-VARIATION OF RESPONSE FOR RODS OF SAME LENGTH, DIFFERENT DIAMETERS..

dB re 10 pV 137

1 I I I 1 I 1 1 1 1 I 1 1 1 1 I I I 1 (a)

2183 -50 t (11) 4073

- 40

- 30

-20

(cm)

1 I 1 1 1 i 1 1 I 1 1 1 1 i 1 1 1111111'11 20 40 60 80 100 120 140 160 180 200 220 240 260 280 111(1[1111111 li1 1 111 1 1 1 1 - '1'1' (b) (cm) fl) 2828 -50 3288

,,, -40 -( \i- /

- 30

-20

III

FIG.5.19 VARIATION OF RESPONSE WITH WAVEGUIDE LENGTH 20 40 60 80 100 120 140 160 180 200 220 240 260 280

FIG.5.20 VARIATION OF RESPONSE WITH WAVEGUIDE LENGTH 139

5.7. Investigation of Energy Transfer Mechanism between Acoustic Medium and Waveguide

5.7(a) Contribution of plane end-face of a rod

A study of many previous graphs, both for radial and axial pressure distributions, measured by means of a semi-immersed rod, indicates that the accelerometer signal resulting from a disturbance in the rod, depends much more on the pressure acting on the lower end-face than that on the curved sides. This is most apparent when considering that for a probe and the end of a rod traversed along the same line, the responses obtained were roughly parallel. If the curved sides provided the major con- tribution to the waveguide disturbance, the disturbance would increase with the length of the immersed part of the rod.

In furtherance of our knowledge on this subject an exper- iment was devised so that the relative contributions which the end-face and the curved sides make to the overall waveguide disturbance, could be estimat ed. This section deals with the end-face contribution.

The rod (W4) used was of stainless steel, dimensions: (136.5 cm long x 1.27 cm (1") diam).

Since it was necessary to ensure that only the end-face of the rod came into contact with the water in the tank, the total curved area was isolated from the water by means of a dural tube and a rubber bung as shown in Fig. 3.6. The rubber bung fitted tightly between the base of the tube and the rod in order to keep out the water. In this arrangement the rod was surrounded by a medium (air) of very low characteristic 140

impedance (41 c.g.s.) compared with that of the acoustic medium (1) (water, 1.5 x 105 c.g.s.). Thus the pressure field surrounding the tube was unlikely to affect the rod along its curved sides because of the acoustic mis-match. The reason for choosing a rubber seal between the rod and the tube was because the rubber, being very compliant, was unlikely to transmit to the rod, any waves that might be excited in the tube.

Now, with only the end-face of the rod in the reverber- ant pressure field, the system was traversed radially and in the z-direction as before so that response curves could be obtained. Recordings were obtained for radial positions along the lines (5-45, 0°, 60) and (5-45, 180°, 60) in the frequency range 1-10 KHz and for z-varying positions along (0, 0°, 15-120), with the source transducer, driven at 60 volts, situated at (10, 90°, 60).

Recordings were also obtained with the tube removed so that for each tank mode a direct comparison could be made between responses with and without the tube. Some of these response curves are represented in Figs. 5.21, 5.22, 5.23. In nearly every case the response for the end-pressure contribution to the rod disturbance is distinctly greater than the response for the unshielded rod, which suggests that the pressure field acts on the curved area in such a way as to oppose the pressure acting on the end-face.

There is quite good correlation between the z- and radial responses depicted in Figs. 5.21, 5.22, 5.23, though not for the mode at 4400 Hz. The z-variation, however, does show clearly the characteristic improvement in response when using the shielded rod . There are no positions with common co-ordinates in the z- and radial distributions though there are two reciprocal positions 141 for cross-correlation:

o o z-variation, (10, 0 , 60) with source at (10, 90 , 60) radial variation, (10, 180°, 60) with source at (0, 0°, 60)

The best agreement is at the tank frequency 4070 Hz.

5.7(b) Contribution of cylindrical boundary of rod

The obvious requirement for this experiment was to exclude the lower end-face of the rod from the water. To do this a short piece of 1" diameter rubber tubing, together with a disc-shaped piece of rubber, was glued with i Durofix' to the rod to form an "air-gap". (See Fig. 5.24). By means of this simple device, an acoustic mis-match was created between the reverberant field

Waveguide

FIG.5.24 "AIR-GAP"

Rubber in the water and the end-face of the rod, without loading the rod significantly.

The results for z-variations of response are displayed in Figs. 5.25, 5.26. The tendency is for the presence of the air-gap to produce a reduction in the "normal" wavegude response (approximately 5 dB for four of the modes). What is also sig- nificant is that as the rod is lowered into the water the increase of curved surface area subjected to the pressure field, does not 142

appear to bring about a change of response. Hence, the z- variation no longer bears any resemblance to a sine function.

143 dB re 10 pV (9.0 mV/g) I I 1 T

1-'25 ------

15 2180 Hz

I I 1 1 1 8=0 Z=60 8=180

-40 A/4 -35

r 25 ..... j____§...... i, ,' 3287 Hz (cm) § . 5L -f , , 1+.---- , 1 i I I , (r) 40 30 30 20 10 0 10 20 30 40 1 I I I 1 I I ; I ...----4L...., . (cm) 30 474 ?\,1 -_ 25 / ___.4 i -s k. 1 _\...,,i, 2820 Hz \ A / 1 f , , 1 , I I I I 8=0 z=60 0=180

I

FIG.5.21 EFFECT OF PRESSURE ON END FACE OF ROD (RADIAL VARIATION) 144 la re 10 IN (9.0 mV/g) r

-35

L 20 4070 Hz

0=0 Z=60 0=180

waveguide alone - 25 § - 20A/1/

4400 Hz

I I Mit I

e=o Z=60 0=180

-30 /- -25 i' 4' '''-n's\ "'''f fr -+- 4642 Hz

1 I

FIG.5.22 EFFECT OF PRESSURE ON END FACE OF ROD (RADIAL VARIATION)

la re 10 pV (9.0 mV/g) r A (' -25

-20

-15 2180 Hz I I (z) 10 20 30 40 50 60 , 70 80 90 100 110

(cm) -35

-30 (i)" -4, -25 / 4070 Hz

I I I I (:)' I 1 1 10 20 30 40 50 60 70 80 90 100 110

30 .25

-20 • ,§

FIG.5.23 EFFECT OF PRESSURE ON LOWER END FACE OF ROD (AXIAL RESPONSE) dB re 10 pV (9.0 mV/g)

3280 Hz ., . -60 . --- ±,... ..

2180 Hz -50 • ,i

-40

3280 Hz -30

2180 -20 - 1 f t

- 10

(z) 10 20 30 40 50 60 70 80 90 100 110 (cm) II I I I I I I I I I I I I I I I I I I I I I .

-PN FIG.5.25 EFFECT OF SPATIALLY VARYING PRESSURE ALONG ROD (Z-VARIATION) rn idB re 10 UV (9.0 mV/g)

o

-50

4063 Hz -40

-30

-20

-10

(z) 10 20 30 40 50 60 70 80 90 100 110 II I I I I t i

FIG.5.26 EFFECT OF SPATIALLY VARYING PRESSURE ALONG ROD (Z-VARIATION) 148

5.8. Effect of Bending a Waveguide

In the PFR it may be necessary to bend the practical waveguide in order to locate it in the required position. Therefo'i.e it was relevant to attempt an experiment to invest- igate the variation of waveguide response with the degree of bending.

The waveguide (W13) chosen was a stainless steel cylind- rical rod of dimensions: length, 136.3 cm; diameter, I" (0.3175 cm). The apparatus depicted in Fig. 3.1 was used with the lower end of the waveguide in one of the two positions: o o (10, 180 , 60) or (5, 180 , 60), and the source transducer at (0, 0°, 60) driven at 30 volts.

Frequency spectra were obtained on the level recorder in the usual way, first with the rod straight, then with it bent through angles 5°, 10°, 15°, 20°, etc. up to 100°, at an arbit- rary position above the water surface. The bend was such that the radius of curvature was large compared with the rod diameter, thereby avoiding distinct "kinks" with accompanying changes in cross-sectional shape. In this way it was thought that vibra- tions could "negotiate" the smooth curve in the rod with little attenuation.

The responses for the various tank modes do not appear to follow any particular pattern, so it is difficult to interpret these results, shown in Fig. 5.27. The modes at 3290 Hz and 5044 Hz exhibit a general fall-off of response with angle of bending, indicating progressive attenuation, though for the latter there is an increase between 5° and 10°. In contrast, the modes at 2180 Hz, 4402 Hz and 4642 Hz exhibit responses that are roughly 149

constant for angles up to 100°.

For practical purposes it can be considered that for a rod bent in this way the spectrum does not change significantly from that obtained for a similar straight rod. However, for very abrupt "kinks" (a sharp 90° bend was made in a similar i" rod) the attenuation of the responses for most of the low-frequency modes was so great as to cut them out completely. Thus the rod was effectively a high-pass filter. 150 la re 10 pV (8.5 mV/_)

-40' 3290 Hz 1.1 foi (5, 180°6 60) a 1/1000 I (10, 180 , 60) A N -30 2180 Hz

I,- 20 —14 I (Degrees of arc)

4076 Hz

-30

N

1 1 1 I I I 10 20 30 40 50 60 7? 90 1 i (1)

A25 4404 Hz 0 I 1 I I I 1 I I (Degrees of arc) I I I I I I I I

a = 1/625 5044 Hz A 440

A

4642 Hz }-30 f

'17 I I I

FIG.5.27 RESPONSE AS A FUNCTION OF DEGREE OF BENDING OF AN I" DIAMETER STAINLESS STEEL ROD 151

5.9. Comparison of Responses for Similar Waveguides of Stain- less Steel, Copper, Brass and Duralumin

In nearly all waveguide experiments so far, stainless steel rods have been used because of their suitability for use in the reactor. But it would also be of interest to have some knowledge of the responses characteristic of other elastic rods, even though they may not meet the practical requirements.

The rods chosen in addition to stainless steel, were cylinders of brass, copper and duralumin, each with the dimensions: length, 150 cm; diameter, 1" (1.27 cm), and radial variation of response was obtained in exactly the same manner as in section 5.3. Results obtained for these four waveguide rods are shown in Fig. 5.28. The figures clearly show that no one rod had a better response than the other three at all resonance frequencies of the tank. A possible explanation may be that for each rod the velocity of wave propagation is different, resulting in a different series of resonance frequencies. The tables in Appendix III show measured and corrected resonance frequencies of the rods (W6, W7, W8, W9), and from a careful study of these it is apparent that the best response in each figure corresponds to the rod which had a longitudinal resonance nearest to the tank frequency under consideration. Some of these resonances are very close together, notably those near the tank frequencies 2249, 3373, 4628 and 4902 Hz.

Another feature of the results, not recorded here, is that at certain tank modes the response for some rods was signif- icantly high whilst for others there was no detectable response. 152 dB re 10 pV (8.5 mV/g)

(t, Brass -30 ti 2180 Hz Duralumin Copper

20 T------1

T-4-4 Stainless Steel: 1" I i" 1 (r) 5 10 15 20 25 30 35 40 (cm) I I

3290 Hz

-50 Stainless Steel

Duralumin -40

-30

-20

(r) 10 25 30 35 40

-40 4628 Hz Duralumin NNN\I NN.

-30 Bras s - / A

Copper 1//'/1 -20 {

FIG.5.28 RESPONSES FOR SIMILAR WAVEGUIDES OF DIFFERENT MATERIALS 153

5.10. Dependence of Probe and Waveguide Response on Power of Source

This section deals with the linearity of the detecting system. It would be expected that if the voltage applied to the source exciting resonances in the experimental tank was increased by (say) 10 dB, then the detected waveform amplitudes would increase also by 10 dB.

Before investigating the response of an immersed waveguide with change of source transducer voltage, the response of a dir- ectly driven rod was considered. Using the apparatus illustrated in Fig. 5.1, the disc-shaped transducer was used to drive a series of stainless steel rods in the frequency range 1-10 KHz, at a range of driving voltages. The analysis of the resulting level recorder traces shows (Fig. 5.29) that for each longitudinal res- onance, the variation of received signal bears a linear relationship with the variation of transducer voltage.

Fig. 5.30 shows typical variations of probe and waveguide response for the source transducer voltage varying from 1 volt to 100 volts r.m.s. 154

dB re 10pV 70 wz. r12- 90 60 35 314 14. 50 0 1161 0 40 A3) 30

0 20

0 10

10 100 (volts)

FIG.5.29 DEPENDENCE ON APPLIED VOLTAGE FOR DIRECTLY-DRIVEN ROD.

dB re 10pV 70 40- 60 322v-Th z. ?I 7I$I V-be •T.----1-:::J)------50 'LZI4 sz,111. ------404°66

4b----- 30 0 21.46 , (la4 I - / 'd 20 - - - \wieV11 - - ..-.- _ -. - - -.- _ - _ - 10 - -_,-- --

10 100 (volts)

FIG.5.30 PROBE AND WAVEGUIDE RESPONSE AS A FUNCTION OF SOURCE POWER. 155

5.11. Effect of Variation of the Angle of Immersion

In section 5.3 we have discussed the response of a wave- guide which is moved from one location to another in the tank. Observations showed that the response was primarily a function of the pressure existing at the lower end of the waveguide, since the pressure existing at the lower end of the waveguide varied according to the functions governing the configuration of the sound field. For example, in the z-direction the waveguide was found to have a sine response which was parallel to the corres- ponding pressure probe response thereby showing no dependence on the length immersed.

To verify that the length immersed does not modify sig- nificantly the response, a further experiment was carried out in which the end of a in diameter stainless steel rod (W11) was located at a constant position as the rod was tilted. In this way, the immersed length was increased but, for any given mode, the lower end remained subject to a constant pressure. Fig. 5.31 shows that for all the resonance frequencies except one (3570 Hz) the response remains constant within the experimental error of ± 2 dB.

For the case where the end of the rod was at position (0, 0°, 60) the immersed length could be increased by only 15 cm before the tilted rod touched the tank, but with the end of the rod at (0, 0°, 20) this immersed length could be increased by up to 25 cm.

