Serial No. 120 DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, SUPERINTENDENT PHYSICAL LAWS UNDERLYING THE SCALE OF A SOUNDING TUBE By WALTER D. LAMBERT Geodetic Computer Special Publication No. 61 PRICE, 5 CENTS Sold only by the Superintendent of Documents, Government Printing Officei Washington, D. C. WASHINGTON GOVERNMENT PRINTING OFFICE 1920 Monograph Serial No. 120 DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY w E. LESTER JONES, SUPERINTENDENT PHYSICAL LAWS UNDERLYING THE SCALE OF A SOUNDING TUBE By WALTER D. LAMBERT Geodetic Computer Special Publication No. 61 PRICE, 5 CENTS Sold only by the Superintendent of Documents, Government Printing Office, Washington, D. C. WASHINGTON GOVERNMENT PRINTING OFFICE 1920 PREFATORY NOTE. The use of the sounding tube, or depth recorder, has long been recognized as a convenient and rapid method for getting approxi- mate soundings in depths of 100 fathoms or less. It has never been considered an instrument for accurate surveying, and, indeed, the depths shown by two tubes of different patterns thrown overboard at the same point, or even by two tubes of the same pattern, have often exhibited surprising discrepancies. 1 Moreover, it does not appear that the method used in graduating tubes in current use has ever been published in detail. It is the purpose of this paper: (1) To provide as correct a scale as possible for the sounding tubes of the new Coast and Geodetic Survey type, the scale to be computed on assumptions definitely stated and by a stated formula; (2) to provide a method of correcting the depths as read directly from the scale for variations in temperature and in atmospheric pressure, so that the tube may be used in surveying as an instrument the precision of which is comparable with that of the lead; (3) to provide a com- pilation of physical data likely to prove convenient in a further study of the subject. This paper was nearly ready for publication two years ago, when the war delayed its completion. In the meantime, the first draft of the manuscript was read by the late Dr. K. A. Harris, of this Survey, and several suggestions were derived from his memoranda. Many of the tables were prepared with the help of the Section of Tides and Currents. 1 The more extreme discrepancies are due partly to accidents in the working of the tubes, partly to irregularities in their bore. 2 D, of J* N» CONTENTS Page. Prefatory note 2 General statement 5 General principles of the sounding tube 6 Approximate formulas 7 More accurate formulas and corrections: Departure from perfect gas 8 Vapor pressure of water 9 Correction of the reading of the tube for departure from assumed normal conditions 11 Amount of compressed air dissolved in the water 13 Absorption by the walls of the tube 17 Change in volume of the trapped water 18 Correction for variation in the density of the water and for the acceleration of gravity 20 Correction for fluctuations of the water surface 21 Situation of the point whose depth is recorded 21 Numerical data and construction of the tables 23 Summary of conclusions 25 Note 1. Rate of absorption of air by water 26 Note 2. Cer-tain questions in mechanics connected with the sounding tube 28 Note 3. The heat evolved by the compression of the air in the tube 29 Note 4. Compressibility of sea water 32 Schedule of mathematical notation: English characters 33 Greek characters 35 Examples in the use of the tables 35 Tables: Table 1. Miscellaneous physical data 37 Table 2. Density of sea water 38 Table 3. Vapor pressure of sea water 39 Table 4. Compressibility of sea water 40 Table 5. Absorption of atmospheric gases by sea water 40 Table 6. Pressure of sea water at various depths 41 Table 7. Table for scale of sounding tube 42 Table 8. Special table for scale of Coast and Geodetic Survey tube 43 Table 9. Corrections to sounding tube readings for temperature and pressure. 45 Table 10. Effect of absorption of air on the scale of a sounding tube. 45 ILLUSTRATIONS. Figure 1. — Diagram showing position of point where depth is recorded 22 Figure 2. — Diagram illustrating absorption of gas by a liquid 26 3 Digitized by the Internet Archive in 2011 with funding from The Library of Congress http://www.archive.org/details/physicallawsundeOOunit PHYSICAL LAWS UNDERLYING THE SCALE OF A SOUNDING TUBE. By Walter D. Lambert, Geodetic Computer. GENERAL STATEMENT. The idea of measuring the depth of water by the pressure it exerts is one that must have occurred to inventors, even in early times. It has been embodied in many forms of apparatus, 1 none of which, however, were ever widely and continuously used until Lord Kelvin (then Sir William Thomson) put his depth recorder on the market in 1871. Improvements in detail have since been made, but the fun- damental principle remains the same. The Tanner-Blish tube, now used by the United States Navy, was patented in 1899. The main advantage of these contrivances is that the sounding can be taken while the ship is under way. The volume of air follows, approximately at least, Boyle's law that the volume is inversely proportional to the pressure, and there- fore at great depths and pressures the volume is so small and varies so little in absolute amount, even with considerable change of depth, that the scale divisions become very minute, and, therefore, this type of instrument has seldom been used for depths over 100 fathoms. It is believed that by increasing the size of the instrument the sound- ing tubes of the United States Coast and Geodetic Survey type could be made useful up to 250 fathoms. To avoid the difficulty mentioned above, various devices have been invented, among them the following: (1) Apparatus similar in principle to an aneroid barometer. (2) Apparatus for measuring pressure by the change in volume of sea water itself or of some other fluid. (3) Apparatus in which the gas to be compressed has an initial pressure of several times the prevailing atmospheric pressure. (4) Apparatus in which, instead of measuring the variable volume of air under compression, a constant volume of more or less com- pressed air is allowed to expand till the pressure is reduced to the prevailing atmospheric pressure. The expanded volume is a measure of the original compression. None of these forms of apparatus has attained any wide use among navigators or hydrographers, and in this paper only those sounding tubes are considered in which the pressure of the water is exerted against air initially at atmospheric pressure. 1 Ericsson invented in 1836 a device similar to Sir William Thomson's. 5 6 U. S. COAST AND GEODETIC SURVEY. GENERAL PRINCIPLES OF THE SOUNDING TUBE. All sounding tubes are alike in fundamental conception, which is that of balancing the pressure of water against the pressure of com- pressed air. The tube is full of air of the same constitution, tempera- ture, and pressure as the air near the surface of the water. It is inclosed in a protective metal case in which it is dropped to the bottom where the depth is to be determined. The tube being open at the lower end, the pressure of the water compresses the air in the tube and the water enters to occupy space left as the air recedes under compression; from the relative amounts of air and water in the tube the water pressure, and therefore the depth, may be deduced. The fundamental differences between tubes of different makes lie in the method of ascertaining the amount of water in the tube. There are tubes in which the water rising in the tube leaves a record of the maximum height reached by it (which corresponds to the lowest depth). In one type the inner surface of the tube is of ground glass, which is wetted by the rising water and remains wet when the water falls in the tube as the latter is raised to the surface again. It has been found that the wetted area sometimes spreads upward a little in the tube after the water has withdrawn. This effect is probably due to capillary action in the roughened surface of the tube. In another type of tube the inner surface is coated with a substance that changes color when touched by sea water, so that the latter leaves an automatic record of the height to which it has risen. In the new United States Coast and Geodetic Survey tube the opening in the tube is closed by a valve and spring, the resistance of the latter being equal to the pressure of 1 fathom of water. When the depth of the tube is greater than 1 fathom, the water pressure overcomes the resistance of the spring behind the valve and the water enters and continues to enter until the tube has reached its lowest point. When the tube is pulled up again, the pressure outside the tube is less than the air pressure within, and the valve remains closed, thus trapping the water. The water is thus brought to the surface in the tube, and its amount may be measured in any convenient way. The way pro- vided for ordinary surveying purposes is by means of a graduated rod of uniform diameter. The rod, which is of known dimensions, is plunged into the tube until the water in it is about to overflow. The depth to which the rod must be inserted gives the amount of air space in the tube, and thus the depth. The depth may be read directly from the scale marked on the rod, or by marking the point to be measured with a sliding marker provided for the purpose, and laying the rod along a suitably graduated scale. For accurate experimental work the water could be measured by weighing it with care. This paper has been prepared with the new Coast and Geodetic Survey tube especially in view. Most of the considerations are, how- ever, of a general nature and will apply to any tube in which the SCALE OF A SOUNDING TUBE. 7 volume of atmospheric air is used as a measure of the water pressure. Whenever the point under discussion does not concern sounding tubes of older types, attention is drawn to the fact. APPROXIMATE FORMULAS. To graduate the scale of a sounding tube, a relation must be found between the volume of the air in the tube under compression and the corresponding depth. A fair approximation to the desired relation is readily obtained. Assume that the air at the surface exerts a pressure 1 Pi and has the absolute temperature t v As the tube de- scends this air is compressed by the water pressure, and the tempera- ture is thereby raised. However, the whole apparatus will quickly take on the temperature of the surrounding water, 2 and we shall assume that the compressed air in the tube has, in fact, the tempera- ture of the water at the point whose depth is to be measured. Let this temperature be t 2 on the absolute scale and let 8 be the mean density of the water between the bottom and surface, and let p 2 be the pressure due to the weight of the water. Then p 2 =--Ugh. (1) In this equation g is the acceleration of gravity, and 1c is a constant dependent on the units used. Suppose that the air obeys the laws of a perfect gas; then at the surface p.v^Rt^or, v^-- 1 - (2) In this equation R is the gas constant, its value depending on the units used. At the depth li, where the total pressure is p t +p 2 an d the temperature t 2 , the corresponding volume v 2 is given by m^_ m ( The ratio of the volume occupied by the compressed air to the original volume of the tube is, therefore, 3 v i *t Pi + ltty~h J For brevity in this publication the word pressure is used in the sense of intensity of pressure or force per unit area. 2 For the contrary case see note 3, p. 29. 