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Journal of Rese arch of the Nati onal Bureau of Standards Vol.^ 59,^ No.4,^ October^19 57 Resear ch Paper 2795
Speed of Sound in Water by a Direct Method 1
Martin Greenspan and Carroll E. Tschiegg
The speed of so und in di st illed water wa,s m eas ur ed over the temperature ra nge 0° to 100° C with an acc ur acy of 1 part in 30,000. Th e resu l ts are gi ven as a fifth-d eg ree pol y- nomia l a nd in tab les. The water was contained in a cylindrica l tank of fix ed leng t h, te rmi- nated at eac h end by a pla ne transducer, and the en d-to- e nd time of flight of a pu lse of so und was d ete rmined from a m eas ur ement of t he pul se -r e pet i tion frequency re quired to set th e success ive echoes into t im e co incidence.
1. Introduction
The speed of sound in water, c, is a physical
property of fundam ental interest; it , together with the dens it y, d ete rmines the adiabatie compressibility, a nd eventually the ratio of specific heats. Th e vari- ation with temperature is anoma lous; water is the o nl y pure liquid for which it is known that the speed of sound does not decrcase monotonically with temperature. There is also a practical interest in c in that wat er is used as a standard liquid for the calibration of instruments that measure the speed of so und in liquids automatically , both in the laboratory and in the field. In fact , it was in co nnection with the cali- bration of su ch "velocimeters" [1] 2 that our interest in this work was first aroused. In the first place, the available data scatter widely, as recent s um - maries [2 , 3] cle arly show. In man y cases, the di s- crepancies far exceed the claimed accuracy or at least the precision of the method s, even when the methods compared ar e the same. In the second place, there exists no se t of data that gives a s mooth variation with temperat ure over any considerable range. In particular , the best of these data yield calibration curves for OUT velocimeters which ar e bad ly curved instead of st raight (as they should be), and about which the data sc atter irregularly, but reproducibly. The results here presented are free of these objections.
2. Method
At the top of figure 1 is a sch ematic of the appa- ratus. The samp le is confined in a tube of which the ends ar e plane, parallel, electroacou st ic trans- ducers, quartz crysta ls in thi s case. If the l eft -hand cr ysta l, say, is excited by a short pul se from the blocking oscillator, the oscilloscope, which meas ur es the voltage on tne right-hand cr ysta l, will show a received pulse and a series of echo es, as indicated in idealized form on the lin e below (fig. 1). The pulse repetition frequency of the blocking oscill ator is controlled by a sine-wave oscillator, and if this frequency were adju ste d so that each blocking oscil- lator pulse coincided with the first received pulse of the next pre ceding cycle, then the oscillator period would equal the time of flight of the pul se. However, I 1 'b is work was supported in p art by the Omce of Nava l R esearch und er contract NA-oTU"-7<J-4S. , Figur es in brackets indi cate the literature referen ces at the end of this paper.
as the two pu lses have different shapes, the accurac.,' with which the coincidence could be set would be very poor. In ste ad , the oscill ator is run at about half this frequency and the coincidence to be set is that among the first received pulses corres ponding to a particular el ectr ical pul se, the first echo co rr espond- ing to the electrical pulse next preceding, and so on. Figure 1 illu strates the uccessive si gna ls co rr espond- ing to three electrical input pu l es. Th e input pul ses fall halfway between the pulses for which the coinci- den ce is se t, so that they do not tend to overload the amp lifier or distort the osci ll oscope tra ces. Th e period of the oscillator, when prop erly se t, multi- plied by twice the l engt h of Ul e tank , is the speed of s ound in the sample. The oscilloscope trace actua lly looks like that shown in the in se t (fig. 1). The firs t cy cle corre- s ponds to sound reH ec ted from the inn er faces only of the transducers, whereas the succeeding cycle co rr espond to s ound reflec ted. one or mor e times from an outer face. Therefore, the coincidence is set by maximizing the peak on eitllcr the first or second balf-cy cl e; the same res ult is obtained in either case but the sec ond half cycle is easi er to use because it is bigger. What we are mea urin g h ere is the speed co rr esponding to the fi1" t arrival of the si gna l ; in a nondispersive liquid this is the same as the ph ase veloc ity. It is true that the co in cidence is made at a
n
n I
I ~I n FI GU RE 1. Schematic of method. 'fhe t hr ee lower lines s ho w in idealized form the eve nt s co rrespond ing to t hr ee sneccssh'e elect ri cal pulses. Th e short , t hick line represents Lhe input pulse.