With a rod immersed vertically in the tank's sound field the response obtained at any frequency is critically dependent on the rod length: for a longitudinal resonant length the response is a maximum. 156

When the rod is inclined in the sound field the con- figuration of the spatially-varying acoustic pressure along its length is such that the rod may be subject to flexure. However, there was no direct evidence for forced flexural vibrations in this experiment. As for the case of the vertical rod the res- ponse at each tank frequency was observed to be a -function of the pressure at the lower end-face.

It is considered that for the inclined rod the sound field is unlikely to excite strong forced flexural vibrations because the system is not sufficiently asymmetiical. The accel- erometer was sensitive only to vibrations in the axial- or z- direction and may not have been sensitive to flexural vibrations, which are characterized by particle displacements that are functions of r, 8, and z. This is because the particle displacements may be such that tension on one side of the neutral axis and compression on the other, cancel out. 157

1dB re 10 pV IdB re 10 pV (Hz) (Hz) 40 3285 35 -3570 3285 -3650 3570 2180 30 3650 25 J 2180 65 70 25 30 35 40 (cm)

(Length of Immersion)

(Hz) (Hz) 35 4070 43 25 30 - , -4395 -4640 -4640 ---- 4395 -2825 4070 25 4325 2825 20

65 70 25 30 35 40 (cm)

FIG.5.31. VARIATION OF RESPONSE WITH LENGTH OF IMMERSION OF WAVEGUIDE: LOWER END IN FIXED POSITION 158

FOOTNOTES

1. Approximate values for Characteristic Impedance (pc) at 18-21°C: 5 (i) stainless steel = 39.5 x 10 c.g.s. 5 (ii) brass = 29.0 x 10 c.g.s. 5 (iii) copper = 30.0 x 10 c.g.s. 5 (iv) duralumin = 13..8 x 10 c.g.s. 5 (v) water = 1.5 x 10 c.g.s. (vi) air = 41 c.g.s. 2. The values for Q1 and Q2 would normally differ but here they are the same since they have been computed from corr- ected values in the same ratios. The difficulty was that all the parameters for a particular waveguide were calculated from one set of measurements. Hence the computation of Q2 in the program was really only a check on Ql. 159

CHAPTER 6 FURTHER QUALITATIVE EXPERIMENTS

6.1. Introduction

In Chapter 5 we have investigated in detail the behaviour of solid cylindrical rods as acoustic waveguides in the tank. In this chapter we consider more complex types of waveguide, but most of the results presented are of a qualitative nature.

Firstly, comparisons were made between the responses of a solid rod and a tube with and without an end-piece, each of the same material and with the same overall dimensions.

In a second experiment finned tubes, normally used in reactor heat exchangers because of their increased conducting area over conventional plain tubes, were used as waveguides. The app- lication of this study lies in the possible use of existing tubes in the reactor as waveguides, without the need to introduce a specially- designed waveguide. Doubtless there will be disadvantages to this proposal but the study is nevertheless worthy of a little attention. In this brief investigation, an attempt was made to estimate the effect of removing the finning.

It was also considered relevant to analyse the waveform from an accelerometer rigidly attached to the wall of the tank, and make comparisons with waveguide responses.

Lastly, a few observations were made with the most complex system used during this project, that of solid copper spiral waveguide. 160

6.2. Comparison of Responses for Rods and Tubes of the same length and material

The aim of this experiment was to find out if there was any distinct advantage in using a tube in the place of a rod as a detector of sound in the tank. A rod and a tube of duralumin, each 150 cm long and 2.54 cm (standard 1") diameter, were used to obtain radial response distributions along the lines (5-45, 180°, 60) and (5-45, 180°, 100), recordings being obtained at 5 cm intervals.

The tube, with a wall-thickness of 0.17 cm, was used both with and without an end-piece: end-piece was stuck into the lower end of the tube with "Durofix", as shown in Fig. 6.1, after the tube had been machined slightly shorter in order to retain the same overall length of 150 cm.

FIG.6.1

The experiment was carried out as in all previous sound detection experiments, using the apparatus depicted in Fig. 3.1, and was divided into three parts: (1) The lower end of the 1" dural rod was traversed along the radial lines (5-45, 180°, 60) and (5-45, 180°, 100); 161

(2) Part (1) was repeated using the 1" dural tube, with the lower end open; (3) With an end-piece of dural rigidly stuck into the tube to exclude the water, the procedure was repeated a second- time.

A qualitative analysis of results revealed the following generalities:

(i) For all tank modes considered the responses of the rod and the tube (with or without the end-piece) had a direct dependence on the pressure at their lower ends. This was evident from the radial response distributions which were Bessel functions whose shape for each mode was parallel (see Fig. 6.2) to the distribution obtained by measuring the pressure with a probe.

(ii) At all tank modes at which the 1" solid rod was forced into vibration, the response was less than that for the tube (whether with or without the lower end-piece). This appeared to be true even when there was a resonance frequency of the rod closer than one of the tube to the tank modal frequency under consideration.

(iii) (a) If the pressure p at z and the pressure p at z 60 60 100 100 were the same, the response at z for the closed tube (A ).would be 60 100 greater, but only marginally so, than the response at z 100 (A100 ). (See Fig. 6.1). i.e. if p = p , A > A . This may be 60 100 60 100 because at z there was less pressure on the curved side opposing 60 the pressure on the end-piece. (b) If the pressure at z (p ) was greater than the pressure 60 60 at z (p ), then the response was greater at z than at z , 100 100 60 100 i.e. if p > p , A > A . 60 100 60 100 162

(iv) For all modes except that at 4385 Hz, the closed tube response at z was as good or better-than that of the open-ended 60 tube, i.e. (A > ) 60)closed 60 open

(v) For all modes except those at 4625 Hz and 2180 Hz, the open tube response at z was better than that of the closed tube, 100 i.e. (A ) > ) 100 open 100 closed'

Whilst observations in (i) and (ii) were well substantiated, those in (iii), (iv) and particularly (v) were not. The responses of the open and closed tubes were, at some tank modes, so close as to be for practical purposes equal (see Fig. 6.2). This suggests that their resonance frequencies were very close but this could not be verified experimentally because only the resonances of the closed tube were measured.

The important point to emphasise is that the tube response did not alter significantly when used first without, then with, an end-piece. A further point is that the well-defined Bessel distribution indicates, as for the rods used in Chapter 5, that the lower end of an open or closed tube acts as the detector part of the waveguide. Despite the additional "collecting" area obtained by the use of the end-piece the resulting vibrations must still be transmitted up the tube wall. The strain produced in the tube wall by the end-pressure may be modified by that produced by the spatially-varying pressure along the curved walls. It must also be considered that when the water is inside the tube the pressure at any depth acts on both the inside and outside of the wall, thereby tending to oppose the end-generated strain more. 163 idB re 10pV (8.5 mV/g)

- 60

- 50

- 40

- 30

3275 Hz 20 z = 100, 0 = 180 5 10 15 20 25 30 35 40

Radial distance (cm)

- 60

1" Tube - 50

40

- 30 1" Rod

3275 Hz - 20 z = 60, 0 = 180 5 10 15 20 25 30 35 40 I I I I I I I I

FIG.6.2 RADIAL RESPONSES OF DURAL ROD AND DURAL TUBE. 164

6.3. Response of Finned Tube Waveguides

The "Estuco" tubes used in this experiment were developed for their application in British Nuclear Power Station boilers to give a weld as strong as the tube metal with a good thermal junction between fin and tube. The fin in the form of strip was helically wound and high frequency resistance welded to a plain tube. (See Fig. 6.3)

FIG.6.3

The experiment consisted of using a piece of this finned tube as a waveguide in the tank, and comparing the response with that of a similar tube with the finning removed. An end-piece of steel was rigidly fixed with "Durofix" into the upper end of each of these tubes as a base for the accelerometer.

The dimensions of the finned tube were as follows: overall length of tube, 150 cm; outside tube diameter, 2.2. cm; inside tube diameter, 1.5 cm; standard fin thickness, 0.0889 cm; fin height, 1.27 cm; number of fins per inch, 7. 165

Four types of waveguide were used: (i) Fins, no lower end-piece (ii) Fins, lower end-piece to exclude water (iii)No fins, no lower end-piece (iv) No fins, lower end-piece

In four separate experiments these waveguides were trav- ersed along the line (5-45, 180°, 60).

The tube without fins was of the same length (150 cm) as the dural tube used in the previous experiment but because of its different resonance frequencies, its responses could not be expected to be the same. For all tank modes considered the response of the dural tube was significantly better (up to 30 dB) at all points than that of the steel tube.

However, the object of the present investigation was to compare the responses of (i), (ii), (iii), (iv). The results of response (dB) as a function of radial distance of the lower end of each waveguide showed no consistency, except for the very general observation that the presence of the lower end-piece in cases (ii) and (iv) produced a better response than in (i) and (iii). (See Fig. 6.4).

The Bessel distributions typical of radial responses for the rods and tubes previously used was not nearly so well defined for the finned tube. For some modes of the tank the predicted dis- continuities were only just discernible below the general response level. With the fins there was a greatly increased coupling area between the pressure field and the waveguide but it is thought that this is unlikely to produce a significant strain in opposition to that caused by the end-pressure. However, the velocity of 166 propagation of longitudinal waves v'(E/p) may well be modified as compared with those along the plain cylinder. The effect can be approximately considerered as a loading of the medium, i.e. increase in density (and also damping), without significant change of elastic response. In consequence, the velocity will become appreciably less. 167 dB re 10 UV (8.5 mV/g)

-50 f Fins, no end piece (ii)A Fins, end-piece (iii)0 No Fins, no end-piece (iv)0 No Fins, end-piece -40

-30

_20 1

_10 0 = 180° 2827 Hz (cm) Z = 60 I 5 10 15 20 25 30 35 40

(cm)

-50

-40

-30

-20

-10 0 = 180° 4070 Hz Z = 60 1

FIG.6.4. RESPONSE OF "ESTUCO" FINNED TUBES 168

6.4. Use of an Accelerometer to Investigate Tank Wall Vibration

Of the three methods of sound detection mentioned in Chapter 3, two of these, probes and elastic waveguides, have- received considerable attention. For completeness we should include a brief discussion on detection of tank wall displacement, in order to have some idea of how this response compares with that of the semi-immersed waveguides.

In order to make direct comparisons between the vibrations initiated in a semi-immersed waveguide by a sound source, and the vibration of the tank walls, it was necessary to use the same detector in each case. This detector was an "Acos" accelerometer of sensitivity 8.5 mV /g peak, which was mounted on the tank r.m.s. wall with "Durofix". Figs. 6.5(a),(b) show the "wall response" for the accelerometer mounted at a position given by 0 = 180° and z = 60 cm. At such a position, half way between the top and the bottom of the tank, the displacement of the tank wall should be of greater amplitude than, for instance, near the base, since at the base-wall junction the tank structure is more rigid and there- fore resistive to vibration.

Fig. 6.5(c) shows the "waveguide response" for a 1" diameter stainless steel rod of length 150 cm with its lower end at position (10, 180° 60), when for Jo-type modes the pressure approaches a maximum, thereby driving the rod into large-amplitude vibrations.

It is clear that in each spectrum the dominant peaks repre- sent the natural modes of the tank, but the amplitudes of some of the corresponding peaks are of different orders of magnitude.

Below 10 KHz the "wall response" is for almost all the 169 detectable tank modes greater than the waveguide response, except for that at 3287 Hz. At this frequency, which is close to a longitudinal resonance frequency of the vibrating system (rod + accelerometer) the peak corresponds to a level of approximately 5 mV r.m.s., which represents an acceleration of the end of the rod of 577 cm sec-2 i.e. 0.59g.

Between 10 KHz and 20 KHz the general level of response for both rod and tank wall is about the same.

In addition to the main peaks of the wall response there are minor peaks, barely resolvable in the recorded spectrum, which may be the result of coupling of vibrations in the water to vib- rations in the tank wall. At the frequency range used in this project, these vibrations of the tank wall are likely to be flex- ural or circumferential. However, wall vibrations characterized by expansion and contraction of the wall thickness may also occur but at much higher frequencies (> 394 KHz) than those in the "unimodal" frequency range.

We can conclude from these and previous observations that for waveguides of lengths "tuned" to particular modes of the tank, the waveguide is a more efficient detector since in this state the waveguide and tank resonate in sympathy.

When the waveguide is not tuned to a. tank mode the wall vibrations of the tank can produce larger outputs than the wave- guide vibrations, but this will depend on the siting of the accel- erometer, and in the general case, on the rigidity of the tank structure.

170 V V 0p 10p 10

(a) e e dB r dB r

50 411 40 V 30

(b) 100p e 20 B r d 10

2 5 10 (KHz)

c0 ce) V V 10p (c) 100p

e e r dB r dB

FIG.6.5 COMPARISON OF TANK WALL RESPONSE AND WAVEGUIDE RESPONSE. 171

6.5. Spiral Waveguide Characteristics

6.5(a) Experimental Procedure

The use of a spiral waveguide in the place of a straight rod or tube effectively opens up a completely new realm of study. For this reason, therefore, we cannot at this time attempt to analyse any quantitative measurements. Only a minimal amount of work has been done on propagation of waves in spirals, and in order to make comparisons with the present work some of this will be discussed later.

We have shown quite conclusively that the forced vibrations observed in semi-immersed rods or tubes are caused by the acoustic field configurations in the tank. Therefore an elastic waveguide of almost any geometry could be expected to be forced into vibration in this way.

The method used to investigate the behaviour of a spiral was of a comparitive nature and designed only to provide a very approximate indication of response. This method involved the use of a piece of 1" diameter straight copper rod, 3 metres in length, which was first used as a waveguide in the normal way, then coiled into a spiral and used with the lower end at the same location as before. Figs. 6.7(a), (b), (c) show the resulting frequency spectra. Despite the noise level, the spectrum for the straight rod (a) does contain peaks representing most of the prominent low-frequency tank modes.

The spiral, which comprised 10 turns, had dimensions which were approximately as follows: length of rod forming spiral (0, 210 cm; length of spiral, 13 cm; mean spiral diameter (2R), 6.5 cm.