3 The following reasoning may make the point clearer: Suppose for the moment that R is so chosen and that v is the volume in cubic centimeters occupied by 1 gram of air under standard conditions (temperature O °C, pressure 1 atmosphere). If the pressure be made p\ and the temperature h, each gram will occupy vi V cubic centimeters, and the tube containing V cubic centimeters will contain — grams of air under these conditions. The air in the tube being now subjected to pressure p\ + p° and temperature ti, each gram V Vi occupies volume vi cubic centimeters, or —grams of the compressed air in the tube will occupy V— cubic t'l Vi Vo Vo centimeters. — is, therefore, the fraction of the total volume that is occupied by the air. 1— — is the Vi Vi fraction of the volume occupied by the sea water. 8 U. S. COAST AND GEODETIC SURVEY. This simple calculation omits several rather obvious considerations : (1) At depths still within the range of the instrument the behavior of the air departs perceptibly from that of a perfect gas; (2) vapor tension of sea water has not been allowed for; (3) the fact has not been considered that the water in the tube may dissolve some of the compressed air in the tube, thus diminishing its apparent volume. MORE ACCURATE FORMULAS AND CORRECTIONS. DEPARTURE FROM PERFECT GAS. Within a range wider than is here needed the behavior of air under different conditions of temperature ft), pressure (p), and volume (t>) may be represented by Van der Waals's equation: (p+fy(v-b) = Rt. (5) In this equation a and b are quantities small in comparison with p and v. The numerical values of a and b depend on the gas under consideration and the units used in measuring p and v. In deriving equation (5) Van der Waals assigned the following physical interpre- tations to a and b : b represents four times the volume of all molecules concerned, and -5 is the pressure due to the mutual attraction of the molecules, which Van der Waals takes as inversely proportional to the square of the volume of the gas, so that a is the factor of propor- tionality. To square with the observed facts over wide ranges of temperature and pressure, a and b must be made variable, but for the compara- tively limited ranges involved in the present case it is quite sufficient to take them as constant. In the computation of the tables the values of a and b used were deduced such as to represent the results of physical experiment over about the range of conditions to which the normal sounding tube would be subject; that is, Van der Waals's equation was used as an empirical formula. The experimental data were taken from Winkelmann's Handbuch der Physik, volume 1, part 2, page 1259, and the values deduced are on p. 25 of this paper. Equation (5) by a simple transformation gives the value of p when v is known, but for our purposes it is better to have v given in terms of p. This would require the solution of a cubic equation; but by taking advantage of the smallness of a and b, which are taken as small quantities of the first order, equation (5) may be transformed into the following approximate form, which retains small quantities of the second order. The equation is readily derived by the method of successive approximations. Rt . , a T 2ab a 2 ~| ,_ N v= J +h -m + lim-(myf- (6) SCALE OF A SOUNDING TUBE. 9 The term in square brackets, even when multiplied by p, is nearly always negligible in our calculation, except when extreme refinement is required at great depths. VAPOR PRESSURE OF WATER. Equation (5) or (6) applies to dry air only. The pressure of the water vapor is added to the pressure of the air and is the same as if the latter were absent. Let us define the quantity jh more strictly and let it signify the pressure at the surface due to the air only and let us call p\ the pressure of the water vapor in the atmosphere immediately above the surface, be that what it may. At sea it will probably be close to the tension of saturated vapor cor- responding to the temperature of the air. When the air is com- pressed by water pressure, the water vapor is made to occupy a smaller space, and the air undoubtedly becomes over-saturated and the vapor condenses, the condensed moisture being added to the sea water in the tube. Even if the air in the tube were initially quite dr} T , it would take up moisture from the sea water in the tube and become saturated. The water given up by the air, or taken up by it, replaces or is replaced by an equal amount of water from outside of the tube, but in any case the amount is very small. The tension of saturated vapor over a salt solution is somewhat lower than it is over pure water. (For numerical values, see p. 39.) The pressure on the water may also affect the saturation tension of the water vapor above it, but only by a very minute amount. The pressure to be used in equation (5) or (6) for computing the volume under conditions at the surface is p l only. Let p r denote the pressure of saturated vapor above water corresponding to the salinity and temperature of the sea water in question. The pressure that compresses the air is : Total atmospheric pressure -I- pressure due to weight of water = Pi+p\+p 2 ) which is equal to the counterpressure exerted by the air and by the saturated vapor: that is, it equals p 2 +p' ', so that the air pressure by itself without the vapor tension, which is to be used in equation (6) to determine the volume of the compressed air, is p = Pi+p\+p 2 — p'- Using the more exact relation (6) in the same way as equation (1) was used to obtain equation (4), we have, instead of (4), the more exact relation v, \p 1 +p\+p 2 -p' RtjV'Kp, BtJ°) = ilp 1 +p\+p 2 -p'~\IW 2 ~M) Pl T L 1 VW 2 ~sd^J (7) With an assumed set of conditions of temperature, etc., and with the depth as the only variable, the formula (7) is readily adapted for tabulation, being less complicated than it looks. 152573°— 20 2 10 U. S. COAST AND GEODETIC SURVEY. It may be written in which v 2 _ C x ^1 V2 + *\2 ~ /pf\3 m equation (6). For depths less than 110 fathoms the effect of this term is only a few units of the fifth place of decimals, and may be allowed for by making C 3 slightly variable. The correc- v tion term to — is + or f 2ab a 2 1/ p 2 + C 2 \ l(Rt 2 r w 2 yj\ v, J + [w^(^] ( ^ +Q f' (lla) This term is to be added to the expression (8) for —, or may be allowed for by combining it with <7 3 . The quantity p 2 is nearly proportional to the depth. The small correction terms proportional to the square of the depth are worked out later. (See p. 24.) When there is a valve spring of known strength, as in the tubes of the Coast and Geodetic Survey pattern, its effect may be allowed for by noticing that it acts like the vapor tension of water (p f ) in reducing the resistance that must be sustained by the compressed air alone. To allow for the spring, formula (10) may be rewritten 2 = p 1 +p l , -p f s, (lib) in which s is the pressure of the spring expressed in the same units as the other quantities. A method practically equivalent to the foregoing is to compute the table using (10) in its original form, but taking p 2 in (8) for the depth 7i — n instead of for depth h, and to tabulate the results under depth 7i, in which n denotes the number of units in the depth of water exerting a pressure equal to the resistance of the spring. In the tubes of the Coast and Geodetic Survey pattern n = 1 fathom. SCALE OF A SOUNDING TUBE. 11 CORRECTION OF THE READING OF THE TUBE FOR DEPARTURE FROM ASSUMED NORMAL CONDITIONS. When the conditions vary much from those assumed and it is desired to make accurate allowance for that fact, as would be the case in experimental work, it will probably be most satisfactory to compute directly from formulas (7) to (11a). However, for the degree of accuracy required under ordinary working conditions, differential formulas suffice. These may be derived from the simpler basic formula (4) instead of from the more complex formulas (8) to (11). The effect of the omitted terms would seldom amount to more than 0.5 fathom for depths less than 100 fathoms. For completeness, however, the formula derived from the basic formulas (8) to (11) is also given. From (4), lo g Gr) =1 °g *2~log *!+log Pt-log (p t +Jc8gh) Differentiate this expression, keeping — constant. <7^ dt x dp x dpi+ledgdh *2 ^i Pi Pi + kdgh ' which solved for dh gives The quantity t^- is the height of a column of water exerting a pressure p x and is about 5J fathoms, since p x is always about 1 atmosphere. In deriving (12), dt l7 dh 2 , dt 1} and dp x were treated as infinitesimals. The corresponding finite increments Ah, At 2 , etc., may be substituted for them without serious error so long as -r~i -—> etc., are small. In accordance with a well-known principle, increased accuracy will be obtained by adding to each original quantity half the correspond- ing increment; that is, by replacing h by 1i-\--k" > t 2 by 2 2 + ~o~ 2 ' e ^ c - In this way (12) becomes Equation (13) gives a second approximation after an approximate value of Ah has been found by any method. ^VEY. 12 U. S. COAST AND GEODETIC SURVEY. A simplified form which will give this approximate value with sufficient accuracy, and may usually be used to give the definitive correction, is obtained by omitting the increments on the right-hand side; and by replacing both t 2 and t x by T, equal to a mean between them(V=^- 2 ). This gives M= T( 7fc+ S) (A<2_At?v> corresponding to dt x > 27° C., also 1 > yq ana Pi 1 oTv , corresponding to one inch difference in the barometer reading. The additional terms are (1) n£r* n ' m the correction for t 2 , (2) the factors 1 + WtS and 1 — g 1 p 1 , which in the approximate formulas are each replaced by unity, and (3) the terms in C 3 in the corrections for t x and Pu which together with the vapor pressure terms d QV ' —p') are omitted in the approximate formulas. The terms in -pj] 1 . p affects the correction for t 2 by about 1/20 L Z 1 of its amount in the extreme case supposed or by nearly half a fathom. The factor 1 + p 