Lime one-fourth or three-quarters of the transducer period lat er Lhan the til!le o~ fir st arrival, b y. which time there is opportumty for sOlll;d travclmg .by paths other t han the shortest to affect the 10e~ t JOn of the maximum. How ever, the result s ar e lllde- pendent of wh et her til e fir st or sec ond half c'y cl ~ is used · they ar e also not affected by s ub st ltutmg crystal s of twice the thickness, or by changing the diamet ers of tlle tank, or of the hot electrodes. The se results lead us to believe that the error introduc ed by th is maximiz at ion technique is negligible. Th e question ha s been examined also in another way. Suppose a coin ciden ce to hav e been mac~ e at
frequency j; oth ers can then be ma.de at submultlJ?les
of j. At the frequency / /2 for lI1 st an ?e, t he. fir st received pulse c orr es pondmg to a partI cular mput pulse co incides with the seco nd echo (not the first, as before) co rr es ponding to the electrical pulse ne xt pr ecedin g, and so on. Effectively , the sOllnd pul~ e is timed over a path twice as long as before. It IS found that t he m eas urement s at j and ncar j /2 are s ub st antially id entical , so that the error in qu estion is less tha n, or at mo st comparabl e to, the e xpen- m ental error of the time m eaS llrement.
- Appa r a t us
3 .1. The Dela y Line
Th e disasse mbl ed dela y lin e is shown in the photo- g raph , figure 2. Th e length of the tank i s about 200 mm , and the bore about 13 mm. Th e filling holes are sealed by plugs having T eflon gaskets; a s mall hole in one plug provid es pre ss ur e.release. Th e tank is of a chromium steel 3 which , after h eat treat m en t, ta k es a good optical fini sh.. Because t hi s stee l is not so corrosion r es ista nt as th e nickel- chromium sta inless steels, the bore of the tank was h eavily gold plated. Th e end s of the tank are opti cally flat and parallel to within less than 1 f.1. To these ends are car efully wrung the 0. 8 -mm thick x-cut quartz crystals, which also are optically flat. Th e cap s, wh en bolt ed on, clamp the cr ysta ls through neopren e O-rings. A coaxial ca bl e pa sses through a seal in each cap , and the ce nt er c ondu ct or mak es contact with the outer (hot ) electrode of the cr ysta l through a light sprin g. Th e outer electrode is a 9 mm circle of aluminum- backed pr ess ur e-sens itiv e adhesive tape. Th e inner (ground ) electrode is of fired-on gold and is about 12 mm in diam eter. Contact is mad e through a light gold-plated h elical spring which touches the electrode around the edge and be ar s on a shoulder ma chined into the bor e. Th e inner el ect rodes and s prin gs are unn ecess ar y if th e sa mpl e ha s high con- du ctiv i ty or a high dielect ri c con st ant ; they are usually omitted for water and aqueou s so lution s of sa lt s. Figure 3 is a sch emati c drawing of one end of t h e assembly. Th e l engt h of t he tank was meas ur ed at 20° C, and th e coe ffi cient of thermal ex pan sion of the steel was measured on a sa mple cu t from th e same bar as 3 F ir tb - Ster ling t yp elB -4 40A.
FI GU R E 2. Delay line, disassembled. Abo ve tb e tank ar e tb e pl u gs whicb close th e filling holes. and at th e e nd s ar e tb e cap s tb rougb wh ich pass th e elect ri cal cables a nd w bl cb a lso clam p t he crystals seen in th e for eg round.
FIGURE 3. Schematic oj one end of the tank, showing th e crystal and cap assembly.