172

With the axis of this spiral horizontal so that one straight arm was at right angles to the other arm as depicted in Fig. 6.6(a), the resulting spectrum, Fig. 6.7(b) was found to exhibit low fre- quency cut-off. This configuration of the spiral appeared therefore to act as a high pass filter. In the range 13-20 KHz the responses of the horizontal disposed spiral and the original straight rod were found to be very similar.

(a) (b)

FIG.6.6

When the straight arms of the spiral were bent as in Fig. 6.6(b) so that they became in line, the spectrum obtained was found to contain peaks at the frequencies of the lower-order tank modes, including one for the third detectable mode at 2180 Hz. It is quite clear from this spectrum that the general response above about 6.5 KHz for the vertical spiral is higher than that for the straight rod. During this recording (Fig. 6.7(c)) it was necessary to reduce the 2603 amplifier gain in order to keep the trace on the paper, otherwise no response would have been visible above 17 KHz.

6.5(b) Discussion

The observations noted in this experiment could be subjected 173

to a good deal of debate so no clear-cut conclusions can be reached.

We must recall that the accelerometer used was sensitive only to vibrations in its z-direction, e.g. longitudinal vibrations in a waveguide. We have argued elsewhere that this type of detector may not be sensitive to flexural vibrations of a waveguide owing to equal and opposite vibration amplitude about the neutral axis. This flexure would, in any case, occur at a tank resonance fre- quency as a result of an asymmetrical pressure field about the waveguide axis. The pressure field surrounding a straight slender rod only becomes significantly asymmetrical when the rod is inclined but for a spiral whose axis and arms are vertical, the individual coils are inclined according to the pitch angle (a), given by: 1 a = tan (p/2TrR), where p is the pitch (distance between adjacent coils). Thus flexure is, in theory possible.

We have noticed in previous sections that the response of a waveguide is critically dependent on the acoustic pressure exist- ing at its lower end, and that the spatially-varying pressure along the waveguide influences the overall response. Therefore, the tendency of the pressure at the immersed end of the horizontal*plane, thereby producing torsion in the vertical arm. This is a plausible explanation for the low-frequency cut-off. But there is another possibility: if the overall effect of the "curved-side pressure" is to oppose that of the "end-pressure" (see Fig. 5.21-23), a reduction of longitudinal vibration amplitude would result. There is also the possibility that longitudinal vibration attenuation could in part be due to mode conversion in the spiral: energy in a longitudinal mode of the spiral rod could, as a result of the cur- vature, be converted into energy of torsion on flexure.

In contrast, the tendency of the pressure at the immersed spiral s Pr. . K,..(0 CA.) , would be t-0 vi brace sp)ra ).,1 Viorizort

174 V 10p

dB re

(a) 3m straight copper rod (10,0°60)

1 2 5 10 (KH ) 50 0

40

V 30 10p

e 20 B r d 10

0 (b) spiral, axis horizontal (10,0° 60) V V 100p 10p e e r dB dB r

F O

(c) spiral, axis vertical (10,0°60)

FIG.6.7 SPIRAL WAVEGUIDE RESPONSE 175 end of the vertical spiral, Fig. 6.6(b), would be to compress the coils together. It appears that this could be responsible for the improved response over that of the horizontal spiral despite abrupt "kinks" where the straight arms meet the spiral.

It is interesting to conjecture why the vertical spiral response is better than that of the original straight rod, not- withstanding the mechanisms of attenuation already discussed. A possible reason could be that the spiral as a whole, in contra- distinction to the rod from which it is formed, is also forced into vibration.

A spiral can be considered as an extension of the case of a simple smooth curve. Experiments using a bent rod (Section 5.8) indicated that any change in response for a particular tank mode was due to attenuation at the bend. For a spiral with a very large diameter compared with the rod diameter, the bend per cm length would be small and we could expect that any attenuation of forced vibrations as a result of curvature would be small. However, for a smaller diameter spiral this attenuation could be expected to increase.

Other authors have studied propagation in spirals using far less complex "driving sources" than the present one, which is believed to be novel. Filipczynski (1962), using a magnetostrictive transmitting probe head (in contrast to our pressure field source), investigated attenuation and dispersion in spirals both with a resonance method and with a pulse method. He had noted in previous work that for a straight rod which was progressively bent, the amplitude of reflected pulses increased with decreasing radius of curvature of the rod, as in Fig. 6.8. The ratio a/A rod radius to wavelength, was 1/20. This result is similar to two of the

176

curves in Fig. 5.27 for the response of a bent rod semi-immersed in the experimental tank, where the abscissa represents "degree of bending", which is easier to measure than radius of curvature. At these tank resonance frequencies 3290 Hz and 5044 Hz the values of a/A were respectively 1/1000 and 1/625.

Filipczynski derived a simple expression relating the group velocity of the waves in a spiral, and frequency, using the wave number k, where:

k = [(w2R2n/E) - 1]i

where R is the radius of the spiral.

He then used the relation v = R(w/k) to determine the "circumferential phase velocity of the wave". v 2 v = v [ 1 / ( -(- 2.) ) o

where v o = (E/p)i.

The "circumferential group velocity" was introduced by the relation vg = R.dw/dk giving the group velocity:

2 1 v [1 o wR

If the radius of the spiral tends to infinity, the wave velocity tends to vo, the "bar" velocity of waves propagating in a straight rod withdispersion. If w decreases, there is a decrease in the group velocity (and a corresponding increase in phase velocity) which becomes zero at a critical frequency v : c

V = V/27111 c Below this frequency there is no propagation as can be seen in

-Ou

177

Fig. 6.9. The critical frequency vc for the copper spiral was 87.1 KHz.

Filipczynski's resonance method has a parallel with the present experiment, in which forced sinusoidal vibrations were observed, since he also used a sinusoidal signal. Furthermore he used a steel spiral made from rod 0.3 cm diameter whereas here we have used a copper rod of similar diameter, 0.3175 cm. (The spiral radii were respectively 2.3 cm and 6.5 cm). With this method he was able to measure the group velocity of waves in the spiral and consequently demonstrate its dispersive properties.

It is pertinent to record some of Filipczynbki's theory because it predicted accurately his experimental results. He calculated the velocity of wave propagation in the spiral by meas- uring two adjacent resonance frequencies of the system, knowing the length (2.) of the spiral rod.

The condition for a resonance to be established is that: n.v nA p 2, = - 2 2v where n is an integer.

If the next resonance occurs when the frequency is increased by Lv then the following relationship is true: dv p • Ay X' 1,_ + dv 2, = (n + 1 1- = (n + 1) Y 2 2(v + Av)

v increasing because the spiral is dispersive. P Thus,

1/ [(1/v ) - (v/v2)(dv /dv) = ( n >> 1) P 178 where n >> 1 Since v = R.dw/dk, then

dv/v ) dv 1 dk p v v dw. dv dv g p

v = 22..Av g

The results obtained were as shown in Fig. 6.9 clearly displaying the dispersive effect of the spiral compared with the absence of dispersion (in this frequency range) for the straight rod.

More recent work on propagation in spirals has been carried out by Langley (1968), who propagated pulses in various modes by siting transducers and receiving strain gauges as shown in Fig. 6.10. He too compared responses of straight rods and spirals with par- ticular reference to the effects of curvature on the flexural mode of propagation.

The work of Filipczynski and particularly that of Langley, who compared his results with the theory of Timoshenko (1921), serves to emphasise the complexity of the underlying theory of the spiral immersed in water and subjected to a sound field. R (cm) mm 179

30 R 20

10

0 1 2 3 4 5 FIG.6.8 AMPLITUDE OF REFLECTED PULSE AS A FUNCTION OF RADIUS OF CURVATURE OF ROD. -1 xln5

V • 0 • • • • - • Straight rod g • 0 • • • 0 • 0 0 • o o Spiral 4 o • 0 • 3 •

2 •

1

0 20 40 60 80 100 120 140 160 180 (KHz) FIG.6.9 DISPERSION IN A SPIRAL. 2 3 Pick-up for 1,2 Pick-up for 3

Pick-up for 3

Pick-up for 1,2

FIG.6.10 SITING OF TRANSDUCERS. 180

CHAPTER 7 INTERPRETATION OF RESULTS AND GENERAL CONCLUSIONS

7.1. Forced Vibrations in a Semi-Immersed Waveguide

7.1(a) Discussion of the Problem

Before considering the theoretical aspects of the work it will be helpful to discuss the actual mechanism of the problem. To recapitulate, we have a sound source which emits waves of sufficient amplitude to excite the natural modes of vibration of a water-filled cylindrical tank. For simplicity we have chosen as a sound source a spherical transducer emitting con- tinuous sinusoidal waves into the water.

Experiments have revealed that at the frequencies of these natural (or normal) modes, a disturbance is induced in a semi-immersed elastic waveguide. In contradistinction a disturbance does not appear at each longitudinal resonance fre- quency of the waveguide, unless these coincide with a natural frequency of the tank. To clarify this further, a disturbance will be induced in the waveguide (irrespective of its length) by a tank mode, but if the waveguide length synchronises with a natural frequency of the tank, the 'forced' disturbance will be a maximum.

If the frequency of the driving transducer is held steady at a tank frequency, the maximum amplitude of the disturbance (or forced vibration) remains unchanged owing to the static pressure field set up. The displacements of the ends of the waveguide resulting from this disturbance are not easily measurable. 181

However, acceleration, which is proportional to the square of frequency, can be more easily measured and so is preferred to displacement or velocity measurement. In its application an accelerometer was attached to the upper end of the wave- guide and produced a sinusoidal output in response to the driving force. In order to maintain this (observed) sinusoidal output, energy from the reverberant field must be continually supplied to the waveguide to overcome frictional resistance.

7.1(b) Mechanism of Driving Force

White (1958), in his theoretical paper on the coupling of spherical waves in a liquid to an immersed waveguide (see Chapter 1), assumed that the "coupling factor" was the same at all points along the guide. It is very unlikely that the coupling efficiency is the same at all points in the present case, owing to the spatially-varying pressure existing at the normal modes of the tank. The fact that the accelerometer output was observed to be sinusoidal shows that the waveguide is subjected to a time-varying (periodic) driving force. However, since the pressure amplitude varies with position, the force on the waveguide is effectively a variation of static pressure over its cylindrical surface and at its lower end-face. The experiments have indicated that the predominant driving force originates at this end-face.

It is reasonable to suppose that the periodic driving force acting on the waveguide is a function (*) of the total pressure P to which the waveguide is subjected, and the total t immersed area A. When 'tuned' by adjustment of the oscillator frequency to a particular resonance of the tank this driving 182 force will remain steady (i.e. the peak amplitudes will not decay) since the pressure-field is reverberant. If at a tank resonance the source frequency is v, then waves of frequency v are coupled to the waveguide. These excite vibrations resulting in strains which perturb the free ends of the waveguide with frequency v also.

Thus we can say that the driving force is some function of total acoustic pressure, total area, frequency and time:

F = tp(Pt, A, v, t) (7.1)

We now consider a cylindrical waveguide (for conven- ience a solid rod) divided into elementary lengths in such a way that the pressure over the curved surface of each res- ulting sub-cylinder is constant with respect to both z and 0. Because of the distinct pressure variation existing at each tank mode the elementary lengths will not necessarily be the same, and they will be subjected to different pressures. Suppose these uniform pressures are 41, 42, 43 .... acting on the respective elementary areas 6A1, (SA2, SA3 .... resulting from the division of the cylindrical waveguide into elementary lengths 621, 62.2, 62,3 .... Fig. 7.1 shows an 'exploded' diagram of these elementary cylinders.

For small-diameter waveguides (a << X) it can reasonably be assumed that the 0-dependent pressure gradient around each of these sub-cylinders is zero, though this assumption may not be valid for very large-diameter guides. Even for the largest- diameter waveguides used in this work (2 .54 cm) this condition is true except possibly where the end of the waveguide is situated close to a pressure minimum.

183

If we suppose that the pressure po sin 2irvt across the lower end face of the waveguide is uniform at any instant, then the force acting over the cross-sectional area is: p0ra2 sin 2Trvt.

In the simplest configuration of the pressure field surrounding the rod, in which there is only particle motion parallel to the rod axis, the total force at some instant t can be written as the sum of the end-face and curved-face contributions, i.e.

n F(t) = p ra2 sin 2irvt + I fdp . SA . sin(2rvt + o r r=1 r (7.2) where 4 r is the phase angle between force on the end-face and force on the curved sides of each element.

It is important to realize that the responses due to each part of this force equation do not necessarily add to give an overall response. This has been well-substantiated by experiments described in section 5.7 in which the response due to pressure acting on the lower end-face only was found to be greater than that due to the summation of the r.m.s. pressures acting over the end-face and over the curved surface.

Sp i S ----Sp 1 1 1

Sp 6, ---dp FIG.7.1 2 I 2 2

Sp d ,Q, ---- Sp 3 1 3 3 184

7.1(c) Damping

Any loss of vibrational energy would be primarily due to frictional resistance, and the rate of loss is a measure of the 'damping capacity' or 'internal friction' of the rod material. Several methods of measurement have been listed by Stephens (1959) who included 'measurement of some response of a system, e.g. a rod specimen, when in forced oscillation under a periodic force whose frequency is varied'. The method used to obtain a value for the damping capacity in this case was to measure the amplitude response at the frequency of a tank mode (v ) and at the 3 dB-point frequencies (v1,v2) as r shown in Fig. 5.3. The damping capacity is then given in terms of the logarithmic decrement,

r where is the Q, A = Tr(v2-v1)/v vr/02-vd or Quality factor, of the resonance. Thus A = IT/Q.

From the table in Appendix III showing the Q's for tank modes the logarithmic decrement can be obtained. At the fre- quency (3285 ± 1)Hz the stainless steel waveguide has a Q of 219, whereas the copper waveguide has a Q of 142.8. It is a matter of common observation that the damping in copper is higher than that in stainless steel.

7.1(d) Equation of motion for forced vibrations

We can attempt to compare the present theory with the classical theory of forced vibrations if we restrict our attention to the case when the end-face of the rod only is exposed to the sound field. In this case the periodic driving force should be simply F sin wt and the equation of motion is therefore:

MX + RX + Sx = F sin wt 185- where M is the equivalent mass of the waveguide; S is the elastic restoring force per unit displacement; and R is the frictional resistance per unit velocity.