2 j- 2 iorp 2 + C 2 = 20 atmospheres, affects the correction for air temperature by less than j^ of its amount, or less than 0.03 fathom. The terms in C 3 amount to about tt^ of the total 1 100 corrections for air temperature and air pressure, or about 0.1 fathom and 0.04 fathom, respectively. The factor l—2 1 p 1 has less effect than the factor 1 + p^rV The effect of vapor-pressure terms dip/ — p') is practically independent of the depth. The vapor pressures are functions of the temperatures of the air and water, respec- tively, but do not vary proportionally to the temperature over any considerable range, and so have not been included with the tempera- ture terms. An inspection of Table 3 will show that Px'ovp' , particu- larly the former, might vary by 20 mm. of mercury = Wo atmos- phere corresponding to 0.14 fathom. AMOUNT OF COMPRESSED AIR DISSOLVED IN THE WATER. It is difficult to make a satisfactory allowance for the amount of compressed air dissolved in the water, because in practice there is never time for the process of solution to be completed. In order, however, to ascertain the maximum possible correction due to this cause, we shall work out the effect of absorption of air by water when continued till the latter is saturated. *The vertical bars indicate numerical value regardless of algebraic sign. 14 U. S. COAST AND GEODETIC SURVEY. At a given temperature the mass of gas absorbed per unit volume of water is, by Henry's law, proportional to the pressure of the gas; and since the density of the gas is very nearly inversely proportional to its pressure, the volume of gas absorbed per unit volume of water, or the ratio between the volume of the gas absorbed and the volume of the absorbing water, is, within wide limits, independent of the pressure of the gas. At ordinary temperatures water will absorb about 2 per cent of its bulk of air. (See p. 40 for more precise data.) When there is no shaking together of air and water, the process of saturating the water with the air is a decidedly slow one. (See note 1, p. 26.) Just how much this process would be hastened by the motion and jarring of the sounding tube incidental to taking a sounding, it would be impossible to predict from theory. The cor- rection to be applied in practice could be determined by experiment only for a rough average for ordinary working conditions. Table 10 (p. 45) shows the limiting values (1) when no absorption of air has taken place and (2) when the absorption of gas is complete. The scale to be used in practice would be very nearly that of the first limiting case for small depths and perhaps less near for larger depths for which the graduation is unfortunately more sensitive. The two scales on sounding tubes in general use are the Thomson scale and the Parmenter scale. The former scale is used on the tubes designed by Sir William Thomson (Lord Kelvin) and the second on the Tanner-Blish tubes made in the United States by D. Ballauf . No statement of the formula by which the Thomson scales were origi- nally made has been found. One of the scales was measured and the measurements examined to see whether any allowance was made for absorption, but apparently there was none, or, if any was made, it was very small. Lieut. Parmenter of the U. S. Navy, who computed the scales for the Ballauf tubes, has left an unpublished memorandum of his method. Unfortunately it is rather hard to follow. Appar- ently, however, allowance was made for absorption. His statement is "That at 100 fathoms the absorption of air was 0.13 per cent, and at 5 fathoms the absorption was 0.04 per cent." It is not clear whether 0.13 per cent of the original air space is intended or 0.13 per cent of the volume of water in the tube, the latter mode of statement being in accordance with our usual conception of the physical phenom- enon. A study of the figures of his table would seem to indicate that 13 per cent (not 0.13 per cent) of the air space itself may be meant. This would be about 0.13 inch at 100 fathoms. This allow- ance for absorption seems somewhat excessive. Tests of the new United States Coast and Geodetic Survey tubes under various pres- sures were made at the Bureau of Standards, and the results of these tests were examined to discover the amounts of absorption. SCALE OF A SOUNDING TUBE. 15 The tests, however, were not made with this phenomenon especially in view, and nothing definite could be learned from them except that, even at pressures equivalent to 100 fathoms of water, the effect of absorption on the size of the air space is probably less than 1 millimeter (0.04 inch), even when considerable time is allowed. The tubes, however, were not used under working conditions, but simply put into the testing apparatus and subjected to the required pres- sure. There was none of the moving or jarring incidental to the usual process of taking the sounding. A careful preliminary test of the new Coast and Geodetic Survey tubes made by the Coast and Geodetic Survey in Florida Straits in May, 1919, indicates that a scale computed on the assumption that absorption may be neglected gives good results, even at depths of 100 fathoms and more. Lieut. Parmenter's statement of the amount of the absorption is given as the result of 36 tests on the U. S. S. Prairie in 1900. Details of these tests are not given, so they can not be examined to see if the apparently large absorption can be otherwise accounted for. How- ever, it may be that Lieut. Parmenter's table is substantially correct on this point. Further experiment on this point is desirable. The method by which the table for the second limiting case, that of complete absorption, was computed is as follows: We may assume that the water, being continuously subject to the air pressure p 1 of about 1 atmosphere, has already absorbed the corresponding amount of gas. This assumption is found to be approximately correct. (See Krummel, Handbuch der Ozeanogra- phie, vol. 1, p. 296.) Under compression some of the air is absorbed into the water, which would tend to reduce the pressure; but the latter is maintained by the inflow of water to take the place of the air absorbed, and this process continues until equilibrium is attained. The earlier forms of sounding tube communicated freely with the water outside without any check valve intervening; and if unlimited time were allowed, there was no reason why all the air might not be absorbed into the water of the tube and ultimately be diffused into the surrounding ocean. In the United States Coast and Geodetic Survey tube this diffusion into the outside water is prevented by the valve. Even with this form of apparatus, the volume of air might be so small that in time the water in the tube alone would absorb all the air. This can not happen, however, at depths of less than 250 or 300 fathoms, and for such depths the tube is not used. In computing the volume of air remaining after absorption is complete it will be assumed that Henry's law is exact and that the density of the air is the same as for an ideal gas. These approxima- tions are amply accurate for the purpose in hand, since they are in 16 U. S. COAST AND GEODETIC SURVEY. themselves near the truth and are used merely to calculate a small correction, and not the principal quantity. Let p 2 be the density due to the pressure of Pt + p 2 . Let m be the number of grains of air absorbed by 1 cubic unit of water under unit pressure, and let Av be the diminution in volume of air due to absorption, so that v— Av is the reduced volume. Let w be the volume of water when equilibrium has been established. The water already contains the amount of air due to the atmospheric pressure p 1} so that the additional mass of air absorbed is that due to the additional pressure p 2 and is given by the equation p 2 Av = wmp 2 (17) and since the water flows in to replace the absorbed air, v— Av + w=V, (18) in which V is the entire volume of the tube. Let pj be the density of the gas at pressure p lf then /^'a±a. . (19) Pi Vi Then from (17), (18), and (19), by eliminating w and p 2 and solving for Av, we get (V—v) mp 2 Av- (£+')'■- mp 2 or, (V-v)^p 2 Av= Pl Vi+V2-~7 1 V2 rnpi Pi The quantity — — is a constant for a given temperature and is equal Pi to the fraction of its volume of air that water will absorb. Put — — Pi equal to a; then (V— v)a Av = i + ;H_ a eo) V2 The numerical value of a is about 0.02. It may be noted that the oxygen and nitrogen of the air are not absorbed into the water in the same proportion in which these gases exist in the atmosphere, the proportion of nitrogen in the dissolved air being less than that in the atmospheric air. The undissolved air is, therefore, of slightly different composition from ordinary atmospheric air, and in strictness its physical constants would be slightly different. This effect is obviously too minute to need further consideration. i SCALE OF A SOUNDING TUBE. 17 The constant a in equation (20) is 0.01942 and is derived as follows: From Table 5 on page 40 it appears that for a salinity 32.5 parts per thousand and a temperature of 10° C, 1 liter of sea water will absorb 6.51 c. c. of oxygen and 12.22 of nitrogen, or 18.73 c. c. in all. This figure refers to the volume at 0°. To reduce the volume to 10° with 283 sufficient accuracy for the purpose in hand, multiply it by ^5* which is the ratio of the absolute temperatures. The result is 19.42 c. c. per liter; that is, sea water at 10° will absorb 0.01942 of its own volume of air, or about 2 per cent, as previously stated. The effect of the air dissolved in the water on the vapor tension of the latter is also negligible. (See J. J. Thomson, Application of Dynamics to Physics and Chemistry, p. 173.) The effect of the dissolved air on the volume of the dissolving water is likewise negli- gible. The increase in volume of the water is about equal to the volume that the air would occupy under a pressure of 2,500 atmos- pheres. (See Winkelmann, Handbuch der Physik, vol. 1, pt. 2, p. 1521.) In the tubes of the new Coast and Geodetic Survey pattern there is a quasi absorption that can be easily guarded against. In measur- ing the amount of water contained in a tube it i ; s inverted and the air bubble rises through the w T ater. In this way minute air bubbles may be formed and remain in the water for some little time, thus increasing its apparent volume. To remove the bubbles the water may be thoroughly stirred with a fine wire. ABSORPTION BY THE WALLS OF THE TUBE. There is still to be considered the possibility that, under the pres- sure up to 20 atmospheres, the water or compressed air might perme- ate the walls of the tube to an extent sufficient to vitiate the measure- ments. This possibility is to be feared much more in the case of brass than of glass, which has long been used in accurate physical experiments at high pressure. With reference to brass or bronze the United States Bureau of Standards furnishes the following information. The rate of air or water passing through brass or bronze would depend upon many f actors f of which the following might be named : Composition of material ; condition of material, cast, rolled, forged, etc.; temperature; pressure; depth immersion; etc. The bureau has been unable to find any specific data on the rate of flow of air or water hrough brasses or bronzes. Carpenter and Edwards (Proceedings Institution of Mechanical Engineers, 1910, p. 1597) state that from their investigations, a pure copper-aluminum bronze contain- ing from 9 to 11 per cent of aluminum has the best ability to withstand high pressures (14 to 20 tons per square inch). This material when properly cast did not leak until just before rupture; the original article gives the necessary precautions which should be taken in casting. This alloy is also only very slightly attacked by fresh or salt water. 