- 'rank ; 2. cap; 3. pl ug sealed wi th T eflon O-ring (th ermoco up le p asSf'S t h ro u gh pre ss ur e relief t u be); 4. qua rt z crystal wrun g on to e nd of tank (s pring s m a ke conta ct to elec trod es) ; 5, neopren e O· rin g; and 6, coaxial ca bl e.
was th e ta nk and h eat -t reated together with it. From these data, the leng th of th e so und path is known to better than 2 part s in 10 5 at any tempera- tur e b et ween 0° and 100 0 C. It i s, of course, necessary t hat t he crysta ls b e wrun g down with great care so that the fringes disappear all around the periphery, to achieve this accuracy. Th e clamp- ing g asket s mu st b ear dir ec tly o ver the conta ct ing surface and not s pr ead out over the un supported ar ea, else th e cryst al will benel. With these pr e- ca ution s, the delay line ma y be disasse mbl ed and reassembled repeatedly with reproducible results. If the cry sta ls ha ve been properl y wTung on and clamped, they ca nno t be removed by hand after several da ys, but mu st b e soaked off.
- -- J
.
. 04 I. _ • '" . 0 2 - "- ~.^.^ .-
vi ...J 0. '.. ' ~ 0 - /:)2 (^) -. Vi w Q: -.0^4
-.0 6
0 10 20 30 40 50 60 70 eo^90 TEMPERATURE. °c FIGURE 4. Deviations, r, of eq1wtion 1 from the data.
putcr SEAC by the method of l east squares, to a fifth-degree pol ynomia l,
(1)
The reduction in the resi dua l sum of squares over a fourth -degree polynomial, due to fitting the f-ifth - degree term , was statistioolly significant at a prob- ability level less than 0.005, and the deviations of the data from the fifth-degree pol ynomia l showed no statistically significant indication of l ack of random- ness. The deviations are plotted against tempera- t u re in figure 4.
The va lues of a, in eq (1), for c in meters per second
(m/s), and T in degrees C, are: a (^) o = 1,402.736; a[ = 5.03358; a2= - 0.0579506; a3= 3.31636 X 10 - 4 ; a4 = - 1.45262 X 10- 6 ; and a 5=3. 0449 X 10 - 9 • The stand - ard deviation of the measurements is 0.0263 m / s, or about 17 ppm. Estimated standard deviations of the va lues of c predicted by eq (1) were calcul ated for five representative temperatures. The results are given in table l.
TABLE l. E stimated standard deviation (s. d.) of values of c predicted by equation 1
Temperature
° C o 10 50 70 100
m/s
- 0114 . 0065 . 0058 . 0062 .0 14 5
s. d.
ppm
Tab le 1 and figure 4 make it clear that eq (1), togeth er with the listed constants, provides a satis- factory interpo l ation formula, and the errors intro- duced by its usc are small relative to the possible systematic errors of measurement (sec section 5). The values given in tab les 2 and 3 were calculated from eq (1) by SEAC. Table 2 gives the speed of sound in meters per second for each degree C from 0 to 100, and table 3 gives the speed of sound in feet per second at inter - va ls of 2 deg F from 32 ° to 212°. In each case, the differences, which are listed for convenience in inter- pol at ion, were calcul ated from a table hav ing more
significan t fig ures, so that on account of rounding-off errors, the tabu l ated differences in some cases differ by one unit in the l ast decimal place from the dif- fer'ences of the tabulated values of c. It is believed (sec section 5) that the systematic errors do not exceed 1 part in 30,000. The tab l es should, there- fore , be used in the following manner. In tab le 2, linear interpolation should be performed to the nearest 0.01 m/s and th e final resul t rounded off to the nearest 0.1 m / s. The error will then not exceed one-half unit ill the l ast place, i. e., 0.05 m /s. Linear interpolation in table 3 will yield errors that do not exceed 2 un its (0.2 fps) in the l ast place.
5. Discussion
Fo llowing is a l ist of the known possible sources of error and an estimate of the upper limit of each error. 5.1. Freque n cy
As al ready stated, the frequency was measured by co u nting cycles for 10 sec; the total count was
about 75 ,000. The inherent error is ± 1 count, but
in all cases the mode of at least three independent readings, of which, at worst , two were the same and the third different by one, was taken as the observed va lue. The counting error can thus be as great as 1 part in 75 ,000, but as it is random, the effect on the final results is negligible, as indicated in section 4. The 10 -sec time base was obtained by division from a I-Me crystal oscill ator which is stab le to 2 parts in 10 7 per week, and which was compar ed with signals from W, VV 01' from a local precision standard. The errors due to inaccuracies in the time base are, therefore, al so negligibl e.