The complete solution to the equation of motion, as elaborated by Stephens and Bate (1966) is:

_ R R2 ) x T exp ym- . t) sin - 4,42 . t - al \M

+ Q sin wt - tan 1 . w (174 - w2q] (7.3)

This result, when applied to the waveguide problem, shows that the action of- the pressure field is to excite two harmonic motions. The wave represented by the first term decays expon- entially leaving a steady-state simple harmonic motion of constant amplitude and with the frequency of the driving force, that is, the source frequency v. Thus, the equation can be written simply as:

x = F sin(wt - (P)/wZ (7.4)

The particle velocity is then given by:

= F cos(wt - (1))/Z and hence,

Z = F cos(wt - 4))/X = Stress/Particle Velocity (7.5) where Z = i[(S/w - Mw)2+ R2 ] , which is the mechanical impedance of the waveguide at the angular frequency w.

Skudrzyk (1958) also obtained this result in a study of the vibrations of a system with a finite or an infinite number of resonances. The study was based on the general differential 186

equation (7.3) of a mechanical system, and dissipation was taken into account by introducing complex elastic constants. The motion of a system with many resonances, such as the waveguide, was given in terms of the velocity distribution excited by a force of total magnitude F:. 1 x = F E - — (7.6) n Rn + iwMn + 1/iwK

where w is the natural frequency of the nth mode, and K is the n compliance of the system (1/S).

7.1(e) Rate of energy supply to the waveguide

If the curved surface of the waveguide is shielded from the pressure field, the work done by the driving force on the waveguide in moving it through a displacement dx is:

dW = (F sin wt) . dx (7.7)

and if this displacement occurs in a time dt then, dx dW = F sin wt • dt . dt

= F sin wt• . cos(wt - (I)) . dt (7.8) Z

The average rate of energy supply per cycle is given by: F2 dW = y- . sin wt . cos(wt - 4,) (7.9)

which, after mathematical manipulation, can be derived to be: 2 at = R ) (7.10) which is the acoustic power factor analogous to cos (I) in the electrical case (and where R = Z sin 4,). This can also be written in terms of the driving frequency v, and the amplitude (X) of motion, i.e. maximum displacement (x). From eqn(7.4), 187

X = F/wZ, when sin (wt - (p) = 1. Hence,

dW = 1 R (42 dt

Stephens and Bate derived an identical expression for the kinetic energy of the driven system (i.e. the waveguide in this case). This indicates that the amplitude and phase of the system are such that the energy lost per cycle by overcoming frictional resistance is recompensed exactly by energy supplied per cycle by the driving force. Hence the greater the frictional resistance, the greater the system is damped and correspondingly more energy has to be supplied per cycle to maintain constant amplitude of vibration. At resonance energy is absorbed in establishing a maximum amplitude of displacement. Results of the experiment described in section 4.11(b) using a stainless steel rod to investigate the variation of response with rod length, clearly show that at each longitudinal resonance the response increases enormously.

Since the sinusoidal waves in the reactor experiment were continuous, the phase of the waveguide displacement with respect to the driving force is of little importance. All measurements were taken at distinct resonance frequencies of the tank, at which the longitudinal waveguide response for any given length was a maximum. Thus, no analysis was done "off" resonance where the discrete pressure field configurations broke down and the corres- ponding waveguide response was greatly reduced. Only in a very few instances were a tank frequency and a waveguide resonance frequency co-incident; this naturally produced a much larger disp- lacement in the waveguide than when it was driven "off resonance", (see Fig.6.5(c)). Hence the important parameter is the magnitude of the acceleration, since this can be related to the pressure in the calibration procedure. 188

7.2. Correlation of Acoustic Pressure and 'Waveguide Acceleration'

From the practical point of view, one of the most import- ant aspects of this work is to be able to relate the observed acceleration of a waveguide to the magnitude of the pressure that forces the waveguide into oscillation.

Because of the complexity of the coupling mechanism of acoustic energy to a semi-immersed elastic waveguide, it would be difficult to estimate the total pressure producing a given acceleration. However, since the response depends on the pressure at the lower end-face, we can correlate acceleration with this "end-pressure".

The principle of this correlation technique in practice was to measure the response of a pressure probe at known co-ordinates of the tank and compare these with the responses of a waveguide whose lower end was situated at the same relative co-ordinates.

The probes used were calibrated against a PZT-4 cylindrical probe (P3) which was fabricated at the National Physical Laboratory and had a sensitivity of 1pV/pbar (across 3200 pF) at a level of -124 dB re 1 volt, as explained in section 3.8(d).

The accelerometers used were pre-calibrated by the man- ufacturers and had sensitivities of either 8.5 or 9.0 mV /g r.m.s. peak' Thus with acceleration and pressure in terms of voltage the two parameters could be correlated, and a waveguide could be calibrated.

It was found convenient to express this calibration in terms of the signal level in dB relative to 1 volt when the sen- sitivity was 1pV/pbar, e.g. the calibration of P3 was -124. 189

Simply this means that if the calibration of a certain waveguide is say -x dB, then at this level the sensitivity is 1pV/pbar, and this can be related to the acceleration in terms of 'g'.

The calibration figures given in Fig. 7.2 give average values for each tank mode considered, taken from a number of graphs of response variations in the z- and radial directions. These graphs appear in Chapter 5. The method used was to obtain an average response for a measured distribution and compare this with the average response of the calibrated probe. The average value was given by

x = 1— E x N r=1 r where N is the number of points at which recordings were made (corresponding to the number of points in a distribution) and xr is the response level (of each point) in dB.

It is instructive to consider an example in order to illustrate the calibration procedure. Suppose that for the pressure probe P3 at a position (r, 8, z) the signal level of response is 1 mV. Since, at -120 dB re 1 volt the sensitivity of this probe is ipV/pbar, then to produce a signal of 1 mV a pressure of 8/5 mbar is required. Next suppose that for the lower end of a certain waveguide in the same position the received signal is 100pV. This corresponds to a 'waveguide acceleration' of g/90 if the accelerometer sensitivity is 9.0 mV r.m.s. gpeak' Hence an r.m.s. pressure of 8/5 mbar produces an acc- eleration of g/90. To represent this another way, a pressure of 144 mbar produces an acceleration of 'g'. 190

(Hz) P1 P2 P3 W2 W3 W4 W5 W11

1403 -113 -124 1728 -125(?) -124 2180 -111 -104 -124 -129 -130 -129 3280 -112 -124 -136 4070 -110 -124 -132 -132 -133 -133 -133 - 4620 -108 -124 -133

Values given are dB relative to 1 volt

FIG.7.2. CALIBRATION VALUES FOR PROBES AND WAVEGUIDES 191

7.3. Proposals for Further Study

Many aspects of the foregoing work could be developed into more detailed studies. On the other hand problems which have not been attempted could be carried out to substantiate the results so far obtained.

Several proposals of such areas of study are as follows:

1. Wire Waveguides

In an ideal flexible string, the only restoring force is due to tension, but in a wire elastic forces are involved and both longitudinal and flexural motion can occur. Thus, a wire has stiffness and like a rod, is capable of propagating elastic disturbances. Several types of wire waveguides were tried in the present set of experiments and they were found to behave in much the same way as stiff rods, i.e. they were forced into vibration by the pressure fields in the tank. It appeared that the wires used, which included a central copper conductor from a co-axial cable, behaved as high-pass filters, attenuating or completely suppressing the responses at many of the lower- order tank modes, a factor which may have significance in the practical application. The real difficulty with this scheme was the lack of a suitable vibration detector. The "Acos" acceler- ometer whose mass was of the same order of magnitude as that of the wires, was of little use for further experiments since it affected considerably the effective inertia of these wires. Many of the accelerometers on the market are either too massive or of too great a contact area. The most suitable one (at a prohibitive price), manufactured by the Endevco Corporation, has a mass of 0.5 gm and a sensitivity of 1 mV/g In any future development of 192 this work, it may be advisable to use either a non-contacting detector or a specially modified record-player pickup with a small area of contact.

2. Phase Measurements

It would be of interest to measure any possible difference in phase between the voltage across the source transducer and the received accelerometer signals, in order to advance the theory set out in the previous section. Reference to a paper by Webb (1956) would be relevant to this work.

3. Horn Adaptor

Since the end-pressure on the waveguide is so significant a large cross-section at the lower end but tapering towards its upper end might lead to increased sensitivity. Of the various types of horn the conical horn might be preferable from the point of view of ease of construction.

4. Further Investigation of the Energy Transfer Mechanism

One method of investigating the contribution to waveguide response by the end generated waves would be to compare the "normal" response with that obtained by an immersed transducer driving the end directly at a frequency of a tank mode. To do this adequately the requirement for the experiment would be a transducer of about the same diameter as the waveguide (preferably a rod), below which was a compliant air-cell to create an acoustic mismatch. With this arrangement working ideally there would be no end-stress with no excitation of the transducer. The response obtained with a voltage applied at a tank resonance could then be directly compared with that of a "free" rod. 193

5. Pulse Propagation in Waveguides

Since the sound field in the boiling sodium bath is perhaps more intermittent than continuous, it would seem desirable to investigate the pulse response of the detecting waveguide. With this in mind a project is at present under way at Imperial College to study pulse propagation in fluid-filled waveguides. This could be followed by work using a broad-band source to sim- ulate conditions in the PFR or a real bubble source produced by an immersed heater element.

6. Practical Design

The object of the research described in this thesis has been to present information about the behaviour of simple types of waveguide semi-immersed in a reverberant sound field. Future workers may now be in a better position to design a waveguide which can be used in a liquid metal-cooled reactor. The use of guides of varying dimensions and shapes, with bends and tapers, surrounding tubes, different vibration detectors, etc. can lead to a final decision about the most appropriate design. However, further work using a liquid sodium rig will be necessary to invest- igate temperature effects since sodium temperatures of the PFR, normally at a maximum of about 600°C, will in event of local boiling, rise to about 880°C.. It has been shown by Measures (1965) that only if a slender rod is suddenly heated will a stress wave be propagated as a result of the inertial forces restraining natural expansion. The temperature conditions in the reactor are such that there could be a sudden temperature rise, i.e. local boiling, but this does not necessarily mean that the waveguide would be sub- jected to a thermal stress since it may be well away from the boiling location. 194

APPENDIX I CALCULATION :OF TANK RESONANCE FREQUENCIES

The following theory has been developed to show that the equation for the natural resonance frequencies of a water-filled tank is the same as Morse's (1948) frequency equation for a hard-walled room. The theory also shows that the number of resonance frequencies possible in any bandwidth depends on the number of Bessel function solutions of the type J (x) = 0 which m comply with the boundary conditions.

The solution of the wave equation in cylindrical co-ordinates is given by:

os cos r (m4)). J A (E- . exp (-2n ivt) (AI.1) P = csin sin Wiz) . c where p is the pressure, 4 is the characteristic function, c is the velocity of sound, and r and z are radial and axial distances respectively.

The frequency v is given by:

v = )/((i) + wp/2n (AI.2)

For a cylindrical water-filled (or liquid-filled) tank of radius 'a' and depth 'P.', the water surface can be described as the z = 0 plane, and the tank base as the z = .2, plane. (See Fig. 2.6). Since this is a pressure-release case in which there is zero pressure and consequently maximum displacement at the tank walls (converse of the hard-wall case), then p = 0 at all the boundary surfaces. 195

Thus p = 0 at z = 0, since when the sine function is chosen (to comply with the mathematics): sin(0) = 0.

From equation AI.1 it is clear that for p to be zero at z = 2;, then wzt/c = nz7, where nz = 0, 1, 2, 3

ircn Therefore, w (n = 0, 1, 2, .) z = k z

But the pressure must be zero at the cylindrical boundaries also and for equation AI.1 to comply with this condition, then:

w r 0 at r = a m k c

w a 2nv a r m,n For this to be true, - Ira c c m,n

(or a 2v a/c) m,n m,n where Tra is the solution of J (11-a ) = 0 m,n m m,n wca m,n Therefore, wr - (AI.4) a

From equation AI.2:

(am, n (AI.5) 2 [( nz2i2 a

(Tra ) = 0 can be found in tables of Solutions for Jm m,n (1) Bessel functions.- In order to obtain values of am n, that is, sol- utions for which J (a m mn) = 0, it is necessary to divide these 196 tabulated values of Tr.

Let j represent the nth solution of the mth Bessel m,n function J (Tra and a represent the nth solution of the m m,n) m,n mth Bessel function J (a ), then the characteristic values a m m,n m,n for a cylindrical water filled tank are the solutions of Jm(amn, ) = 0 given in the following table:

1 2 3 4 5 6 7 JO j0,n 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711 21.2116 a 0,n 0.7655 1.7511 2.7546 3.7543 4.7527 5.7522 6.7519 J 3.8317 7.0156 10.1735 13,3257 16.4706 19.6195 22.7601 1 jl,n a 1.2197 5.2428 6.2451 7.2448 1,n 2.2331 3.2383 4.2417 J 5.1356 8.4172 11.6198 14.7960 17.9598 21.1170 24.2701 2 j2,n a 1.6347 2,n 2.6793 3.6987 4.7097 5.7168 6.7217 7.7254 J 3 j3,n 6.3802 9.7610 13.0152 16.2235 19.4094 22.5827 25.7482 a3,n 2.0309 3.1070 4.1429 5.1641 6.1782 7.1883 8.1959 J 4 34,n 7.5883 11.0647 14.3725 17.6160 20.8269 24.0190 27.1991 a4,n 2.4154 3.5220 4.5749 5.6073 6.6294 7.6455 8.6577 J 5 j5,n 8.7715 12.3386 15.7002 18.9801 22.2178 25.4303 28.6266 a5,n 2.7921 3.9275 4.9975 6.0416 7.0721 8.0947 9.1121

This table gives the first seven solutions for the first six Bessel functions. These are all "exact" values but as n increases the difference between consecutive solutions tends to Tr, e.g. the difference between j07 and j06 is 3.1405.