152573°— 20 3 18 U. S. COAST AND GEODETIC SURVEY. The possibilities of error from this source would be diminished by giving as little time for the pressure to act as may be consistent with other requirements. The experience of the Coast and Geodetic Survey does not indicate that the error arising from the permeability of the tube is serious, but no data have been obtained as to the exact amount or the rate of permeation. CHANGE IN VOLUME OF THE TRAPPED WATER. There is one particular in which an accurate calculation for a tube of the Coast and Geodetic Survey type differs from the calculation for a tube of the recording type mentioned on page 6. In the latter type the height of the water in the tube is recorded automatically in situ; in the Coast and Geodetic Survey tube the volume of the water is measured at the surface and under surface conditions of temperature and pressure (but see p. 20), whereas we are concerned with its volume under the conditions that prevailed at the depth measured. As on page 16 let w denote the volume of the water at the given depth, v the volume of air in the tube, and V the entire volume of the tube. Call Aw the total increment of w due to change of temperature and decrease of pressure experienced in going from the given depth to the surface and put Aw = A x w + A 2 w, (21) in which A x w is the increment due to change in temperature and A 2 w the increment due to the decrease of pressure. These increments are so small that they may be computed independently. Since v + w= V= constant, Aw = —Av. Let d 1 and 8 2 denote, respectively, the densities of the water at air temperature t x and water temperature t 2 . Then, since the mass of the water is unchanged by the change in temperature, wb 2 = (w + A 2 w)d l or A 2 w = w ^ 2 ~ dl) - (22) Since the 5's are nearly unity, we may put in rough calculations A 2 w = w(S 2 -\). (23) We have further, A t w = Wfxp 2 , ix denoting the coefficient of compressibility for the temperature and salinity of the water, and p 2 the water pressure. The vapor pres- sures J?/ and p' are much too small to need consideration. (See SCALE OF A SOUNDING TUBE. 19 Table 3, p. 39.) Since — (formula (8), p. 10) is the ratio of the volume occupied by the compressed air to the total volume of the tube, we have v=^ 2 Fand Therefore, Aw = A 1 w + A 2 w = w ^1' 2 + ^^ V(l -^)(m2> 2 + ^^)- (24) The corresponding correction to 7i, All, is found by *- _ A ,^ =Aw ,[r|(|)]=^(^ + A^). (25) dJi\vJ The minus sign is used before Av because we are reducing back from the volumes at the surface to the volumes under water, whereas the A's have been denned as the changes caused by going in the op- posite direction. w( — j is essentially negative. Ah may be sepa- rated into Ajt+Aji. where l-J M" d/vX *"?*' < 2G ) dh\vj which is the the correction due to change in pressure, and -j ^2 ^STSft*^ (27) bh\vj which is the correction due to change in temperature. A^ has been allowed for in computing Table 8 for the scale of the Coast and Geodetic Survey tube. Its value is small, being less than 2 fathoms for a depth of 100 fathoms. A 2 h has not been allowed for in computing Table 8, as a further correction of the same sort has to be introduced when the actual temperatures differ from the standard assumed temperatures, and the introduction of part of the necessary correction into the table itself would complicate rather than simplify matters. The quantity -^ — - is a function of the temperature, but 20 U. S. COAST AND GEODETIC SURVEY. is not a linear one even approximately, so that this term does not lend itself to combination with the other temperature corrections in i_2s equations (13) or (16). The quantity -\ / \ * s tabulated for each dJiyvJ depth on pages 42-44. Although .the expression for kt ( — ) T»o«rlil"ir n^ rtarhipprl frnm fimiatinna (R\ anr] f90^ in n?9/>i.iftfl it. ia r can readily be deduced from equations (8) and (20), in practice it is more convenient to get its numerical value from the differences in the v tabulated values of — by some one of the formulas connecting finite differences and derivatives. Under certain circumstances it may not be necessary to make the correction for change in volume of water due to change in temper- ature. If the difference in temperature is not great and the meas- urement of volume is made very promptly, the water may be as- sumed to have maintained its original temperature. If this assump- tion is made, it will be advisable to keep the graduated rods that are introduced to measure the volume at somewhere near water temperature. On the other hand, if the correction is to be applied, sufficient time should be allowed for the water to take on the tem- perature of the air. CORRECTION FOR VARIATION IN THE DENSITY OF THE WATER AND FOR THE ACCELERATION OF GRAVITY. Tables 7 and 8 were computed with a standard surface density of water equal to 1.025 and a standard acceleration of gravity (# 45 ) equal to theoretical gravity at sea level in latitude 45°. When act- ual conditions depart much from the assumed conditions, as when soundings are taken in fresh or nearly fresh water, a correction must be applied. The water pressure is proportional to the product (density) X- (ac- celeration of gravity) X (depth), or, with the notation previously used, p 2 = UgJi. (28) For a given reading of the tube scale, p 2 is constant, although each of its factors may vary. Logarithmic differentiation of (28) gives =-(M> By substituting finite increments for differentials, and denoting by A 3 h the correction for difference between the actual and standard values of density and gravity, we get SCALE OF A SOUNDING TUBE. 21 AS is to be taken so that 1.025 + A<5 shall represent the mean density from the surface down to depth h for the actual distribution in depth of salinity and water temperature and a pressure of 1 atmosphere; A5 can be deduced from Table 2 for the assumed conditions. No allowance is to be made for increase in the density of the water due to the pressure of the water above it, since this effect is small and has already been allowed for with sufficient accuracy (p. 24). The term — may be dropped, except in computations of unusual refine- rs ment. In such cases take ^i= -0.0026 cos 2 dt dx 2 (36) Fig. 2. — Diagram illustrat- ing abscrplion of gas by a liquid. (See Winkelmann, Handbuch der Physik, vol. 1, pt. 2, p. 1446.) The quantity Jc is a con- stant, the coefficient of diffusion. It is re- quired to build up a solution of this equation P=f(x, t), in which f(x, t) must satisfy the following conditions: (1) f(x } °° ) =p s ; in which p 8 is the gas density in the liquid when the latter is saturated. (2)f(x, 0) = except when x = 0, in which £ case f(x, t) is indeterminate. when x = l for all values of t 7 since the further end of the tube is closed and there is no flow of gas across it. To simplify the printing of exponential quan- tities with complicated exponents we shall use the notation exp (z) for e z , z standing for any expression simple or complicated and e being the base of the natural logarithms. *> U -0 A simple particular solution of (36) is p = exp( — 7ccH)sm ex (37) or p = exp ( — Jcc 2 t) cos ex In (37) c is an arbitrary constant. If we choose the sine function and put e = -sr, where n is an odd integer, we satisfy condition (3) ; and SCALE OF A SOUNDING TUBE. 27 by making use of the Fourier expansion, . f. sin y sin 5y , e e . , OQ> . 1=- sin yH — g-M =- ° (38) we can build up a solution of (36) that satisfies the require* 1 condi- tions; namely, 1 /-25K 2 A . 57rz "I) + 5 ex pv— if-; sm_ 2r J) (39) The mean gas density over the whole column of liquid p m is defined i r z Pm = T I P also when t = co , p m becomes p 3 . Winkelmann gives the following numerical values of k in c. g. s. units for temperatures of 16° C. (Handbuch der Physik, vol. 1, pt. 2, p. 1450.) Coefficient of diffusion Gas. k. C0 2 0.0000159 N 0000200 0000187 N 2 0000156 CI 0000127 NH 3 . . : 0000128 For a numerical example let us take Z = 50 centimeters, which, is rather shorter than the column of liquid in the sounding tube when the latter is used at a considerable depth. Let us consider also the case of nitrogen, 'which has a larger coefficient of diffusion than any other gas in the preceding table. Accordingly, 7t: = 2xlO~ 5 . In JiTT 2 t order for -j™- to be of such size that p m may approximate even roughly 5 X 10 7 to its limiting value p s , i must be large. Take t = » — seconds, or 7T" over 58 days; by computing from (41) we find that, even with this large value of t, p m is less than one-fourth (23 per cent) of the possible density of saturation. 28 U. S. COAST AND GEODETIC SURVEY. The values of the coefficients of diffusion given in the preceding table are for pure water. Although they appear to be somewhat larger for sea water, absorption, whether of fresh water or of sea- water, when effected by diffusion alone, is an extremely slow process. In sea water, under natural conditions, absorption is aided by convection currents due to differences of temperature or to differ- ences of salinity produced by evaporation. It is also supposed that particles of dust act as nuclei of condensation and as carriers for minute quantities of gas, and so hasten absorption at lower depths. It has been suggested also that there are nuclei of condensation of a electrolytic nature present in sea water which act as carriers and may serve to account for the observed difference in the rate of diffusion between air into sea water and air into pure water. 