5 .2. Len gth of Pa th
The length of the tank across its polished end s at 20 ° C was determined within IlL, i. e., 5 ppm. Ther- mal expansion measurements were made at 20 °, 60 °, and 100° C; the lengths at intermediate tempera- tures were calculated by quadratic interpolation. The maximum absolute error in the therma l expan- sion coefficient is estimated at 0.2 ppm; th is accu- mulates to 4 ppm at 0° C , and to 16 pp m at 100 ° C.
T A BLE 2. Sp ee d of sound in water, metric units
l' c '"
T c Ll. T c Ll. l' c^ A
---- ---- ----^ --------
°C mls mls °e mls^ m!s^ °e^ mls^ mls^ °C^ mls^ mls 1, 400 + 1,400+^ 1,500+^ I,^ .500+ 0 2.74^ ------ -- --^25 97 .00^ 2.71^50 42.87^ 1.^12 75 55 .4^5 -0. I 7.71^ 4.97^26 99.64^ 2.64^51 4 3.93^ 1.07^76 55.40^ -. 2 ] 2.57 4.86 27 '2.20 2.56 52 44.95 1.^02 77 55.3 1^ -. ~ 17.32^ 4.75^28 4.68^ 2.49^53 45.92^0 .9^7 78 55.18^ -.] 4 2l.^96 4.64^29 7.10^ 2.41^54 46.83^ .92^79 55.02^ -.
I 5 26.50^ 4.53^30 9.44^ 2.34^55 47.70^ .87^80 54.81^ -. 6 ~O.^92 4.43^ 3 1^ II.^71 2.27^56 48.51^ .82^81 54.57^ -. 7 3.1.24 4. ~2 32 13. 9 1 2.20 57 49.28^.^77 82 54.30^ -. 8 39.46 4.22 33 1605 2.14 58 50.00 .72^83 53.98^ -. 9 43.58 4.12 34 1 8. 12 2.07 59 50.68^ .67^84 53.63^ -. 10 47.59 4.02 35 20. 12 2.00 60 5 1.^30 .63^85 53.25^ -.39^ I II 51. 51 :1.92 36 22.06 1.^94 61 5 1.^88 .58^86 52 .82^ -. 12 55.34 3.82 37 23.93 1. 87 62 52.^42 .53^87 52.37^ -. 13 59.07 3.73 38 25.74 1.81 63 52.91^.^49 88 51.88^ -. 14 62.70 3.64 39 27 .4 9 1. 75 64 53.35^.^45 89 51.35^ -. 1.1 66.25 :3.0.1 40 29.18 1.^69 65 53.76^.^40 90 50.^79 -. 16 69.70 3.4f> 41 30.80 1.^63 66 54.11^ .36^91 50.20^ -^. 59
I
17 73.07 3.3, 42 32.37^ 1.^57 67 54.^ 4~^.^31 92 49.58^ -. 18 ifi .3 .1) :J.28 43 33.88^ 1.^51^68 54.70^ .27^93 48 .92^ -. 19 79.55 3.. 19 44 35.33 1.^45 69 54.93^ .23^94 48.23^ -. 20 82.66 3.11 45 36.72^ I.^39 70 55.^12.^19 95 47.50^ -. 21 85.69 3.03 46 3R06^ 1.^34 71 55.27^. 15^^96 46.75^ -.