We can obtain an approximate relationship giving the higher-order solutions a0 n by considering the following formula given by Morse: a) (w a 3m( r 2c cos r 2m + 1 a r 4

197

For the tank, p = 0 at r = a, therefore from equation AI.1, if p = 0 then:

J ( wra 1 = 0 ra — c and hence, ( w a r 2m + 1 cos . n = 0 c 4

w a r 2m + 1 . n = (2n - (n = 1, 2, 3 4

In general,

w a r (2m + 1) 1. + (2n - = (•-• n ) ir - 4 7 2

ire or since w a . a • r m,n a • —2 + n - } m,n

But for the J o function, m = 0 w a r •• (n - 1)n or a = n - 1 (n = 1, 2, 3 o,n

So for our purpose it is more convenient to use a general formula worked out by Watson (1922) to compute the higher order zeros of J (x): o 1 = $ + 31 + 3779 (AI.6) j o,n 80 38483 1536005 where 13 = (n - 1)n , n = 2, 3

The results are correct to 5 decimal places, but with m = 1, joo. = 2.406 which is correct to only 2 places.

The general formula for the nth root of Jm(x) = 0, given also by Watson, is: 198

t - 1 4(t - 1)(7t - 31) x = a 8a 3(8a) 3

32(t - 1)(83t2 - 982t 3779) 15(8a)5

- 64(t - 1)(6949t3 - 153855t2 + 185743t - 6277237) 105(8a)7

(AI.7) where t = 4m2 ; a = —4 (2m - 1 + 4n)

This formula pertains to the pressure release type problem such as the water-tank.

The following FORTRAN IV program was executed on the Imperial College IBM 7094computer. It reads in the solutions of J (a ) = 0 m mn and the appropriate boundary conditions (governed by the diameter of the tank and the depth of the water) to evaluate the natural resonance frequencies of the tank up to 10 KHz. These frequencies are then sorted into numerical ascending order and written out. •

The flow chart shows that the frequencies, given by equation AI.5, are computed using firstly the exact values in the tables, then the values from the general formula. The program is arranged so that the first seven frequencies are computed (from the exact tables) for the function J , followed by all the other values (from the formula) o for J , before going on to J1. Similarly, all the values for J o 1 are computed before continuing with J2. The higher the order of the Bessel function the fewer will be the number of resonance frequencies associated with it (up to 10 KHz in this case), since these functions "flatten out" as shown in Fig. AI.1.

199 1.0

.8

.6

.4

.2

-2

-4

6 0 1 2 3 4 5 6 7 8 9 10 11 12

FIG.AI.1

The number of resonance frequencies v in a given m,n bandwidth is given by the maximum number of solutions of a m,n shown in Table I. Since the functions J (x) have an infinite m number of real positive roots and the negative roots are equal and opposite to the positive roots, then theoretically there are twice as many frequencies satisfying the boundary conditions than there are represented in this table.

In the second part of the program the number of frequencies in 100 Hz bands between 1 KHz and 10 KHz are calculated, and a histogram of "Number of Modes per Bandwidth" is plotted against "Bandwidth" using the IBM CALCOMP graph-plotting facility.

FOOTNOTE

See Jahnke, E. and Emde, F (1945). Tables of Functions with Formulae and. Curves 200

TABLE I

a a a0,1 a0,2 a0,3 a0,4 a0,5 a0,6 a0,7 0,n-1' 0,n.

a a a a a a a 1,1 1,2 1,3 1,4 1,5 1,6 1,n- ' al,n.

a a a a a 2,1 2,2 2,3 a2,4 2,5 2,n-1' a2,n.

a a3,1 a3,2 a3,3 a3,4 a3 ,n-1' 3,n.

a a a a a 4,1 4,2 4,3 4,n-1' 4,n.

a a a 5,1 5,2 5,n-1' a5,n.

a a a 6,1 6,n-1' 6,n.

a . a m-2,1 • am-2,n-1' m-2,n. a az m-1,1 rl,n. NB The en's" are different in each line

a 201 START FLOW DIAGRAM FOR PROGRAM BESSEL (Continued over Page) CLEAR ARRAYS ALPHA & FRQ M = order of Bessel function, i.e. J J J etc. 0' 2' READ C. TEMP N.= an integer to refer -to a partic- ular solution. BL = the solution of a given function \WRITE HEADINGS/ ALPHA = BL/Tr

KOUNT=0 J=0

\READ M,N,BL/

COMPUTE ALPHA M=1,N<6 ? Yes No I=1 Yes M=1 .N>7 T No COMPUTE FRQ M=2,N<6 ? Yes --s- No Yes GFRQ > 10K ?> Yes M=2 N>7 ? No No KOUNT=KOUNT+1 M=3,N<6 . Yes No J=J+1 Yes M=3 N>7 ? No FREQ(J)=FRQ M=4,N<6 ? Yes -- No 1=1+1 Yes M=4,N 7 ? No COMPUTE 'BL M=5,N<6 ? Yes --a- No L=M-1 Yes M=5,N>7 ? N M=6,N<6 ? Yes N Yes No M=6,N>7 ? ;No M=M+1,'N=1 I=1,y=1 Yes M>6 ? cont'd

202

Continued

WRITE FREQ SUBROUTINE ORDER

(Calculated Order) (sorts FREQ into RETURN numerical order)

CALL ORDER SUBROUTINE CALC I=0 ADD (I)=0 P=1000 (adds number of Q=1100 modes/100Hz b/w)

WRITE FREQ. (Ascending Order) —No FREQ>P FREQ

Yes WRITE KOUNT ADD(I)=ADD(I)+1

CALL CALC P=P+100 Q=Q+100 COMPUTE AFRQ (Abscisdae) I=I+1 No Q -10K ? Yes CALL GRAPH SUBROUTINE GRAPH (plots histogram WRITE ADD using Calcomp plotter) STOP COMPUTE Y RETURN (Ordinates)

RETURN 203 -MGDES- UF CYL1-NuRICAL -WATER-FTULED-TANK CALCULATED FROM BESSEL SOLUTIONS WHICH FIT BOUNDARY CONDITIONS WATER DEPTH-7E-170.0 CM TANK RADIUS = 45.5 CM SOUNDPNEITY IN WATER AT 20.0 DEG C -=148400.0 CM/SEC RESUNANut FREQUENCIES IN- ASCENDING ORDER (HERTZ 1393.1 1757.2 2082.9 2033.1 2235.9 2342.1 2342.3 2719.8 2719.9 2736.6 2737.0 2770.5 2931.4 2938.7 2939.0 3120.9 3173.9 -31-74. -3247.7 3248.0 3334.2 3369.1 3369.7 5413.5 3535.3 3535.8-3636.5 3636.8 3676.2 3676;3 3693.8 -37885.2 3796.0 3796.5 3846.0 3914.4 3987.2 3989.0 4092.3 40a2.5 4037.0- 41286 4130;3 4133.5 4134.0 4209.5 4209.6 4215.3 4353.9- 435'5.- 6 4402.2 4412.8 4504.7 4530.7 4531.1 4534.4 4540.9 4568.5 4563.7 4595. C 4599.3 4651.1 4652.7 4659.2 4637.7 411a2 4722.4 4746.8 4763.5 4763.5 4777.1 4860.0 4916.6 4920;6 4973.2 4173.6 5007.4 5008.8 5020.3 5083.4 5093.6 5101.7 5104.4 5128.0 5181.6 5185.4 5190.9 5198.7 5203.4 5215.6 5312.5 5317.0 5331.0 5311.6 5352.5 5195.7 5411.1 5412.4 5423.3 5450.1 5450.4 5453.2 5489.4 5503.6 5507.2 5597.3 5619.3 5619.4 5638.3 5656.6 5703.3 5716.7 5728.0 5731.9 5776.3 5801.0 5826.0 5831.5 5852.4 5153.6 5873.3 5875.2 5876.7 5901.0 5909.8 5909.3 5935.6 5953.0 5953.3-7 6020.9 6035.7 6060.8 6063.3 6119.4 6142.6 S1541;2 6152.0 6157.2 6170.6 6170.7 6238.0 6244.5 6253.5 -62-5?4.4 6277.7 6279.9 6232.2 6285.4 6308.1 6310.5 6323.4 &124.5 6360.6 6395.8 6399.3 6434.6 6453.9 6475.9 6476.3 643,3.3 -6495.4 3495,4 6519.1 6522.8 6523.1 6546.1 6600.0 6601.7 6634.8 6650;7 6663.9 6669.1 6681.9 6723.2 6726.1 6733.5 6733.7 6740.0 6777.9 6784.3 6815.0 6818.0 6819.0 6828.1 6832.7 6337.6 6857.3 6859.8 6865.8 6868.3 6915.3 6944.8 6993.1 7006.1 7014.4 7014.7 7026.9 7074.6 7075.3 7080.0 7031.1 7091.4 7081.) 7086.5 7086.6 7112.0 7151.3 7152.8 7157.4 7161.6 7162.5 7176.1 7190.3 7191.9 7193.1 7194.6 7208.5 7213.0 7235.9 7294.7 7305.4 7305.6 7331.4 7332.4 7346.1 7371.1 7180.6 7391.8 7401.8 7424.0 7429.9 7475.9 7486.2 7496.6 7524.3 7526.1 7562.4 7565.1 7555.4 7571.3 7576.7 7580.1 7534.9 7593.6 7600.1 7621.0 7642.8 711160.2 -7671.9 7678.9 7691.5 7632.0 7682.0 7687.8 7700.8 7702.6 E7705.3 770-n7 7715.2 7720.1 7730.2 7775.1 7779.4 7783.9 7794.0 7300.7 7843.5 7949.3 7959.9 7859.9 7860.8 7869.3 7834.4 7884.5 7892.1 7901.3 7915.7 7954.0 7954.1 7969.4 797C.9 7994.2 797.4 8016.3 3023.6 8057.7 8064.6 8063.9 8075.3 3083.0 8084.1 8092.7 8125.6 8125.3 8131.6 8134.7 8135.6 8151.1 815).3 8167.4 8173.2 8175.2 8135.0 8187.4 8200.9 8206.8 8?43.1 3245.1 3265.5 8272.5 8276.3 8279.4 8280.8 8280.8 8293.1 8332.2 3344.4 8351.1 8358.3 8360.2 8373.4 8400.7 8401.6 8419.5 3421.0 8439.3 8444.1 8457.3 8458.0 846E.9 9469.0 3491.5 9484.8 3504.0 8511.3 8516.9 3579.6 8532.3 8533.8 8541.4 8542.9 8543.4 8572.1 8597.7 3610.9 8611.1 8616.3 8623.3 8625.3 8638.0 8638.7 8673.9 8687.7 8693.9 3694.1 8700.5 8705.6 8707.9 8711.1 8716.5 3718.4 8746.2 8749.6 8752.9 8764.3 8777.1 8816.1 8824.6 8329.0 8829.5 9348.3 9877.1 8877.2 8877.9 8882.2 8887.3 13983.5 8900.3 8902.3 8902.6 8912.9 8915.7 8919.4 8937.1 9951.6 8951.6 3952.4 5954.5 8956.3 8971.0 8976.3 8984.9 8985.6 9019.5 9034.3 9039.9 9041.5 9057.8 9058.0 9085.6 9090.9 9091.5 9107.4 9117.2 9118.6 9135.6 9145.0

204

-914-8.1 9149.1 9150-.3 9158.5 9165.2 9166.6 9194.5 9202.4 920-8.3 9227.9 9229.7 9238.3 9240.5 9268.6 92-6-8.8 9271.20 -19278.0 9290.2 9307.5 9314.2 9318.3 9320.0 9330.5 9342.3 4343;2 9352.4 9352.4 9358.6 9381.1 9383.2 9385.8' 9387.1 9391.5 9399.4 9400.9 9404.4 9413.5 9461.7 9468.7 Li- 9472.9 9-473.6 9484.7 9485.9 9485.9 9496.7 9497.1 9502.0 9505.5 -951C.7 9511.4 951-3.3 9514.5 9533.6 9535.3 9541.4 - 9W,--4 9573.47:79574.6 9582.9 9586.8 9595- 9617-.9 9618;-8 9645.3 9650.5 -9650.6 9652.8 9654.1 9656.1 96727686.75 96419.9 9693.0 9696.8 9701.1 9701.1 9707.5 - 9 /35.7 9752.43- 9754.5 9755.3 9766.4 9766.8 9778.8 9781.1 818. 1- -97:9- 7 9802.0 9810.1 9822.0 9330.7 9830.8 98413-.6 9848.8 9860.--2 9868.2 9868.6 9870.2 9871.7 9871.8 9876.9 1.6 913-79-i41 -9890.8 9894.8 9914.8--- 9929.7 9939.6 9946.1 9964.3 -9971.8 9984.- 0 999T.1 0.0 0.0 0.0 0.0 MIAL NUMBER -JW MUtit-S, = 524

_ _ _

CD cv R Gooq PLOT. GOOD: FRIENDS. AND FULL OF E)(PEURTI ON WJLLIRM :SHRKESPERRE M (HENRY IV, PPM' I ) CNJ V 1.51 iI

0,

W'

O

ti cpO

'b.or) 20.00 140..00 60.00 . 60.00 100.0,0' . 120.00 .:. 140.00 1.60.00 180.00 ,Ff5EOLJENCI CX10? , 206

APPENDIX II PROGRAM TO COMPUTE "EFFECTIVE VESSEL RADIUS"

The concept of "Effective Vessel Radius" has been fully explained in Chapter 4. The following Subroutine merely utilizes the re-arranged frequency equation (2.5) for cylindrical symmetry, to give the radius (RAD). This is done for various depths of water, taking into account the water temperature and the corresponding velocity of sound.