1 In the sounding tube absorption would be hastened by all the influences just mentioned, and probably much more by the motion of the water itself in entering the tube, and by the other motions and concussions incidental to the process of taking a sounding. The in- formation available leads us to conclude that if the correction for the quantity of air absorbed be omitted the resulting error in depth will be small, and that it will be better not to attempt to make the cor- rection until further tests have been made. NOTE 2.— CERTAIN QUESTIONS IN MECHANICS CONNECTED WITH THE SOUNDING TUBE. The problem has been treated hitherto as a statical problem; that is, as if either the tube descended to the bottom with extreme slow- ness or else as if the valve were not released to admit the water until the tube had reached its lowest point, and as if, furthermore, the water lost all momentum immediately after passing the valve opening. It is found that the tube will descend to a depth of about 105 fathoms in about 45 seconds. A freely falling body would cover this distance in about 6 seconds. The buoyant effect of water will dimin- ish the apparent acceleration of gravity and increase the time of descent, but not greatly. Most of the difference between 45 and 6 seconds is to be explained by the resistance of the water. This resist- ance implies a pressure which would be additional to the statical pressure of the water. Formulas for the motion of a body subject to a constant acceleration and to a resistance proportional to the square of the velocity (Newton's assumption, which is probably nearly correct) may be found in many works on mechanics. 2 1 For an account of diffusion under natural conditions and of experiments dealing with the rate of diffu- sion of gases through liquids, also references to the literature of the subject, see Krummel, Handbuch der Ozeanographie (vol. 1, p. 298). 2 Some idea of the total resistance encountered may be derived from the following general considerations. If a body of mass to in a resisting medium like water is acted on by a constant force like gravity and by the resistance of the medium, which is some function/ (v) of the velocity v, it being understood that resistance SCALE OF A SOUNDING TUBE. 29 In this case, however, the acceleration is not constant, since water is continually entering the tuhe, thus diminishing the buoyant effect of the water outside. The problem is thus rendered very complicated and requires for a numerical solution further experimental data, especially on the rate at which water under a given pressure would pass through the valve opening. It might be found that the opening was so large that, owing to the pressure arising from the motion, an amount of water would enter the tube in excess of the amount determined for the depth. If the valve opening is too small, it will require an excessive time to estab- lish equilibrium between the pressures inside and outside the tube On this account care should be taken not to raise the tube too soon to the surface. The matter seems to be one for experimental investigation. When the tube reaches the bottom, it is traveling with considerable velocity and may be stopped with more or less suddenness, depending on the nature of the bottom. The first effect of the shock would be to throw the water in the tube against the valve at the bottom, thus preventing the entrance of any more water. Possibly when the water in the tube rebounds from the lower end there might be a chance for more water to enter, but, if the valve opening is small, this does not seem likely. Much would also depend on how nearty instantaneous is the adjustment of the amount of water in the tube to the depth. If the opening is small, this adjustment would require time, and it would be well to investigate the effect of letting the tube remain at the depth to be sounded for a longer or shorter period. NOTE 3.— THE HEAT EVOLVED BY THE COMPRESSION OF THE AIR IN THE TUBE. In all formulas for the volume occupied by the compressed air it has been assumed that sufficient time had been allowed for the air to take on the temperature of the surrounding water. As the tube in practice is lowered and quickly raised again, it may be of interest to estimate how much heat must be given out by the tube in this short time in order that our assumption may be justified. The amount of heat evolved in compressing a gas depends on how the compression is brought about. We shall calculate the amounts on three simple suppositions. The formulas used will be found in almost any elementary book on thermodynamics, or may be readily increases with velocity, then the velocity -will increase more and more slowly as a certain limiting velocity V, called the terminal velocity, is approached. The resistance/O) depends, among other things, on the size and shape of the body. When the acceleration is practically zero as the terminal velocity is ap- proached, terminal resistance/( V)=mg, g being the acceleration of gravity. This gives the limit to which the resistance or total pressure on the body approximates. What would be the pressure intensity on a particular point— for instance, on the valve— can not be estimated, even approximately, without a knowl- edge of the form of body. 30 U. S. COAST AND GEODETIC SURVEY. deduced from the ones there given. The air is treated as a perfect gas, an assumption which greatly simplifies the formulas and gives more than sufficient accuracy for the purpose in hand, which is illustrative only. Case 1. — The compression is so gradual that the heat of compression is absorbed by the water as fast as it is involved and the air is all the while at the temperature of the surrounding water. This is called isothermal compression. Denote by J the mechanical equivalent of heat, by W the amount of work done compressing the gas isothermally at temperature t 1} from volume v t and pressure p x to volume v 2 and pressure p 2 , and H' the heat given out in the process. Then „, W Rt t , p 2 p x v x , p 2 p t v t , v t , -: In this case the heat given out is less than will be given out on any other admissible supposition. Case #.— Suppose the air to be surrounded by matter impervious to heat, so that heat of compression is retained. Suppose that the air retains its heat till the pressure p 2 is attained, and that the non- conducting layer is then removed. The temperature of the air, and the corresponding amount of heat and final volume attained under pressure p 2 will depend on how the pressure is applied. If the pres- sure is increased gradually, so that the volume and temperature are at every instant adjusted to the pressure, the compression is called adiabatic. If t" 2 and v" 2 denote the absolute temperature and the volume corresponding to pressure p 2 and if & denote the ratio of the specific heat of the air at constant pressure C p to its specific heat C at constant volume, C YJ that is, Jc=jf- the formulas for adiabatic compression are * 2 " = U° / V l Y (44) 2/ When the nonconducting layer is removed, the air cook at constant pressure p 2 to the temperature t t of the surrounding water and in so doing shrinks from volume v 2 " to v 2 , the volume it finally attained in Case 1. In so doing the heat II" given out is expressed by R" = C»{t 2 "-t>> -A^ft)"- 1 ] (45) SCALE OF A SOUNDING TUBE. 31 Case 3. — In this case, as in Case 2, the heat is to he retained by a nonconducting layer, but the pressure, instead of being increased gradually, is suddenly increased from jh to p 2 . When the volume v 2 " f and the temperature t 2 rn corresponding to pressure p 2 nav e been reached, the nonconducting layer is removed, the air cools to the temperature t 1} and shrinks to the volume v x of Case 1. The formulas for II'" , the heat given out, are H" f = C p [t 2 "' — t 1 ] -jCPi-A) ' (48) As a numerical illustration we take Pi—1 atmosphere, p 2 = 20 atmospheres, corresponding to a water pressure of 105 fathoms nearly, and ^ = 273° absolute = 0° C. If the C. G. S. system of units be used, we must take the pressure in dynes (1 atmosphere = 1.0132 X 10 6 dynes), and the unit volume will be that of a gram of air under standard conditions = 773.5 c. c. We have further, by experiment, <7 P = 0.238, and fc=g=1.4; T 1.0132 X10 6 X 773.5 . Qf)v1n7 ,,_. J= 97Qvn9Qg/ T~\ = 4 - 22 x 10 ergs- ( 49 ) and from theory, : 273X 0.238 a) The values of II will be expressed in gram calories per unit volume of air, and, to find the heat given out by each cubic centimeter of air at atmospheric pressure, we must divide the respective values of II by 773.5. Case 1. (Isothermal compression. Compression gradual, heat given out as fast as produced.) Each cubic centimeter of air at atmos- pheric pressure gives out enough heat to raise the temperature of 1 gram of pure water by 0.°072 C. Case 2. (Adiabatic compression. Compression gradual, heat is retained till pressure of 20 atmospheres is reached and is then given out.) The temperature of the air is raised by 369.°5, but each cubic centimeter of air at the original atmospheric pressure gives out only enough heat in cooling to raise the temperature of one gram of pure water by 0.°114 C. 32 U. S. COAST AND GEODETIC SURVEY. Case 3. (Compression sudden; heat is retained till equilibrium is reached at a pressure of 20 atmospheres; heat then given off.) The temperature of the compressed air is raised by 1482°, but each cubic centimeter of air at the orignal volume gives out in cooling only enough heat to raise the temperature of 1 gram of pure water by 0.°456 C. Sea water has a specific heat slightly less than pure water (0. 936 for water of salinity 32.5 parts per thousand, density 1.026 at 0°) ; and if instead of reckoning by grams of pure water we reckon the heating effect in a cubic centimeter of pure water the figures in Cases 1, 2, and 3 will be, respectively, 0.°077, 0.°122, and 0.°487. The actual heating effect evidently lies between Case 1 and Case 3, probably much nearer to the former. In any event, it seems plain that, in spite of the limited time spent in taking a sounding, the air can take on practically the temperature of the surrounding water, as has been assumed. NOTE 4.— COMPRESSIBILITY OF SEA WATER. The mean coefficient of compressibility /* of a substance between the pressures P and P+p is defined by the equation H = 50 In this equation v and v are the volumes corresponding to pres- sures P and P+p, respectively. The true coefficient of compressi- bility (ju ) at pressure P is the limit of the mean coefficient as p approaches zero, or 1 dv d n . ,_ 1S "° = -^ = -^ (l °s v) - (51) V D We have also, since — =— , the p's being densities corresponding to the v% Mo= | (logp) =i|. ; (52 ) The most thorough investigation on the compressibility of sea water is by W. V. Ekman, in paper No. 43 of the "Publications de Circonstance" of the "Conseil Permanent International pour l'ex- ploration de la mer" entitled, "Die Zusammendrueckbarkeit des Meerwassers." His final result in his own notation is: l0V= 1 +0 4 000186p ~ [227+28 - 33< ~ - 551P + - 004<3]+ TTOO [105 - 5 -0.87« + 0.02* 2 )]+(^^Y[4.5-0.U-^(1.8-0.060]. (53) SCALE OF A SOUNDING TUBE. 33 In this equation fi represents the mean compressibility between atmospheric pressure and p additional units of pressure, so that when p is zero the pressure is 1 atmosphere. The unit of p is the bar, that is, 1,000,000 dynes per square centimeter. 