I
22 88.63 2.9.5 47 39.34^ 1.^28 72 55.^ :.i7^.^11 97 4 5.96^ -. 23 91.50 2.87 48 40.57^ I.^23 73 55.44^ .07^98 4 5.^14 -. 24 94.29 2.79 49 41.^74 1.17^74 55.47^ .03^99 44.29^ -. 25 97.00 2.71 50 42.87^ I.^12^75 .):,),4.1^ -.01^100 43 .4^1 -.^88 1,400+ 1,.500+^ 1.^ 500+^ 1,500+ --- --- I
TM1L 'E; 3. Sp eed of s01md in water, Engli sh 1U~its
T ..'>^ l'^ A
_.-- -.--
OF (^) fps fps OF (^) fps fps 4, 600+ 4.900+ 30 80 25.7^ 9. 32 2.1 82 34.8 9. :j4 20.3 18. I 84 43.7 8. 3(1 37.9 17.7 86 52.2 8. 38 55.^1 li.2^88 60.5^ 8. 40 71.9 1 6.8 90 68.5 8. 42 88.2 16. :l 92 76.2^ 7. 44 104.1 15.9 94 83.6 7. 4n 119.6 15. 5 96 90.8 7. 48 134.6 1 ,i. I 98 97.7 6. ,10 149.3 14. i 100 "'4.4^ 6. .12 163.6 14.3 102 10.8^ 6. 54 177. 5 1 3.9 104 17.0^ 6. 56 19l. 0 1 3.5 106 22.9^ 5.^ ~ 58 '4.1 13.1 108 28.6^ 5. 60 16.9 12.8 lIO 34.0^ 5. 62 29.3 12.4^112 39.2^ 5. 64 4 1. 3 12.0^114 44.2^ 5. 66 53.0 J I. i 116 4 8.9^ 4. 68 64.4 I!.^4 11 8 53.5^ 4. 70 7,1.4^ II. 0 120 57.8^ 4. 72 86. I 10.7^122 6 J.^9 4.^1 74 96.4 10.4^124 65.8^ 3. 76 106 ..^1 10.1^126 69.4^ 3. 78 116.2 9.8^128 72.9^ 3. 80 125.7^ 9.4^ lao^ in,2^ 3. 4,800+ 5.000+
Thu s, the toLal uncertainty III th e length of the tank is a bout 5 ppm at 20° 0, and increases with
te mp era tU1'c both wa ys; at 0° 0 it becomes abou t
9 ppm and at 100° C, about 21 ppm. Th e question ar ises as to how closely the length of the sound path in the sample, i. e., th e distance bctween th e inner faces of the transdu cer s, approxi- mat es to the l ength of th e tank across th e ends to which th e crysta ls are wrung. Exp erience with developmental models show ed that unless the assem- bly \: e['e very carefully made , with particular
T Ll.^ T^ ..'> ---- ---- ---- o p (^) fps fps OF (^) fps fps 5,000+ 5.000+ 130 76.2 3.3 180 99.2 -1. 132 79.2 3.1 182 98.0 -1. 134 82. I 2.9 184 96.7 -1. 136 84.^ 2. 7^186 95.2^ -1. 138 87.3 2.5 188 93.6 -1. 140 89.6 2.3 190 91.8 -1. 142 9 1. 7 2.1 192 89.9 -1. 144 93. G 1.9 194 87.9 -2. 146 95.3 1.7 190 85.7 -2. 148 96.9 1.6 198 83.4 -2. 150 98.3 1.4 200 81.^0 -2. 152 99. ,I 1.2 202 78.4^ -2. 154 100. ,I 1.0 204 75.7^ -2. 156 lOt. 4 0.9 200 72.9^ -2. 158 102. 1 .7 208 70.0^ -3. 160 1 02.6. 6 210 66.9^ -3. 162 103.0 .4 212 63.7^ -3. 164 !O3.2. 166 103.2. 168 !O3. I -. liO 10 2.8 -. 172 102.4^ -. 174 101. B -. 176 101.^ I^ -. 178 100.2^ -. 180 99.2^ -^ l. 5.000+ ,i.^ 000+
attention to avoidan ce of clamping pre ss ur e too ncar th e un s upport ed arcas of the crystals, the crysta ls mi g ht defle ct enough to cause very sizable error s. Th e present design make s it possible to disasse mbl e and rea ssemble the delay line repeatedly without affect in g the re sul t by a de tectabl e amount; this holds true when th e crystals normall y used , which ar e 0.8 mm thick, ar e r e pla ced by crystals 1.6 mm thiclc It, therefore, app ear s that errors produ ced by mi s pla ce ment or deformation of th e crysta ls arc insignifican t.