The instructions in the program are in the following order:

1. Prints headings. (statements 21, 22)

2. Reads in solutions of Bessel functions, nz (see equa- tion 2.5), water depth, water temperature, velocity of sound, and measured resonance frequency. (state- ment 24)

3. Obtains mm,n by dividing Bessel solution by Tr. (statement 27)

4. Evaluates RAD. (statement 25)

5. Prints results from which Fig.4.6 is obtained, leading to a value for the 'EVR'. (statement 23)

thN UURLE STS

L0MMLN/BLK5/RADI3C)/BLK6/ALPHA(301 -,===A3A4Hijilaagt-430 ) T EMP L 3laioLT3ti ) WRITE16,21) 21 flg#MAXIIIIVIW44=t4A32HCALCULATIU' --A 133HRADIUSILVR) 1-RUM BESSEL SOLUTIONS)

ieZ t-UKMArtill ,TUX-i-TIRWATER UEPTHI2X110HWATER TEMP,2X, 114-HSCUNIFVELOY,2X 1 I5HRES FRQ OF MODE,2X, 314tVit/14X, 24HTEM);-7)(17H(GEG C),5X,12H(CM PER SEC4,9X,4H(HZ),10X, 35AtEMS)) PI=3.141!92E5 K=I -- 24 READ(52D) tiL,RW,XLE(K),TEMP(K),C(K),FRQTK) -1-ORMATTFWTAglKif-1.1an7-5.1,1X,F4.141XpFa.1,1X,Fb.1) 27 ALPHA=BL/PI Z=RAO(K)=SQRT(A8SI(ALPHA)**2/1(2.*FRQ(K)/C(K))**2 -I(RNZ/XLEIK-T)**2))1 --=1F(KOECYBIgMl K=K+1 LO TO 24 26 DO 23 K=1,18 It1519)ITEMP4K),CUK),FRQ(K),RAO(K) 30 FORMAT(IH ,13X,F5.1,8X,F4.118X,F8.1,10X,F6.1,9X,F5.2) RETURN tN 0

(.ALLULAT ILN Ul- tFFELT-TVE VESSEL. RADIUS( EVR ) FREIN BESSEL Sth..U1 IONS = I TY RES FRQ OF M i CM) (121h6 Li (CM PER SEC) (HZ) ILMS) 2- 0 181 5 14(80 1210.0 18.5 141800.0 1388.0 45.48 115.0 13 45-60 110.0 18.5 141800.0 1413.0 45.51 1-Utri- -0- 13.-/--- 1 Ain 1424.0 45-.;T4 100.3 13.-9- 148000.0 1439.0 45.90 95.0 19.0 1-2 813411*=, 1458.0 - 45.96 90.70-- 1-9.0 148000.0 1477.0 46.1T 85.0 19.0 1 4f8=0:0110_ _940_ 45.32 80.0 19.0 148800.0 1520.0 46.96 1413-6153E4-0 ----V407.0 115.0 20.8 148600.0 1416.0 -45.14 110.0 14 EttfULL. 1424.0 45.37 IC 5.0 20.8 148600.0 1433.0 45.64 -0 20. H _245.- 41.Z 115.0 20.8 145600.0 28360- 47.28 - 47.31 1(35. El 20.8 148600.0 2843.0- 47.41 209

APPENDIX III CALCULATION OF "DYNAMIC" VALUES OF YOUNG'S MODULUS AND RESONANCE FREQUENCIES FOR SOLID CYLINDRICAL WAVEGUIDES •

The main routine in the following FORTRAN IV program "calls" the subroutine YMOD and XMOD. The argument of each calling statement allows for a number -(L) sets of data values to be read in, the maximum number being 30.

YMOD The "bar" velocity (VBAR) for each waveguide is computed at each longitudinal resonance frequency. These are summed (SUM) and from the average bar velocity (AVBAR), a value for "dynamic" Young's Modulus (E) is obtained. These computed values of E are slightly lower than the generally accepted values. The frequencies (FRQ1) used in the computation of VBAR are those which are corrected for the masses of the driving transducer and accelerometer. The resonant wavelengths given by An = 2k/n do not include the possible "end-effect" allowed for by Zemanek (1962), who gave the wavelengths as: An = 2i/(n-g), where g is the deviation of n from an integer. The resonant frequencies are given here in the form of vo = vn . An. If the correction is allowed for the following equation, given by Massey (1967) is applicable:

v = (v + Av) . 2k/n o n where Av is the amount by which the ideal resonant frequency is reduced assuming that the actual wavelength is given by An = 2Z/n. The fact that the computed frequencies do not increase by a con- stant amount suggests that this correction may well be applicable. 210

XMOD This subroutine computes Q-values for some of the semi- immersed waveguides at resonance frequencies of the tank. The 3 dB frequencies used are measured values, i.e. no correction • for the accelerometer mass is considered.

List of Waveguides Used

Material Length (cm) (Diameter)

311 W1 Stainless Steel 259.1 g

W2 Stainless Steel 136.5 i"

W3 Stainless Steel 136.5 i"

W4 Stainless Steel 136.5 1" W5 Stainless Steel 136.5 3 It

W6 Brass 150 1" W7 Stainless Steel 150 It, W8 Copper 150 I"

W9 Duralumin 150 i" 3fl W10 Stainless Steel 200 a W11 Stainless Steel 136.5 W12 Stainless Steel 136.5 I" W13 Stainless Steel 136.5 in W14 Duralumin 150 111 211

FLOW DIAGRAM FOR PROGRAM WAVGDE

START 4 \WRITE HEADINGS/

SUBROUTINE YMOD READ DATA (corrects measured CALL YMOD resonance frequencies, WRITE MATERIAL total number = L)

SUBROUTINE XMOD NEW PAGE OR NOT ? (Computes Q of tank CALL XMOD modes as "seen" by waveguide) COMPUTE VOLUME, DENSITY, 7.(mass READ DATA acclr/waveguide) STOP 4 COMPUTES Q CORRECTS MEASURED FREQUENCIES WRITE FREQUENCIES AND 4 A=2L/N RETURN J=1

VBAR(J)=X*FRQ1(J)

J=J+1 SUM(J)=SUM(J-1)+VBAR(J)

No J-L ? Yes AVBAR=SUM(L)/L

Yes B=0 ? B=1 ?

READ 3DB-POINT/ FREQUENCIES

COMPUTES CORRECTED 3DB-POINT FREQUENCIES WRITE DATA/

RETURN

11 U N

* FRQ1 = RESUNANCE FREQUENCY OF WAVEGUIDE ALONE 4tAirtA = CA4-4-U-LATEU_R_LSCNANI__E WAVELLNCTM 4cAV3AR = EAR VELOCITY ----.==amm==..r=a__-__EariA7=Ez=ilmmF,Iria 02 = 0 OF LONGTL RESUNANCE CF WAVEGUIDE ACCELERUMETER ZtreNZIEZ-ZZIEZ-ZZW=ZNIZIMUZZWZMIZZIZ ZrZ•ZZI Z Z Z = Z Z Zer.: •X-I. I Z s 1ST-

FRQ3 FR 2 .FRQ1 LA-M A VBAR LHLI (HZ) (HZJ (CM) tCMISEC)

LENGTH = 136.5 CM,RADIUS = 0.95250 GM/WEIGHT = 3041.0 GM ENS1TY = 1.e2 ACCELERLMETER WT AS A PERCENIAGE OF WAVEGUIDE WT = 0.0283 EL 491641. 89 YUUNGS PLGULLS = 1.926E 12 DYNE/50 GM 148.. 182a.4 1833.5 213.00 500541.22 3568.0 2654.8 3664.9 136.50 500260.81 .345.0 5475.1 5490.2 91.00 49-9606.50 712.0 7295.3 7315.4 68.25 499279.36 97.cYJ 10658.0 1C911.4 10947.5 45.50 498110.96 124 1. 127 2.8 12737.8 39. 0 496775.65 14164.0 14508.7 14548.7 34.13 496475.20 15J55.0 16347. 17648.0 1E071.4 18127.4 27.30 494876.84 11326.0 19196.3 19850.9 24.82 492664.11

(HZ) (HZ) (HZ) (CM) (CM/SEC)

LENGTH = 136.5 CM / RADIUS = 0.79375 CM /WEIGHT = 2120.0 GM ENSITY = 7.85 ACCELEROMETER WT AS A PERCENTAGE Cl- WAVEGUIDE WI = 0.0406 EAR VEL = 49,550.19 WSE-C YOUNGS MCDULIS = 1.942E 12 DYNE/SQ CM 1767.0 1328,7 1835.8- 273.01 5011P6.04 3534.0 3657.4 3671.7 136.50 501186.04 529C. 5474_7 5436-.1 91. 580146.04 7035.0 728C.6 7309.1 68.25 498846.04 9144-._1_ 54.60 49-9484. 2k 10528.0 10895.5 10938.2 45.50 497687.85 ET; 13945.0 14431.8 14488.3 34.13 494414.21 1 -515e.-0 163 ..(1 16372.-0 3e.33 496616.34 17414.0 18021.8 18092.5 27.30 493924.94 1913`.0 9 _9 198 _5 24.82 493398,

kRQ-4 1-WW- FRQI L A-NnA V-BAR (HZ) (HZ) (HZ) (CM) (CM/SEC)

LENGTH = 136.5 CM I RADrUS = 0.63500 CM,WEIGHT = 1353.0 GM

ACCELERLMETtR VAT AS A PERCENTAGE OF WAVEGUIDE WT = 0.0633

• /5 C YOUNGS MCOULUS = 1.950E 12 DYNE/SQ CM 17213. 1823.2 1834.2 273_0 5CC727.30 3454.0 3642.2 3664.r 136.50 500148.09 .6 54813 91. .6 6861.0 1234.9 7278.3 68.25 496745.23 6 10356.0 10920.3 10585.9 45.50 499858.48 12019. 1267'.'1 12750.1 19*00 457252.04 13186.0 14531.2 14624.5 34.1.3 499062.07 15312.0 16146.4 1 243.3 3 .33 4 2714. 17216.0 18154.1 18263.2 27.30 498584.21

STAINLESS STEEL

• 0 (HL) (HZ) (HZ) (CM) (CM/SEC)

LENGTH = 136.5 CM,RADIUS = 0.47625 CM,WEIGHT = 761.0 GM

ACLELER0MLIER WE AS A PERCENTAGE UF WAVEGUIDE WT = 0.1130 BAR VtL - 5 3865.63 CM/SEC

YUUNGS MODULUS = 1.586E 12 DYNE/SQ CM f • 1 1 . r. 24.04 3298.0 361.7 3656.0 136.50 499054.1;2 6638.0 1283.5 7358.5 68.25 502217.46 91 40.0 23 .1 54.60 504184.56 10004.0 109/6.8 11089.8 45.50 504588.07 52 12529.0 13747.3 13888.9 34.13 4(3459.21 ===2:==2,==a-Va=a7:••••=f1r=..==-..- 3,==.%-. 6020443 1715z.0 18319.9 19013.1 21.30 5190/4.05 18888.0 2C12'.7 2())33-a---t---24-482 51)646-29

WUA V Q1 QZ (HZ) (HZ) UHZ) 1CM) (CMTSEC)

LENGTH = 136.5 CM,RAD1APS- = 0.63500 CM,WEIGHT = 1358.0 GM ALLELERLMETIR WT AS A PERCENTAGE OF WAVEGUIDE WT = 0.0633 BAR VEL = 501537.40 CM/SEC YOUNGS MODULUS = 1.976E 12 DYNE/SG-CM • ,e. IS 2.2 4323 3456.0 ?,644.3 3666.2 136.50 500437.70 101.0 101.6 -istlyainswirxwerewraim..-a..-tamm-somi NIMITI1=-"Priltit7 1 ,0 61• o912.0 728S.7 7333.5 68.25 500510.09 58.1 58.1 2-_6 A • 62 ..# 49.3 10360.0 10924.5 1C990.1 45.50 5C0051.55 37.1 37.1 w.....m...lromralw6lotz.,romalrel4Rmik....NER:grwa264,114X1m.7.11 31.EL 13806.0 14558.3 14645.7 34.13 499786.08 35.2 35.2 31v.-11 5a61444154, -0. 0 17428.0 18377.7 18488.1 27.30 504723.85 31.3 31.3 E 'it fr• ts- 5

to (HZ) (HZ) (HZ) (CM)(CM/ S E C)

LENGTH = 259.0 CM I RADIUS = 0.47625 CM,WEIGHT = 1444.0 GM - . ACCELERCMETER NT AS A PERCENTAGE CF V.AVE".;IJI0E WT = 0.0596 VEL = 503384.41 CM YCUNGS MODULUS = 1.983E 12 DYNE/SL; CM

1822.0 1921.1 1932.6 259.00 5C0534.49 112.671264.6 3662.0 3849.7 3871.5 129.50 501355.93 4582.0 4816.8 4844.1 1 3.60 5 1848.80 5504.0 57E6.1 5813.8 36.33 502359.92 6432. 6761.1767974,00 53194.41 1362.0 /739.3 7783.1 64./5 503957.18 32)3. T8718. 8767 4- 57-56 504611.29 9217.0 9689.3 9744.2 51.80 504751.24 wasm'y • 6rzwamomi •Of _ I - I I .10 11106.0 11675.1 11741.3 43.17 506832.24 • FRQ1 LAMA V AR C1 02 IHL) 11121 (HZ) (CM) (OM/SEC)

LENGTH = 15-0.0 CM I RADDYS = 0.63500 CM,WELGHT = 1603.0 GM OiN\JTY 8.44 ACCELEROMETER WT AS A PERCENTAGE OF WAVEGUIDE WT = 0.0536 h(tS E C YUUNG-S MUDIFECS = 4.708E 11 DYNE/SQ CM tat!74mosimk mi swimonot i ir7vorJir lay /M-elreM114W7s11-InliJEW•Tg 537.5 537.5 2149.0 2248.2 2259.7 150.00 338960.17 268.6 268.6 .2 .90 78.7 4296.0 4494.3 4517.4 (5.00 338802.44 18.1 18.1 0 03 6403.0 6658.6 6(32.9 50.00 336646.80 -0.0 -0.0 8615.0 9014.7 9058.9 37.50 339709.38 -0.0 -0.0 J669. 0.0 -0.0 10818.0 11317.4 11375.4 30.00 341263.02 -0.0 -0.0