1 One standard atmos- phere = 1.01323 bare, or 1 bar = 0.98694 atmosphere; t represents the temperature in degrees centigrade; c is a quantity connected with the density 5 of the water at 0° C, in such a way that density at °° c - 1+ i&r On the basis of formula (53) Tables 4a and 4b have been calculated. It seemed more convenient, however, to tabulate the true compressi- bility for p = 0, to make the unit of pressure the atmosphere instead of the bar and to use as one of the arguments the salinity instead of the density at 0°C. This has accordingly been done. In strictness the tabulated true conpressibility, /z , applies only when p = 0; practically, it may be taken for the mean compressi- bility for all pressures within the range of present sounding tubes. To take into account, however, the terms of formula (53) that do not appear in ju , we may proceed as follows: Suppose the change in relative volumes to be expanded in a series of powers of the incre- ment of pressure, p. From (50) and (51), the first term is evidently PIjl ; that is, ~-^-=VoP+a 2 p 2 +a*P* , (54) the a's being coefficients independent of p, but for sea water dependent on the temperature and salinity. A brief table of the values of a 2 is given on page 40. The unit of pressure is the atmosphere. a 3 is very small, indeed; according to (53), it is, when reduced to the atmosphere as unit of pressure, ( 1.76 — r-^— jxlO -12 . If it is desired to compute the change in relative density, the expression to be used is ^^-° = MoP + (a 2 +Mo 2 )^ 2 (55) Po SCHEDULE OF MATHEMATICAL NOTATION. ENGLISH CHARACTERS. a Constant of Van der Waals's equation; definition, p. 8; numerical value, p. 25. a In equation on p. 22 only, special meaning. a 2 Coefficient in formula (55), p. 33, for change of volume under pressure. Numerical values in table, p. 40. a 3 See p. 33. 6 Constant of Van der Waals's equation, p. 8, numerical value, p. 25. 1 This unit is also called the megnbar, the bar being then defined as a pressure of one dyne per square centimeter. 34 U. S. COAST AND GEODETIC SURVEY. b In equation on p. 22 only, special meaning. B Used only in AB, p. 12. c Special meaning, p. 24; different special meaning in note 1, p. 26. Cj, C 2 , C 3 Denned by equations (9), (10), (11), and (lib), p. 10. Cp, C Y Specific heats at constant pressure and constant volume, respectively, in note 3, p. 29. / Used only on p. 24, and defined there. g Acceleration of gravity in general. g Q Acceleration of gravity at sea level. h Depth to which tube is submerged below sea surface; more specifically depth of bottom of air space, see p. 22. Working unit of h, the fathom. H' y H", H"' Quantities of heat, note 3, p. 29 only. J Mechanical equivalent of heat, note 3, p. 29 only. h General meaning, coefficient of proportionality between depth and water pressure; special meaning in note 1, p. 26, coefficient of diffu- sion of gas into liquid; second special meaning, note 3, p. 29 only, is a. l In note 1, p. 26 only, m Mass in general, note 2, p. 28; elsewhere number of grams of air absorbed by a cubic unit of water. n See p. 10, following equation (lib). p Pressure in general. Working unit of pressure the atmosphere. P and p Indicate particular pressures defined in note 4, p. 32. p x Atmospheric pressure at sea surface, exclusive of pressure due to water vapor, also a somewhat different meaning in note 3, p. 29. p\ Pressure at the surface due to water vapor. p / Pressure due to water vapor in the air space of the tube, and assumed to be the vapor pressure of saturation for water of the salinity and temperature of that surrounding the tube. p 2 Pressure due to the weight of the water, and also a somewhat different special meaning in note 3, p. 29. R The gas constant of Charles's law or of Van der Waals's equation. s See equation (lib), p. 10. S Salinity of water, see Table 1, pts. 1 and 2, p. 37. t Time in note 1, p. 26 only, elsewhere temperature. In the formulas temperatures are on the absolute scale unless otherwise stated. ^ Temperature of the air, always on absolute scale in formulas. t 2 Temperature of the water, always on absolute scale in formulas. t 2 /// , t 2 " Temperatures defined in note 3, p. 29. T Defined, p. 12. v To suggest volume in general, and in particular volume of air in tube as on p. 16, except in note 2, p. 28, where v is used for velocity. v Initial volume under compression. Note 4, p. 32. v l Volume of unit mass of air under pressure p x and temperature t x ; denotes a different special volume in note 3, p. 29. v 2 Volume of unit mass of air when tube is submerged ; denotes a different special volume in note 3, p. 29. v/, v 2 '\ v 2 '" Volumes defined note 3, p. 29. V Entire volume of the tube except in note 2, p. 28, where V denotes terminal velocity. w Volume of water in the tube. W Mechanical work equivalent to W . x Two distances, defined where they occur, pp. 22 and 26. z x Defined, p. 12. SCALE OF A SOUNDING TUBE. 35 GREEK CHARACTERS. a Volume of gas which unit volume of water will absorb. /3 Coefficient of compressibility of sea water. 5 Density of sea water in general. 5 Density of sea water at the surface. 5 1 Density of sea water at air temperature and atmospheric pressure. 5 2 Density of sea water at water temperal ure and al mospheric pressure. A Finite increment of quantity following, which see. A p A.,, A 3 Special finite increments, see pp. 18-20. e Base of natural logarithms = 2.71828 n Mean coefficient of compressibility of sea water. See note 4, p. 32. n True coefficient of compressibility of sea water, at atmospheric pressure, see note 4, p. 32. p Denotes density in general in note 4; elsewhere a gas density, particu- larly in note 1, p. 26. p lt p 2 Densities of air due to different pressures, p. 16. Pm> P 8 See note 1, p. 26. a Defined in note 4, p. 32.

v + W + a 2)v 2 } = 1.02375 {4,644 X10~ 8 X 20 + [(4,644 X10- 8 ) 2 - 0.774 X10" 8 ] 20 2 } 36 U. S. COAST AND GEODETIC SURVEY. p — p = 0.0009485, which agrees substantially with the result found by the first method. Example 3. — What is the correction to the depth for the difference of temperature between air and water due to the change in volume of the water with temperature ? (Table 9 gives only the effect of tem- perature on the volume of air.) The first-mentioned correction is needed only in a table in which the water is brought to the surface for measurement. (See p. 18.) Take, for example, air temperature, 24° C. water temperature, 10° C. salinity, 36, and depth, 27 fathoms. In the notation of formula (27), page 19, for 24°, ^ = 1.02442, and for 10°, 5 2 = 1.02775. From Table 8, b7i\vj Therefore, the required correction .i i«, 1.02775-1.02442 A , 1flK v 0.00333 = A 2 h = - 165 X - — 1 Q2442 or A z h== ~ 165 x ~T024~ = fathom. This example shows the necessity of applying the correction in accurate work, as the difference in temperature is by no means extreme, and, for a given difference of temperature, the correction varies directly as (?) &/aY bh\vj which increases rapidly with the depth. Example 4- — Suppose that the expansion of water due to the release of pressure on coming to the surface has not been allowed for in com- puting the scale. (The allowance has been made in Table 8.) What correction must be applied to the heights read from the scale; that is, what is Aji formula (26), page 19 ? Take depth and salinity as in example 3. For depth 27 fathoms, p 2 = 4.90 atmospheres (Table 6) . For 10° and salinity 36,m<> = 4474 X 10" 8 , and, as before, 1 = - 165. b7i\vj Therefore, A 2 ft= -165x4,474 XlO~ 8 X 4.90= -0.036 fathoms. At this depth, the correction is small, but A 2 h increases even more rapidly with the depth than Aji, so that toward the end of the table A 2 7i should not be neglected in accurate work. SCALE OF A SOUNDING TUBE. 37 Table 1. — Miscellaneous physical data compiled from various sourc* 1. Composition of m:\ water. The salinity of sea water is defined as tho number of prams of salts contained in l.OOO prams of sea wafer. The relative proportions of i ho various salts in sea water is almost consl mi the world over, except undei obviously peculiar conditions. The following may bo taken as represent al I vq: Amounts of various salts in 1,000 grams oi sea water; salinity, 85. Name. Common salt Magnesium chloride Magnesium sulphate Calcium sulphate Potassium sulphate — Magnesium bromide Calcium carbonate and traces of other substances. Total. Chemical symbol. NaCl... MgCl 2 . Mg SO, CiiSd,. K 2 S0 4 . Mg Bro. CaC0 3 Amount. 35. 00 Per cent of all salts. Grams. 27.21 77. 75 3.81 10.88 1.66 4.74 1.26 3. 60 .86 2.47 .08 .22 .12 .34 100.00 Since, in a dilute solution like sea water the various salts are partially dissociated into their ions, it is better to give simply the amount of the separate elements and composite ions. Amounts of various elements in 1,000 grams of sea water; salinity So. Name. Chlorine. . . Sodium — Magnesium Calcium Potassium . Bromine . . . Chemical symbol. CI. Na Mg Ca. K. Br. Amount. Grams. 19.32 10.72 1.32 .42 .38 .07 Name. Sulphuric acid ions Carbonic acid ions and traces other matter. Total Chemical symbol. SO4. C0 3 . Amount. Grams. 2.69 .08 35.00 2. The salinity, the density, and the chlorine coutent of sea water are connected with one another by the three following empirical formulas, the last of which is derived from the two preceding: S= 0.030+1.805 CI ffo = -0.069+1.4708 Cl-0.00157 (Cl)2+O.00O0398 (Cl)3 £=978.039 (1+0.005294 sin 2 ^.-0.000007 sin*2 \^ Salinity Temp.,°C/"\^^ 24 26 28 30 32 34 36 38 40 —2 1.01929 1. 01928 1. 01922 1. 01909 1.01892 1.01869 1.01842 1.01810 1.01775 1.01735 1.01692 1.01644 1. 01593 1.01538 1. 01480 1.01419 1. 01354 1. 02091 1. 02089 1. 02081 1.02068 1.02049 1.02026 1. 01998 1.01965 1. 01929 1. 01888 1.01844 1.01795 1.01744 1. 01689 1. 01630 1. 01568 1. 01503 1.02253 1. 02250 1. 02240 1. 02226 1. 02206 1.02182 1.02153 1.02119 1. 02082 1. 02041 1.01996 1.01947 1.01895 1.01839 1.01780 1.01717 1. 01652 1. 02415 1. 02410 1. 02400 1.02384 1.02364 1.02338 1.02308 1. 02274 1. 02236 1. 02194 1.02148 1.02099 1. 02046 1.01989 1. 01930 1. 01867 1.01801 1. 02577 1. 02571 1.02560 1. 02543 1. 02521 1.02495 1.02464 1. 02429 1. 02390 1. 02347 1.02300 1.02250 1. 02197 1.02140 1. 02080 1. 02017 1. 01950 1. 02739 1. 02732 1. 02720 1. 02702 1. 02679 1.02651 1.02619 1. 02584 1. 02544 1. 02500 1.02453 1.02402 1. 02348 1. 02291 1.02230 1. 02167 1. 02100 1. 02902 1. 02894 1. 02880 1.02861 1. 02837 1.02808 1.02775 1. 02739 1. 02698 1. 02654 1.02606 1. 02554 1. 02500 1. 02442 1.02381 1.02317 1. 02250 1. 03065 1. 03055 1. 03040 1. 03020 1. 02995 1.02965 1.02932 1. 02894 1. 02853 1. 02807 1. 02759 1. 02707 1. 02652 1. 02594 1. 02532 1. 02468 1.02400 1. 03227 1. 03217 2 1.03200 4 1.03179 6 1. 03153 8 1.03123 10 1. 03088 12 1. 03050 14 1. 03007 16 1. 02962 18 1.02912 20 1.02860 22 1. 02804 24 1. 02745 26 1. 02683 28 1. 02619 30 1. 02551 1 (Spec. pub. No. 40, U. S. Coast and Geodetic Survey.) SCALE OF A SOUNDING TUBE. 39 '. Table 3. — Vapor pressure of sea water for various salinities and U mperalures. The vapor pressures are givon in millimeters of mercury. For basis ol table see p. 23.) Temp., ° C Salinity. — 1. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. -1. 0. 1. 2. 3. 4. 5. 6. 7. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 4.24. 4.56 4.91 5.28 5.67 6.08 6.52 6.99 7.49 8.02 8.58 9.18 9.81 10.48 11.20 11.95 12.75 13. 59 14.49 15.43 16.43 17.48 18.59 19.77 21.00 22.31 23.68 25.13 26.66 28.26 29.95 31.73 4.24 4.56 4.90 5.27 5.66 6.07 6.51 6.98 7.48 8.01 8.57 9.17 9.80 10.47 11.18 11.94 12.73 13.58 14.47 15.41 16.41 17.46 18.57 19.74 20.98 22.28 23.66 25.11 26.63 28.23 29.92 31.69 in 4.23 4.55 4.90 5.27 5.65 6.07 6.51 6.98 7.47 8.00 8.56 9.16 9.79 10.46 11.17 11.92 12.72 13.56 14.45 15.39 16.39 17.44 18.55 19.72 20.96 22.26 23.63 25.08 26.60 28.20 29.89 31.66 12 It 4.23 4.55 4.89 5. 26 5.65 6.06 6.50 6.97 7.46 7.99 8.55 9.15 9.78 10.45 11.16 11.91 12.71 13. 55 14.44 15.38 16.37 17.42 18.53 19.70 20.93 22.23 23.61 25.05 26.57 28.17 29.85 31.62 4.22 4.54 4.89 5.25 5.64 6.05 6.49 6. 96 7.46 7.98 8.54 9.14 9.77 10.44 11.15 11.90 12.70 13. 53 14.42 15.36 16.35 17.40 18.51 19.68 20.91 22.21 23.58 25.02 26.54 28.14 29.82 31.59 16 4.22 4.54 4.88 5. 25 5.64 6.05 6.49 6.95 7.45 7.97 8.53 9.13 9.76 10.43 10.14 11.88 12.68 13. 52 14.40 15.34 16.34 17.38 18.49 19.66 20.89 22.18 23.55 24.99 26.51 28.11 29.78 31.55 is 20 4.21 4.21 4.53 4.r>:\ 4.88 4.87 5.34 5.24 5. 63 5.62 6.04 6.03 6.48 6. 17 6.94 (i.'.ll 7.44 7.43 7.97 7.96 8.52 8.51 9.12 9.11 9.75 9.74 10.41 10.40 11.12 11.11 11.87 11 86 12.66 12.65 13. 50 13.49 14.39 14.37 15. 33 15. 31 16.32 16.30 17.36 17.34 18.47 18.45 19. 63 19. 61 20.86 20.84 22.16 22.13 23.53 23.50 24.97 24.94 26.48 26. 45 28.07 28.04 29.75 29.72 31.52 31.48 Temp.,°C Salinity. 24 4.20 4.52 4.86 5.22 5.61 6.02 6.46 6.92 7.41 7.94 8.49 9.09 9.71 10.38 11.08 11.83 12.62 13.45 14.34 15.27 16.26 17.30 18.41 19.57 20.79 22.08 23.45 24.88 26.39 27.98 29.65 31.41 26 4.19 4.51 4.85 5.22 5.60 6.01 6.45 6.91 7.40 7.93 8.48 9.08 9.70 10.37 11.07 11.82 12.60 13.44 14.32 15.26 16.24 17.29 18.38 19.54 20.77 22.06 23.42 24.-85 26.36 27.95 29.62 31.37 28 4.19 4.51 4.85 5.21 5.60 6.01 6.44 6.90 7.40 7.92 8.47 9.07 9.69 10.35 11.06 11.80 12.59 13.42 14.31 15.24 16.22 17.27 18.36 19.52 20.74 22.03 23.39 24.82 26.33 27.92 29.58 31.34 30 4.18 4.50 4.84 5.21 5.59 6.00 6.43 6.90 7.39 7.91 8.46 9.06 9.68 10.34 11.04 11.79 12.58 13.41 14.29 15.22 16.20 17.25 18.34 19.50 20.72 22.01 23.36 24.79 26.30 27.88 29.55 31.30 32 4.18 4.50 4.84 5.20 5.58 5.99 6.43 6.89 7.38 7.90 8.45 9.05 9.67 10.33 11.08 11.77 12.56 13.39 14.27 15.20 16.19 17.22 18.32 19.48 20.70 21.98 23.34 24.77 26.27 27.85 29.52 31.26 34 4.17 4.49 4.83 5.19 5.58 5.98 6.42 6.88 7.37 7.89 8.44 9.03 9.66 10.32 11.02 11.76 12.55 13.78 14.26 15.18 16.17 17.20 18.30 19.45 20.67 21.96 23.31 24.74 26.24 27.82 29.48 31.23 36 4.17 4.49 4.83 5.19 5.57 5.98 6.41 6.87 7.36 7.88 8.43 9.02 9.65 10.31 11.01 11.75 12. .53 13.36 14.24 15.17 16.15 17.18 18.28 19.43 20.65 21.93 23.28 24.71 26.21 27.78 29.44 31.19 38 4.16 4.48 4.82 5.18 5.56 5.97 6.40 6.86 7.35 7.87 8.42 9.01 9.63 10.29 10.99 11.73 12. 52 13. 35 14.22 15. 15 16.13 17.16 18.26 19.41 20.62 21.90 23.25 24.68 26.17 27.75 29.41 31.15 4.20 4.52 4.87 5.23 5. 62 6.03 6.46 6.93 7.42 7 95 8.50 9.10 9.72 10.39 11.10 11.84 12.63 13.47 14.36 15.29 16.28 17.32 18.43 19. 59 20.82 22.11 23.47 24.91 26.42 2S.01 29.68 31.44 40 4.16 4.48 4.82 5.17 5. .56 5.96 6.40 6.86 7.34 7.86 8.41 9.00 9.62 10.28 10.98 11.72 12. .50 13.33 14.20 15.13 16.11 17.14 18.23 19.38 20.60 21.88 23.23 24.65 26.14 27.72 29.37 31.12 40 U. S. COAST AND GEODETIC SURVEY. Table 4. — Compressibility of sea watpr. 1 [Table 4a gives IO'Xm, m« being the true coefficient of compressibility at atmospheric pressure, for water of various salinities and temperatures. Table 4b gives IC^X^, «2 being the coefficient of the second term in the formula (55), p. 33 for the change in relative volume, for water of various salinities and temperatures. The unit of pressure in both tables is the atmosphere. Temperatures are in degrees centigrade.] Table 4a.— 10 8 Xm - — - __ Salinity. Temp., ° C. ~~~~~— ________ 5 10 15 20 25 30 35 40 5,106 4,942 4,808 4,701 4,617 4,554 4,509 5,037 4,880 4,751 4,648 4,568 4,508 4,465 4,970 4,819 4,695 4,596 4,520 4,462 4,421 4,905 4,759 4,640 4,546 4,473 4,418 4,378 4,842 4,701 4,587 4,497 4,426 4,374 4,335 4,779 4,644 4,535 4,448 4,381 4,330 4,293 4,719 4,589 4,484 4,400 4,336 4,287 4,251 4 659 5 4* 534 10 4* 434 15 4' 354 20 4 292 25 4' 245 30 4' 210 Table 4b.— 10 8 Xa 2 Temp., ° C. Salinity. 5 10 15 20 25 30 35 -0. 897 -0. 880 -0. 864 -0. 849 -0. 834 -0. 820 -0. 806 .841 .826 .812 .799 .786 .774 .762 .796 .783 .771 .759 .747 .736 .725 .761 .749 .738 .727 .717 .707 .697 .737 .726 .716 .706 .696 .686 .676 .723 .713 .703 .693 .683 .674 .664 -0. 720 -0. 710 -0. 700 -0. 690 -0. 680 -0. 670 -0.660 40 0. 5. 10 15 20 25 30 -0.793 .750 .715 .687 .667 .655 -0. 650 Table 5. — Absorption of atmospheric gases by sea water. [The tabular values are the number of cubic centimeters of gas that can be absorbed by sea water of the given salinity at the given temperature. The volumes are reduced to a temperature of 0°C., and a pressure of one atmosphere. Carbon dioxide (C0 2 ) is absorbed rather freely by sea water, but, since the total vapor pressure of C0 2 is only about 0.0003 atmosphere, its absorption need not be considered in connection with he sounding tube.] ' • 1. OXYGEN. ~~ — -——___ Salinity. Temp., ° C. -— . _^__^ 5 10 15 20 25 30 35 40 —2 c.c. 10.88 10.29 9.03 8.02 7.22 6.57 6.04 5.57 c.c. 10.53 9.97 8.75 7.79 7.03 6.40 5.88 5.42 c.c. 10.18 9.65 8.48 7.56 6.83 6.22 5.72 5.27 c.c. 9.84 9.33 8.21 7.33 6.63 6.05 5.56 5.12 c.c. 9.50 9.01 7.94 7.10 6.43 5.88 5.40 4.96 c.c. 9.16 8. 68 7.67 6.87 6.23 5.70 5.24 4.80 c.c. 8.82 8.36 7.40 6.63 6.04 5.53 5.08 4.65 c.c. 8.47 8.03 7.13 6.40 5.84 5.35 4.93 4.50 c.c. 8.12 7.71 5 6.86 10 6.17 15 5.64 20 5.18 25 4.77 30 4.35 2. NITROGEN. —2 19.45 18.56 16. 60 14.97 13.63 12.54 11.66 10.94 18.83 17.97 16.10 14.55 13.27 12.24 11.40 10.70 18.18 17.37 15. CO 14.13 12.91 11.93 11.13 10.46 17.61 16.77 15. 10 13.70 12.55 11.63 10.86 10.22 16.90 16.18 14.59 13.27 12.20 11.32 10.59 9.98 16.27 15.58 14.09 12.85 11.84 11.02 10.32 9.74 15.63 14.99 13.59 12.43 11.48 10.71 10.05 9.50 15.00 14.40 13.08 12.00 11.12 10.40 9.78 9.26 14.36 13.80 5 12.58 10 11.57 15 10.76 20 10.09 25 9.51 30 9.02 1 See note 4, p. 32. SCALE OF A SOUNDING TUBE. 41 Table 6. — Pressure of sea water at various depths, byJormula(85) pag< [Density 1.025; temp. 10°C; gravity as in lat. 45°.] Depth, h (fathoms). Water pressure, V». (atmos- pheres). Depth, h (fathoms). \\';iler pressure, Vi (atmos- pheres). 1 >epth, h ( fathoms). Water pressure, Pa (atmos- pheres). 1 0. 18142 .36284 . 54426 . 72569 0.90712 1. 08855 1.26997 1.45141 1.63284 1. 81427 1.99571 2. 17714 2. 35858 2. 54002 2.72147 2.90291 3.08436 3. 26580 3. 44725 3. 62870 3.81015 3.99160 4. 17306 4. 35452 4. 53597 4. 71743 4. 89889 5.08036 5. 26182 5. 44329 5. 62475 5. 80622 5. 98769 6. 16916 6. 35064 6. 53211 6. 71359 6. 89506 7. 07654 7. 25803 41 7. 43951 7. 62099 7. 80248 7. 98397 8. 16545 8. 34694 8. 52844 8. 70993 8. 89142 9. 07292 9. 25442 9. 43592 9. 61742 9. 79892 9. 98043 10. 16193 10. 34344 10. 52495 10. 70646 10. 88797 11. 06949 11. 25100 11. 43252 11.61404 11. 79556 11.97708 12. 15860 12. 34013 12. 52166 12. 70318 12. 88471 13. 06624 13. 24778 13. 42931 13.61084 13. 79238 13. 97392 13. 15546 13. 33700 14. 51855 81 1 1.70009 2 42 82. . 14. smc, i 3 43... 83. 15. 06310 4 44... 84. 15. 23544 5 45... 85... 15. 42629 6 46 86. . 15. 60784 47 87... 15. 78939 8 48 ss. 15. 97095 9... 49 89. . 16. 15251 10... 50... 90... 16. 33407 11 51 91 16. 51563 12 52 92... 16.69719 13 53 93 16. 87875 14... 54... 94. 17. 06032 15... 55 95. . 17. 24189 16 56 96 17. 42345 17 57 97 17. 60442 18. 58. . . 98. 17. 78660 19... 59 99... 17. 96817 20... 60 100... 18. 14974 21 61 101 18. 33132 22 62 102 18. 51290 23 63 103... 18.69448 24 64 104 18. 87606 25 65 105... 19. 05764 26 66 106 19. 23922 27 67 107 19.42081 28 68 108 19. 60240 29 69 109 19. 78698 30... 70 110. 19. 96558 31 71 Ill 20. 14717 32 72 112 20. 32876 33 73 113 20. 51035 34 74 114 20.69195 35 75 115 20. 87355 36 76 116 21. 05515 37 77 117 21. 23675 38 78 118 21. 41835 39 79 119 21. 59996 40... 80... 120... 21. 78156 42 U. S. COAST AND GEODETIC SURVEY. Table 7. — Table for scale of sounding tube. [Temperature of air, 15° C; of water, 10°; humidity of air, 100%; surface density of water, 1.025; gravity as in latitude 45 . 1 — is volume of air when volume of tube is taken as unity. than three significant figures and is always greater than unity.] is given to not more Volume of air space Vi Vi h\ vj 1 0. 82630 . 71529 . 63057 . 56376 .50978 . 46521 .42781 . 39596 . 36853 .34465 .32367 . 30510 .28854 . 27369 . 26029 .24813 . 23706 . 22694 .21764 . 20908 .20116 . 19382 . 18699 .18063 . 17469 . 16912 .16390 . 15899 . 15436 . 15000 . 14587 . 14197 . 13827 . 13476 . 13142 . 12824 . 12521 . 12232 . 11956 . 11692 .11440 .11198 .10966 .x0744 .10530 . 10325 .10128 . 09938 . 09755 .09578 .09408 .09244 .09085 .08931 .08783 .08640 . 08501 .08367 .08236 . 08110 61 0.07980 . 07869 . 07753 .07641 .07532 . 07426 . 07323 . 07223 . 07126 .07031 . 06939 .06849 . 06761 . 06676 .06593 .06511 .06432 .06355 . 06279 . 06205 . 06133 .06063 . 05994 . 05927 . 05861 .05796 . 05733 . 05672 .05611 .05552 .05494 .05438 .05382 . 05328 . 05275 .05223 . 05171 . 05121 . 05072 .05023 .04976 . 04929 .04884 . 04839 . 04795 .04752 . 04709 . 04667 .04626 .04586 .04547 . 04508 . 04469 .04432 . 04395 . 04359 .04223 . 04288 .04253 . 