CLIPPER

FRQ3 FR6 FRQ1 LAMDA VBAR Q1 Q2 (HI) (HZ1 HZ) (L-4‘9-1-4:1451

LENGTH 15 . DIUS = 0.31;00 = 1701.5 GM DENSITY = 8.95 -ViAVEGUIOE W = 0.0505 BAR VEL = 3(C715.18 CM/SEC SQ CM - 11(5.0 1226.1 1232.0 300.00 369612.25 391.7 391.7 2.0 24541.3 24E6.2 150.00 369926.81 294.0 294.0 3527.0 3680.4 3698.2 100.00 369821.96 185.6 185.6 4( C.0 4904.4 4 5869.0 6124.2 (3153.9 60.00 369234.77 96.2 96.2 73E0T--500-0-161-035.55 78.2 78.2 8222.0 85/5.6 3621.1 42.86 369477.43 52.0 52.0

10559.0 11018.2 11011.6 33.33 369053.02 29.7 290 .0 -0-.0-- 13015.0 13581.0-13646.8 27.2/ 372185.95 -0.0 -0.0 lt .0 11+9.9 14/21.8- 25. 0 3/3046.23 -0. -0.0 15390.0 16059.3 1613(.1 23.08 3(2394.92 -0.0 -0.0

OP 51. 17(50.0 18522.0 18611./ 20.00 3[2233,61 -0.0 -0.0 10.0 19132..4 ._8_1.761.14- 18.75 311774.87 0.0 -0.0 20140.0 21015.9 21111.1 1(.65 3(2665.36 -0.0 -0.0

216

01 LAM A VBAR el 02 THZI (HZ) (HZ) {CM) (CM/SIC) LENGT-H = r50.0 LM,RADIDS = 0.63500 CM.WEIGHT = 514.0 GM- ALCELERLMETER %T AS A PERCENTAGE CF WAVEGUIDE WY = 0.167-3- to V.T. Z... TGUNGS PICUULIITS = 7.8371 Il DYNE/SQ CM 123.2 123.2 2-918.0 3406.7 3456.6 150.011 518484.86 27.1 27.1 • i 42.2 Aa. 613C.0 7012.5 75.00 533031.99 55.1 55.7 5-1726B,-.12-0 -0.0 0. 9309.0 10645.2 10805.0 50.00 540247-0-8 -0.0 -0.0 -0.0 1242/.0 14216.1 14424.0 31.50 540900.88 -0.0 -0.0 cis:7/mi w;yzce..m.domm-siimairov_^s.-1 -IP) -0.0 16238.0 18575.8 18847.5 30.00 565423.58 -0.0 STAINLESS STEEL FRQ3 FR02 FRQI LAMDA VBAR

DENSITY = 7.83 -77 1Z--W-T-T- 0.07771 BAR VEL = 462224.18 CM/SEC • CM 970.0 1034.3 1041.8 400.00 416717.55 . 2 9C. 2t3X1 3380.0 3604.1 3630.2 133.33 484022.45 .0 .4 5250.0 5558.1 5638.6 80.00 451086.01 .7 7623.0 8128.5 8187.2 57.14 46/840.63 10020.0 10684.4 10761.6 44.44 478294.37 11184.0 11925.6 12011.3 40. 0 48 471 04 12281.0 13101.7 13196.4 36.36 479869.59

(NZ) (HI) il-IL ) (CM) (CM/SEC)

10.0 GM

I) st 19 Y 7 ACCtLEKUMEILK WT AS A PERCENTAGE OF- 1, AVEGUIDE WT = 0.1686

TUUNCS MLUULUS = 1.783E 11 DYNE/SQ CM 14A.5. 1143.2 1617.4 3 . 5 0224.11 159.4 159.4 2905.0 3326.5 3375.5 150.00 506324.40 83.0 83.0

- • 5U0 . • U.00 513819.0558.2 8.2 5966.0 6811./ 6932.3 (5.00 519919.34 52.8 52.8 _I:La 4-1 840 60.00 522_24-..6-0 10.8 20.41 9u4[.0 10354.0 10506.4 50.00 525322.46 19.8 1. 0 UK LUM1N

k vKL/2 1-1W1 LAMUA VBAR Q2 (HZ) (HZ) (HZ} (CM) (CM/SEC)

LLIWAH = 150.0 LM,RAUEUS = 1.27000 CM.WEIGH = 2127.0 GM

ACLELLKLMLIER WT AS A PERChNIAGE OF WAVLGUIDE WT = 0.0404

YUUNt'S MUUULUS = 7.380E 11 DYNE/SQ CM 11.20 412. 412:75 3295.0 3409.6 3423.0 150.00 513443.72 253.5 253.5 a 7 .5 1 tbdc. 6811.0 6837.6 75.00 512820.42 101.3 101.3 • • . ewe 31 34.8--174. 9813.0 10154.4 10194.1 50.00 5C9/03.92 1.0 1.0 115 11 0 .2 11,49.1 42.46 512130.330,0 13193.0 13652.0 13705.3 37.50 513950.15 -0.0 -0.0 .46 46.5 46.5 1653.0 17114.4 17181.3 30.00 515438.27 63.6 63.6 U U 87`]_6 Z1 51478 414 I a. 2 19838.0 2c528.2 20608.4 25.00 515209.73 98.1 98.7

1 qt1 INIKL TANK xtSONANCE tnmtR 3178 FRU UPPER JOB I-KU STAANchss sittL, LtNbiH = 150.0 LM,OIAMEIER = 0.2/0 CM. 3285.0 32/9.0 3294.0 219.0 • 5030.0 M;C8.0 504/.0 129.0

-0.0 -0.0 0.0 -0 .0 0.

-0.0 -0.0 0.0 BRASS, LEN(.31H = 150.0 CMI UIAMLIER = 1.270 (-M. 2172.0 2190.0 121.2 2247.0 3230.0 3251.0 1546 * 66.0 4056.0 4076. 20342L 4391.0 4384.0 4401.0 258.3

LUPPER, LLNallH = 15C.0 CM,DIAMLIER = 1.270 CM. _231. 3271.0 3216.0- 142.8 4068.0 4056.0 4078.0 184.9 .0 4894. 4913.0 25#/4 6C80.0 606/.0 6C93.0 233.8 0. .0 0.0 0.0 -0.0 -0.0 -0.0 0.0 DURALUMIN, LENCITH = 150.0 LM,DIAMETER = 1.270 CR. 1 . C 2165. 21`15. 72.7 3241.0 3236.0 3251.0 216.5 72,2_ 3274-0 3292. 182.3 4 066. C 4055.0 4077.0 184.8 .U.I. 50-047.- 5054.0 111. 219

APPENDIX IV ACCELEROMETERS AND TRANSDUCERS

Accelerometers

The type of unit used in the experiments described in this thesis was an "ACOS ID1003", manufactured by Cosmocord Limited. The following specifications are applicable to the two accelerometers actually used:

Weight, 8.6 gm including the mounting stud (48.A. thread) Sinusoidal Voltage Sensitivity, (a) 9 mV /g r.m.s. peak (b) 8.5 V m r.m.s./gpeak Rdsonant Frequency, approximately 55 KHz In addition the manufacturers claimed the following: Capacitance, 150 pF Charge Sensitivity, 3.7 Coulomb x 10-12/g Resonant Frequency of Element (Base Anchored), 75 KHz Maximum Acceleration, 1000 g Temperature Range, -600C to +1750C

A little detail about accelerometers in general is pertinent here.

An accelerometer consists essentially of a mass which is supported seismically with respect to a case and guided to prevent motion in directions other than the seismic direction. For the ID1003, the sensitive direction was the z-direction shown. The seismic system in general consists of a mass suspended from a case by a "spring" (the piezoelectric element). Displacement of the case is given by x and relative displacement of the seismic mass with respect to the case is given by y. The system shown is linear,

220

damped and with a single degree of freedom (z), having a spring constant k and damping factor C.

In the operating frequency range, which is usually below its resonant frequency the mass undergoes about the same accel- eration as the body of the accelerometer. The mass exerts on the element a force that is directly proportional to the magnitude of the component of acceleration, and the output is a known function

of force. For a given displacement the acceleration varies as the square of the frequency and the velocity increases linearly with frequency. For a case displacement of x = X sin 27rvt in the sensitive z-direction, the relative displacement of the mass with respect to the case, under steady-state conditions, is given by y = Y sin (27ryt - 0, where (15 is the phase angle. The output signals are proportional to y or its time derivative.

At very low frequencies the mass and the case vibrate in phase, but as the frequency increases the displacement of the mass lags behind that of the case until at resonance the lag angle is 90°, and at very high frequencies it is 180°. The phase angle, (I) is given by: 2Cv/C = tan-1 cn (1 - Cv/vn) 2 221

where v is the forcing frequency, is the undamped natural n frequency, C is damping factor, and Cc is the critical damping resistance (= 2i(mk)).

The accelerometer mounting must be sufficiently rigid so as to prevent appreciable relative motion between the acceler- ometer and the vibrating structure. To indicate an acceleration of 10 g to within 1% at a frequency of v Hz, the relative motion between accelerometer and structure must have an amplitude of less than 1/v2, (i.e. < 0.0001" at 100 Hz, < 0.00001" at 1 KHz).

The weight of the accelerometer together with its mountings must be small enough so that the inertia of the accelerometer will not appreciably alter the motion being measured. In the computer program in Appendix III, the weighting of the waveguide caused by the accelerometers (and transducers) is corrected for.

It is interesting to have some idea of the order of mag- nitude of the displacement of the end of a waveguide rod at a known frequency. Using a typical result obtained during the course of the experimentation, we can obtain a value for the rod displacement at the frequency of a tank mode which forces the rod into a maximum oscillation.

Using a i" diameter waveguide and with 30 volts across the source transducer in the water, a trace was obtained on which the response at 16 KHz was approximately 10 mV. Calling this 9 mV for simplicity (since the calibration was 9 mV /g ) the r.m.s. peak -2 corresponding acceleration is 1 g or 981 cm sec . If the displace- ment at time t is x = X sin wt, then the velocity is x = wX sin wt or: 222

x = -w2x = -4Tr2v2x 981 = 472.16.108.x x = 9.8 x 10 8 cm

Obviously there will be a wide range of displacements throughout the frequency range used, so this calculation merely 0 gives an order of magnitude, i.e. approximately 10 A. 223

Piezoelectric Transducers

(a) Specifications

Three piezoelectric cylinders were used as pressure probes in the investigation of sound fields in the tank. These were all manufactured by the Brush Clevite Company Limited, and were made of Lead Zirconate Titanate (PZT) with dimensions as follows: (i) P1 (PZT-4). Outside Diameter, 0.5 cm Length, 0.95 cm Thickness, 0.05 cm (ii) P2 (PZT-5A). Outside Diameter, 0.635 cm Length, 1.27 cm Thickness, 0.051 cm Type Number, 8-4020 (iii) P3 (PZT-4). Outside Diameter, 0.635 cm Length, 1.27 cm Thickness, 0.079 cm.

The source transducer was a Barium Titanate (Ba Ti 0 ) hollow 3 spherical transducer approximately 3 cm in diameter and with a 0.15 cm wall thickness manufactured by United Insulators Company Limited.

The "standard" specifications were as follows: (1) Transmitter (as in these experiments) Frequency at resonance, 81 KHz Resistance at resonance, 440 n Capacitance at resonance, 8000 pF Bandwidth, 10 KHz. Q, 8 Power Efficiency (n = Power radiated into load/Total Power Input), 80% (ii) Hydrophone below resonance Sensitivity, -100 dB re 1 volt/dyne cm 2 Capacitance, 8000 pF Sphere tested to withstand 10,000 p.s.i. 224

(b) General Theory

Mechanical forces produce or are produced by electric fields. A piezoelectric (pressure-electric) substance behaves both dielectrically (like a capacitor) and piezoelectrically (like a mechanical spring with internal stiffness opposing the applied force). The capacitance of such a substance is given by:

C (farad) = Q (coulomb)/V(volt) and the "stiffness constant" by:

h chk = X /xk (Hooke's Law) where Xh is the stress and xk is the strain.

Electric charges produced by straining the material create a voltage across it owing to its capacitor property. The charge per unit area, or polarization, is:

P. Q/A _elk • Xk 1 Conversely, forces are produced by subjecting the material to an electric field and given by:

n e E. hj j where E. is the field strength and ehi ,ik is the piezoelectric stress constant, relating a given field strength vector j to a particular stress component k to a polarization vector i.

These relationships are summed up by the following tensors: 225

c Mechanical Quantities hk Xk

eik,hi Electromechanical Quantities dih dkj

II 9

P. = E.. E. Electrical Quantities where d . ih = piezoelectric strain constants and e = E ch .1 kJ , d ik ih

(c) Transmitters iwt An applied voltage V = V e causes a displacement o current I = iwC V, where C o o o is the static capacitance given by C = C'/(1 + 81c/Tr2) o C' being the measured capacitance and Ice the electromechanical coupling factor.

This is the current that will flow if all mechanical vib- ration of the transducer is prevented. Therefore V/I is the o "clamped impedance". If the transducer vibrates a "motional" current will flow, where

I = I + I total o m

Normally the Acoustic Power Radiated = In . Zm i , where Zm is the motional impedance, or total mechanical radiation impedance. However, at resonance, the LCR - circuit becomes a pure resistance R e and the equivalent electric circuit of the transducer is then a parallel RC - circuit. Optimum operating conditions for the

I I 226 total m

Z -m

crystal are when the impedance Ri of the electric generator is equal to the input impedance Re of the crystal at resonance after C is tuned out. The reactance iX (= -1/iwC ) can be o o o cancelled out by a shunt coil accross the input, then the input impedance is Zr = Re.

I I m total

227

At resonance the power factor is given by:

cos (I) = 1/1(1 w2 c2 R 0 0 2) where $ is the phase angle between and I . The values Itotal m of L and the impedance Z at resonance for the previously mentioned o Barium Titanate sphere have been calculated as follows:

To tune out C at 81 KHz, o 1 L = = 1/[472(81 x 103)2 . (8 x 10 3)J = 0.482 mH o w2 c o o Reactance at resonace: X = 1/w = 246 ) o 0 C0 Total impedance that generator "sees" at resonance

11. -- + 1 Z R e iiiwoCo or Z = Re/V(1 + w2 R2 C2) '-f, 215 0

The frequency characteristic of radiated power is determined by the mechanical Q, which is the frequency of maximum power output at constant voltage. If C is tuned out by L , then R is the o o e only dissipative element in the circuit and:

energy stored in Lo Quality Factor , Q nw L /R n o o e energy dissipated in Re per cycle

The mechanical Q is independent of crystal dimensions and for water, as applicable to the present work, it is approximately 28. Mechanical Q decreases with increased loading, whereas Elect- rical Q increases with increased loading.