04219 - 121 - 117 - 114 - 110 - 107 - 104 - 102 - 99 - 96 - 94 - 91 - 89 - 87 - 84 - 82 - 80 - 78 - 76 - 75 - 73 - 71 - 70 - 68 - 67 - 65 - 64 - 62 - 61 - 60 - 58 - 57 - 56 - 55 - 54 - 53 - 52 - 51 - 50 - 49 - 48 - 47 - 46 - 45 - 44 - 44 - 43 - 42 - 41 - 41 - 40 - 39 - 39 - 38 - 37 - 37 - 36 — 36 - 35 - 34 — 34 - 763 2 62 - 787 3 63 - 811 4 64 - 836 5 65 - 862 6 66 - 887 7 67 — 913 8 68 - 940 9 69 - 966 10 -2233 -1970 -1751 -1566 -1409 -1274 -1158 -1057 - 969 - 892 - 823 - 762 - 707 - 658 - 614 - 575 - 539 - 506 - 476 - 449 - 424 - 401 - 380 - 360 - 342 - 326 - 310 - 296 - 282 - 270 - 258 - 247 - 237 - 227 - 218 - 209 - 201 - 194 - 187 • - 180 - 173 - 167 - 161 - 156 - 151 - 146 - 141 - 137 - 132 - 128 - 124 - 29 - 34 - 40 - 45 - 52 - 58 - 65 - 72 - 80 - 88 - 96 - 105 - 114 - 124 - 133 - 144 - 154 - 165 - 177 - 188 - 200 - 213 - 226 - 239 - 253 - 267 - 281 - 296 - 311 - 326 - 342 - 359 - 375 - 392 - 409 - 427 - 445 - 464 - 483 - 502 - 522 - 542 - 562 - 583 - 604 - 626 - 648 - 670 - 693 - 716 - 739 70 — 994 11 71 -1020 12 72 -1050 13 73 -1080 14 74 -1110 15 75 -1140 16 76 -1160 17 77 -1190 18 78 -1220 19 79 -1250 20 80 81..... -1280 21.. -1310 22 : 82 -1350 23 83 -1380 24 84 -1410 25 85 -1450 26 86 -1480 27 87 -1510 28 88 -1550 29 89 -1580 30 90 -1610 31 91 -1650 32 92 93 -1690 33 -1730 34 94 -1760 35 95 -1790 36 96 -i830 37 97 -1870 38 98 -1900 39 99 -1940 40 100 -1980 41 101 -2020 42 102 -2060 43 103 -2100 44 104 -2140 45 105 -2180 46 106 -2220 47 107 -2260 48 108... —2310 49 109 —2350 50 110 —2390 51 Ill —2430 52 112... —2470 53 113 —2520 54 114.. —2560 55 115 —2610 56 116 —2650 57 117... —2690 58 118 . —2730 59 119. —2780 60 120 —2840 i See formulas (7) to (11a), pp. 9 and 10, and (21) and following, pp. 18-20 SCALE OF A SOUNDING TUBE. 43 Table 8. — Special I able for scale of Coast and Geodetic Survey tube. [Temperature of air, 60° F., of water, 50° F. = 10° C; humidity of air, 100%; surf: * i 1.025; gravity as inlat.45V — is volume of air when volume of tube is taken as unit v. — — ; — r-is given Vi d / i_A oh \7J to not more than three significant figures and is always creator t ban unit v The last three columns give the length of rod of the diameter stated thai tnu I be inserted in order to till the air space, thus bringing the water to the top of the tube. The numbers are computed for a tube 24 inches long and A inch in diameter.] Depth, h (fathoms). Volume of air space dli V'i/ units of fifth deci- mal place. 1-^ Length of rod to bring water »p (inches). J inch diameter. A inch diametei . finch diameter. 1 0. 97561 . 82455 . 71399 .62955 . 56287 .50911 . 40465 . 42733 . 39554 .36815 .34431 .32330 .30481 . 28827 . 27343 . 26004 . 24790 .23683 . 22671 .21741 . 20885 . 20093 . 19359 . 18676 . 18040 . 17445 . 16888 .16366 . 15874 . 15411 . 14974 . 14561 . 14170 . 13800 . 13448 .13113 . 12795 . 12491 . 12202 . 11925 . 11661 . 11408 .11165 . 10933 . 10709 . 10495 . 10289 . 10091 . 09900 . 09716 .09539 .09368 .09203 .09044 . 08890 2 3 4 5 24.016 6 21.722 7 19 s''.") 8 • 18. 232 9 24.302 22. 019 21. 15 J 19.868 18. 728 17.711 16. 799 15. 977 15. 231 14. 551 13. 929 13.358 12.832 12. 345 11. 894 11.475 11.084 10. 718 10. 376 10. 055 9.753 9.469 9.200 8.946 8.700 ■ 8. 478 8.262 8.057 7.861 7. 675 7.497 7.327 7.164 7.010 6.860 6.717 6.580 6.448 6.322 6.200 6.083 5.970 5.861 5. 756 5.654 5.556 5.462 16 876 10 -2549 -2229 -1967 -1749 -1565 -1408 -1273 -1158 -1057 - 969 - 892 - 823 - 762 - 708 - 659 - 615 - 575 - 539 - 506 - 477 - 450 - 425 - 402 - 381 - 361 - 343 - 326 - 311 - 296 - 383 - 270 - 259 - 248 - 237 - 228 - 219 - 210 - 202 - 194 - 187 - 180 - 174 - 168 - 162 - 157 - 151 - 25 - 29 - 34 - 40 - 45 - 52 - 58 - 65 - 72 - 80 - 88 - 96 - 105 - 114 - 123 - 133 - 143 - 154 - 165 - 176 - 188 - 200 - 212 - 225 - 239 - 252 - 266 - 280 - 295 - 310 - 326 - 341 - 357 - 374 - 391 - 408 - 426 - 444 - 462 - 481 - 500 - 520 - 539 - 559 - 579 - 601 15. 70s 11 14 691 12 13. 797 13 13.005 14 12.300 15 11.666 16 24.964 23.798 22. 736 21.764 20. 872 20.049 19. 289 18. 584 17.929 17.318 16. 747 16. 213 15.711 15. 239 14. 795 14.375 13. 979 13. 603 13.247 12. 910 12. 589 12.283 11.992 11.713 11.448 11.194 10. 951 10. 718 10. 495 10.281 10. 075 9.878 9. 687 9.504 9.328 9.158 8.994 8.835 8.682 8.534 11.095 17 10. 577 18 10. 105 19 9. 673 20 9.276 21 8.911 22 8.573 23 8 260 24 7.969 25 7.697 26 7.443 27 7.206 28 6.983 29 6.773 30 6.575 21 6 389 32 6.213 33 6 046 34 5.888 35 5.738 36 5.595 37 5 459 38 5 330 39 5 206 40 5.088 41 4 975 32 4 867 43 4 764 44 4 664 45 4.569 46 4 478 47 4 390 48 4.305 49 4 224 50 4 146 51 4 070 52 3 997 53 3 927 54 3 859 55 3.793 i See formulas (7) to (lib), pp. 9 and 10, and remark on p. 19. 44 U. S. COAST AND GEODETIC SURVEY. Table 8. — Special table for scale of Coast and Geodetic Survey tube — Continued. Volume of air space 02. Vl dhyvxJ units of fifth deci- mal place. 1 v 2 b /v 2 \ dh\vi) Length of rod to bring water to top (inches). Depth, h (fathoms). Jineh diameter. ■rVinch diameter. finch diameter. 56 0. 08741 . 08596 .08457 .08322 .08191 . 08063 .07940 . 07821 . 07704 .07592 .07482 .07375 . 07272 .07171 .07073 .06977 . 06884 .06793 .06705 .06618 .06534 .06452 .06372 .06294 .06218 .06143 .06070 .05999 .05929 .05861 .05794 .05729 . 05666 .05603 .05542 . 05482 .05424 .05366 .05310 .05255 . 05200 .05147 . 05095 . 05044 .04994 . 04945 . 04897 . 04850 .04803 .04757 .04713 . 04669 . 04625 .04583 .04541 .04500 . 04459 . 04420 . 04381 .04342 . 04304 .04267 .04231 . 04195 .04159 - 147 - 142 - 137 - 133 - 129 - 125 - 121 - 118 - 114 - Ill - 108 - 105 - 102 - 99 - 97 - 94 - 92 - 90 - 87 - 85 - 83 - 81 - 79 - 77 - 76 - 74 - 72 - 70 - 69 - 67 - 66 - 64 - 63 - 62 - 60 - 59 - 58 - 57 - 56 - 55 - 54 - 53 - 52 - 51 - 50 - 49 - 48 - 47 - 46 - 45 - 44 - 44 - 43 - 42 - 41 - 41 - 40 - 39 - 39 - 38 - 37 - 37 - 36 - 36 - 35 - 622 - 644 - 666 - 688 - 711 - 734 - 758 - 782 - 806 - 830 - 855 - 881 - 906 - 932 - 959 - 986 -1010 -1040 -1070 -1100 -1130 -1150 -1180 -1210 -1240 -1270 -1300 -1330 -1360 -1400 -1430 -1460 -1490 -1530 -1560 -1590 -1630 -1660 -1700 -1730 -1770 -1800 -1840 -1880 -1910 -1950 -1990 -2030 . -2070 -2100 -2140 -2180 -2220 -2260 -2300 -2340 -2380 -2430 -2470 -2510 -2560 -2600 -2640 -2680 -2720 8.391 8.253 • 8. 119 7.989 7.863 7.741 7.623 7.508 7.396 7.288 7.183 7.080 6.981 6.884 6.790 6.698 6.608 6.521 6.436 6.354 6.273 6.194 6.117 6.042 5.969 5.897 5.827 5. 759 5.692 5.627 5.563 5.500 5.439 5.379 5.320 5.263 5.207 5.151 5.097 5.044 4.992 4.941 4.892 4.843 4.794 4.747 4.701 4.656 4.611 4.567 4.524 4.482 4.440 4.399 4.359 4.320 4.281 4.243 4.206 4.169 4.132 4.096 4.061 4.027 3.993 5.370 5.282 5.196 5.113 5.032 4.954 4.878 4.805 4.734 4.664 4.597 4.531 4.468 4.406 4.345 4.287 4.230 4.174 4.119 4.066 4.015 3.964 3.915 3.867 3.820 3.774 3.729 3.686 3.643 3.601 3.560 3.520 3.481 3.443 3.405 3.368 3.332 3.297 3.262 3.228 3.195 3.163 3.131 3.099 3.068 3.038 3.009 2.980 2.951 2.923 2.895 2.868 2.842 2.816 2.790 2.765 2.740 2.716 2.692 2.668 2.645 2.622 2.599 2.577 2.555 3.729 57 3.668 58 3.609 59 3.551 60 3.495 61 3.440 62 3.388 63 3.337 64 3.287 65 3.239 66 3.192 67 3.147 68 3.103 69 3.060 70 3.018 71 2.977 72 2.937 73 2.898 74 2.861 75 2.824 76 2.788 77 2.753 78 2.719 79 2.685 80 2.653 81 2.621 82 2.590 83 2.560 84 2.530 85 : 2.501 86 2.472 87 2.444 88 2.417 89 2.391 90 2.365 91 2.339 92 2.314 93 2.290 94 2.266 95 2.242 96 2.219 97 2.196 98 2.174 99 2.152 100 2.131 101 2.110 102 2.089 103 2.069 104 2.049 105 2.030 106 2.011 107 1.992 108 1.973 109 1.955 110 1.937 Ill 1.920 112 1.903 113 1.886 114 1.869 115 1.853 116 1.837 117 1.821 118 1.805 119 1.790 120 1.775 Sl'ALl'L OK A SOUNDING TUBE. 45 Table 1>. — Corrections to sounding tube readings for U mperatun and pressure. [Computed for air temperature^ 60°F.= 154° (V, water temperature 50° f.= io° c. Barometer— 30 in. 70.200 cm. See equation (15), p. 12.] 1. CORRECTION FOR TEMPERATURE. Depth from scale, h (fathoms). 5.. 10. 20. 30. ■10. 50. 60. 70. SO. 90. 100 110 Temperature of air minus temperature of water ( Fahrenheil I. —10° Fath- oms. + 1.0 + 1.5 + 2.5 + 3.4 + 4.4 + 5.4 + 6.4 + 7.3 + 8.3 + 9.3 + 10.3 + 11.2 -30° Fath- oms. +0.8 + 1.2 + 2.0 +2.8 +3.5 +4.3 +5.1 +5.9 +6.6 + 7.4 +8.2 +9.0 •20° Fath- oms. +0.6 + .9 + 1.5 +2.1 + 2.6 +3.2 +3.8 +4.4 +5.0 +5.6 +6.2 +6.7 ■10 c Fath- oms. + 0.4 + .6 + 1.0 + 1.4 + 1.8 +2.2 +2.6 +2.9 +3.3 +3.7 + 4.1 +4.5 0° +10° +20° +30° +10° +50 Fath- oms. +0.2 + .3 + .5 + .7 + .9 + M + 1.3 + 1.5 + 1.7 + 1.9 + 2.0 +2.2 Fath- oms. 0.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 Fath- oms. -0.2 - .3 - .5 - .7 - .9 -1.1 -1.3 -1.5 -1.7 -1.9 -2.0 -2.2 Fath- oms. -0. ! - .6 -1.0 -1.4 -1.8 -2.2 -2.6 -2.9 -3.3 -3.7 -4.1 -4.5 Fath- oms. -0.6 - .9 -1.5 -2.1 -2.6 -3.2 -3.8 -4.4 -5.0 -5.6 -6. 2 -6.7 Fath- oms. -1.2 -2.0 -2.8 -3.5 -4.3 -5.1 -5.9 -6.6 -7.4 -S.2 -9.0 + 00° Fath- oms. - 1.0 - 1.5 - 2.5 -3.4 - 4.4 -5.4 - 6.4 - 7.3 -8.3 -9.3 -10.3 -11.2 2. CORRECTION FOR PRESSURE. Depth from scale, h (fathoms). 10. 20. 30. 40. 50. 60. 70. 80. 90. 100 110 Barometer reading in inches. 29.0 Fath- oms. -0.3 - .7 -1.0 -1.3 -1.7 -2.0 -2.3 -2.7 -3.0 -3.3 -3.7 29.2 Fath- oms. -0.3 - .5 - .8 -1.1 -1.3 -1.6 —1.9 -2.1 -2.4 -2.7 -2.9 29.4 29.6 29.8 30.0 30.2 30.4 Fath- oms. -0.2 - .4 - .6 - .8 -1.0 -1.2 -1.4 -1.6 -1.8 -2.0 -2.2 Fath- oms. -0.1 - .3 - .4 - .5 - .7 - .8 - .9 -1.1 -1.2 -1.3 -1.5 Fath oms. -0.1 - .1 - .2 - .3 - .3 - .4 - . 5 - .5 - .6 - .7 - .7 Fath- oms. 0.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 Fath- oms. +0.1 + .1 + .5 + .5 + .6 + .7 + .7 Fath- oms. +0.1 + .3 + .4 + .5 + .7 + .8 + .9 + 1.1 + 1.2 + 1.3 + 1.5 30.6 Fath- oms. +0.2 + .4 + .6 + .8 + 1.0 + 1.2 + 1.4 + 1.6 + 1.8 + 2.0 +2.2 30.8 Fath- oms. +0.3 + .5 + .8 + 1.1 + 1.3 +1.6 + 1.9 +2.1 +2.4 +2.7 +2.9 31.0 Fath- oms. +0.3 + .7 + 1.0 + 1.3 + 1.7 +2.0 +2.3 +2.7 +3.0 +3.3 +3.7 Table 10. — Effect of absorption of air on scale of sounding tube. [Column 1 shows the depth in fathoms. Column 2 shows the relative volume of air at the given depth. Column 2 has been taken from Table 7, and column 3 shows the relative volume of air if the sea water in the tube were saturated with atmospheric gases. 2 Column 4 shows the correction to be applied to the depths read from a scale computed on the supposition of no absorption, when, in point of fact, absorp- tion had gone on to the saturation point.] Depth (fathoms). 5. 10 15 20 25 30 35 40 45 50 V2IV1 No ab- sorption. 0. 5098 .3447 .2603 .2091 .1747 .1500 .1314 .1169 .1053 .0958 V2IV1 Complete satura- tion. 0. 5052 .3363 .2496 .1968 .1613 .1358 .1166 .1016 .0896 .0797 Correc- tion to depth (fath- oms). 0.07 .3 .8 1.4 2.1 3.0 4.0 5.1 6.3 7.7 Depth (fathoms). 55. 60. 65. 70. 75. 80. 85. 90. 95. 100 V2IV1 No ab- sorption. 0. 0878 .0811 .0753 .0703 .0659 .0621 .0586 .0555 .0528 .0502 V2IV1 Complete satura- tion. 0.0714 .0644 .0584 .0533 . 04S7 .0447 .0411 .0379 .03,50 .0324 Correc- tion to depth (fath- oms). — 9.2 -10.8 -12.4 -14.2 -16.1 -18.1 -20.1 -22.2 -24.5 -26.7 1 See equation (7), p. 9. t?=lQ° C; h=15° C; pi+p'i=l atmosphere. s See equation (20), p. 16. O LIBRARY OF CONGRESS # 029 714 149 i Hollinger Corp. P H8.5