In practice, transducers have "losses", which include diel- electric losses, losses due to transducer mounting, losses due to 228 internal damping within the crystal, and capacitive losses along the connecting cable. In the spherical source transducer used here, there was also the possibility of loss due to the compliant joint between the two hemispheres. These losses can be repre- sented in an equivalent circuit

losses

Shunt (Dielectric motional loss resistance resistance)

(d) Non-directional Receivers

The upper limit of the flat response of a transducer is determined by the lowest vibrational mode. The three types of mode for a cylinder are Length, Radial, and Wall-thickness, but there can be coupling between length and radial mode. The lar- gest coupling occurs when length/diameter '4 1.5.

Radial resonance of a long tube is:

1 E V d (1 - a2)p PI where d m is the mean diameter, a is the Poisson's ratio, p the 229

density and E is Young's modulus.

For a pulsating thin hollow sphere,

2 E p (1 + cr) = 7rd 2(1 - a)p p (1 - a) m where p is the rigidity modulus.

More detail on piezoelectricity can be obtained by ref- erring to Hueter and Bolt (1955) and to Fox and Griffing (1949). Further useful information, including practical applications and data tables on piezoelectric ceramics, can be found in the Brush Clevite Bulletins:

66011/B (January 1967). General Characteristics and Application Guide

66017/A (January 1967). Measuring properties of Piezoelectric Ceramics

66019 (January 1967). High Voltage Generation from Piezoelectric Ceramics, by C. Pearcy.

Another publication on piezoelectricity, issued by the Component Division of Mullard Limited, describes the ceramic piezoelectric element "Piezoxide". 230

GLOSSARY

Reverberation

In the general case the amplitude of sound waves dimin- ishes with distance but in an enclosure these waves are reflected by the walls. Superposition of such reflections causes the sound to be of such an amplitude, and therefore of such an inten- sity, as to prolong audibility. This gives rise to the phenomenon of reverberation. The actual duration for a source of a certain intensity depends on the absorption of the walls, and is the time required, after the source has been switched off, for the intensity 6 of the sound to drop by a factor of 10 below its original value, i.e. by 60 dB. It is called the reverberation time (typical values obtained for the present tank, measured by means of a waveguide as a detector, ranged between 0.1 sec and 0.5 sec).

Bar 6 2 The bar is a unit of pressure which is equal to 10 dyne cm . It is approximately equal to one standard atmosphere. The micro- bar (pbar), as used in this work, is the c.g.s. unit of pressure 2 and equals 1 dyne cm .

Acoustic Energy Density at a point is the sound energy per unit 3 volume at that point. The c.g.s. unit;erg cm .

Characteristic Impedance is the density (p) of a medium multiplied by the velocity (c) of sound in that medium.

Specific Acoustic Impedance is the complex ratio of the sound pressure to the sound particle velocity at the point in the medium in which sound waves are being propagated. 231

LIST OF SYMBOLS

A area

"response" of waveguide at depth z in tank

C capacitance

C critical damping resistance c C static capacitance o C' measured capacitance

E energy density; Young's elastic modulus; intensity (power) level

E. field strength 3 F force

F function of variables 1 F function of variables o F function of variables 2 I displacement current o I ''motional" current m J Bessel function of the first kind, of order m m K decay rate; coupling factor

L length

L characteristic length c M mass

N number of normal modes

P peak pressure

P. polarization 1 232

Q quality factor of system; charge

R radius of spiral; expression of room size; reference distance

R resistance e S function of variables; surface area

S function of variables 1 T time

✓ volume, voltage V pp peak-to-peak voltage V peak voltage p root-mean-square voltage ✓r.m.s. W energy

X reactance (at resonance)

Rh stress

Z impedance; axial distance

Zm "motional" impedance

a radius

velocity of sound c hk stiffness constant d m mean diameter eh piezoelectric stress constant f function of variables g acceleration due to gravity

unit vector 233

nth solution of Bessel function, of order m jm,n k wave number (= 27/X); spring constant

1 (X) length, depth

1 , 1 , 1 length of sides (of a rectangular enclosure) x y z m mass

n , n , n integers x y 2 p pressure

✓ distance variable in radial direction t time

u particle velocity

particle velocity; velocity of wave propagation in a waveguide

v o "bar" velocity vg group velocity

v phase velocity p x distance variable y distance variable z distance variable

a nth solution of Bessel function of the first m,n kind, of order m, divided by it

axial strain

ep axial strain proportional to pressure

power efficiency 234

angle

A wavelength

= v/v . rigidity modulus C2 frequency v characteristic frequency c

density

Poisson's ratio

4) phase angle; characteristic function w angular frequency (2wv)

bandwidth 235

REFERENCES

ABRAMSON, H. N. (1957). J.A.S.A. 29, 42. Flexural Waves in Elastic Beams of Circular Cross-Section.

BALTRUKONIS, J. H., GOTTENBERG, W. G., SCHREINER, R. N. (1961) J.A.S.A. 33, 1447. Axial-Shear Vibrations of an Infinitely Long Composite Circular Cylinder.

BANCROFT, D. (1941). Phys. Rev. 59, 588. The Velocity of Longitudinal Waves in Cylindrical Bars.

BATCHELDER, L. (1963). J.A.S.A. 35, 775. Investigation of a Reverberant Sound Field in a Concrete Sonar Test Tank.

BAUER, L., TAMARKIN, P., LINDSAY, R. B. (1948). J.A.S.A. 20, 858. The Scattering of Ultrasonic Waves in Water by Cylindrical Obstacles.

BOLT, R. H. (1939). J.A.S.A. 10, 228. Frequency Distribution of Eigentones in a Three-Dimensional Continuum.

BRITTON, W. G. B., LANGLEY, G. O. (1968). J. Sound Vib. 7(3), 417. Stress Pulse Dispersion in Curved Mechanical Waveguides.

CHREE, C. (1889). Trans. Camb. Phil. Soc. 14, 250. The Equations of an Isotropic Elastic Solid in Polar and Cylin- drical Co-ordinates, their Solutions and Applications.

CHEVALIER, J. (1967) H.C./061-215/1. Service des Etudes et Recherches nuclgaires, thermique et hydrauliques, Electricitg de France, Chatou, France (Oct. 1967). Veine d'essai destine a l'observation comparg due dgbit de cavitation dans l'eau et le sodium liquide.

COOK, R. K., CHRZANOWSKI, P. (1946). J.A.S.A. 17, 315, Absorption and Scattering by Sound Absorbent Cylinders.

DAVIES, R. M. (1948). Phil. Trans. A240, 375.

FAHY, F. J. (1968). Annual Report of Institute of Sound and Vibration. Research (University of Southampton). June 1968.

FILIPCZYNSKI, L. (1962). Proc. Vib. Problems (Warsaw). 3, No. 3(12), 241. Propagation of Ultrasonic Waves in Spirals. 236

FOX, F. E., GRIFFING, V. (1949). J.A.S.A. 21, 352. Exper- imental Investigation of Ultrasonic Gain in Water Due to Concave Reflectors.

HAMMITT, F. G. (1968). La Houille Blanche (France), 23(No.l), 31. Cavitation Phenomena in Liquid Metals.

HARBOLD, M. L., STEINBERG, B. M. (1963). J.A.S.A. 35, 773. Three-Dimensional Model of the Sound Field Surrounding a Rigid Obstacle.

HOLTZ, R. E., SINGER, R. M. (1968). A.I.Ch.E. Journal (June Edn). Incipient Pool Boiling of Sodium.

HUDSON, G. E. (1943). Phys. Rev. 63, 46. Dispersion of Elastic Waves in Solid Circular Cylinders.

HUETER, T. F. (1950). J.A.S.A. 22, 514. Ultrasonic Velocity Dispersion in Solid Rods.

HUNTLEY, N. R., YOUNG, H. C., GRINDELL, A. G. (1966). Cavita- tion Forum, A.S.M.E., p.15. The Cavitation Characteristics of Two Types of Electromagnetic Pumps in Potassium.

KOYAMA, T. (1933). Proc. Phys. Math. Soc. Japan. 15, 369.

KUMAR, R. (1,1966). Acustica, 17, 47. Purely Radial Vibra- tions of an Infinite, Isotropic, Composite, Hollow Cylinder.

KUMAR, R. (2,1966). Acustica, 19, 21. Axially Symmetric _ Vibrations of a Finite Isotropic, Solid Circular Cylinder, immersed in an Acoustic Medium.

LANGLEY, G. O. (1968). Ph.D. Thesis. (University of London).

LAX, M., FESHBACH, H. (1948). J.A.S.A. 20, 108. Absorption and Scattering for Impedance Boundary Conditions on Spheres and Circular Cylinders.

MAA, D. Y. (1939). J.A.S.A. 10, 235. Distribution of Eigen- tones in a Rectangular Chamber at Low Frequency Range.

MACLEOD, I. D. (1966). Private Communication (U.K.A.E.A., Risley).

MACPHERSON, R. E. (1967). Preprint from: Confgrence Inter- nationale sur la Saret6 des Macteurs a Neutrons Rapide, Aix-en-Provence, France, p.19. Techniques pour stabiliser .1t4bullition en vase de m4taux liquides. 237

MASSEY, L. (1967). Ph.D. Thesis. (University of London)

McSKIMIN, H. J. (1956). J.A.S.A. 28, 484. Propagation of Longitudinal Waves and Shear Waves in Cylindrical Rods at High Frequencies.

MEASURES, R. M. (1965). Acustica. 15, 133. Acoustical Propagation of Thermally Induced Stresses.

MEITZLER, A. H. (1961). J.A.S.A. 33, 435.

MORSE, P. M., BOLT, R. H. (1944). Rev. Mod. Phys. 16(2), 69. Sound Waves in Rooms.

PEARCY, C. (1965). Electronic Engineering (Oct. Edn). High Voltage Generation from Piezoelectric Ceramics.

POCHHANNER, L. (1876). J. Reine Angew. Math. (Crelle) 81, 324. Uber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einen unbegrenzten isotropen Kreiszylinder.

POLLARD, H. F. (1962). Aust. J. Phys. 15(4), 513. Resonant Behaviour of an Acoustical Transmission Line.

RAYLEIGH, Lord (1904). Phil Trans. Roy. Soc. 203, 87.

ROE, G. M. (1941). J.A.S.A. 13, 1. Frequency Distribution of Normal Modes.

SINGER, R. M. (1967). Argonne National Laboratory Report, ANL-7337. (Jan. issue).

SMITH, P. W., SCHULTZ, T. J. (1961). Report by Bolt, Beranek and Newman Inc. On Measuring Transducer Characteristics in a Water Tank.

SPILLER, K. H. GRASS, G., PERSCHKE, D. (1967). Atomkernenergie, 12, No. 3/4, 111.

SKUDRZYK, E. (1958). J.A.S.A. 30, 1140. Vibrations of a System with a Finite or an Infinite Number of Resonances.

STEPHENS, R. W. B. (1959). Nat. Trade Press - Nondestructive Testing - 20. The Application of Damping Capacity for Investigating the Structure of Solids.

STROH, W. R. (1959). J.A.S.A. 31. 234. Direct Measurements of the Acoustic Ratio. 238

TIMOSHENKO, S. P. (1921). Phil Mag. 41, 744. On the Corr- ection for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars.

TIMOSHENKO, S. P. (1922). Phil. Mag. 43, 125. On the Trans- verse Vibrations of Bars of Uniform Cross-Section.

WARREN, A. G. (1930). Phil Mag. 9, 881. Free and Forced Oscillations of Thin Bars, Flexible Disks and Annuli.

WATERHOUSE, R. V. (1955). J.A.S.A. 27, 247. Interference Patterns in Reverberant Sound Fields.

WATERHOUSE, R. V. (1958). J.A.S.A. 30, 4. Output of a Sound Source in a Reverberation Chamber and other Reflecting Environments.

WEBB, H. D. (1956). - Electtdnic Design (Jan Edn). Accurate Oscilloscope Phae Shift Measurements.

WHITE, J. E. (1958). J.A.S.A. 30, 65. Spherical Waves Coupled to a One-Dimensional Wave Guide.

WIENER, F. M. (1947).. J.A.S.A. 19, 44. Sound Diffraction by Rigid Spheres and Circular Cylinders.

WOOD, G. M. (1963). Trans A.S.M.E. Ser D. J. Basic Engr. 85(No.l), 17. Visual Cavitation Studies of Mixed Flow Pump Impellers.

WOODWARD, B. (1966). M.Sc. Dissertation (University of London). Acoustic Waveguides.

ZEMANEK, J. (1962). Tech. Report No. XVIII. (University of California) USA.

ZEMANEK, J. (1962). J.A.S.A. 34, 734. Experimental Invest- igation of Elastic Wave Propagation in an Aluminium Cylinder. 239

BIBLIOGRAPHY

BLITZ, J. (1963). Fundamentals of Ultrasonics (Butterworth, London).

HUETER, T. F. BOLT, R. H. (1955). Sonics (Wiley, New York).

KOLSKY, H. (1963). Stress Waves in Solids (Dover Publications, New York).

LAMB, H. (1924). The Dynamical Theory of Sound (Arnold, London). (Also published by Constable, London, 1960).

LAMB, H. (1924). Hydrodynamics (Dover Publications, New York).

MORSE, P. Mi (1948). Vibration and Sound (McGraw Hill, New York). Published previously in 1932, 1936, also a Dover publication.

RAYLEIGH, LORD. (1945). Theory of Sound (Dover Publications, New York).

REDWOOD, M. (1960). Mechanical Waveguides (Pergamon Press, London and New York).

STEPHENS, R. W. B., BATE, A. E. (1966). Acoustics and Vibrational Physics (Arnold, London).

TIMOSHENKO, S. P. (1941). Strength of Materials (Dover Publications, New York).

WATSON, G. M. (1922). A Treatise on the Theory of Bessel's Functions (Cambridge University Press). 240

" All studies here I solemnly defy "

William Shakespeare (Henry IV, Part I)