The velocity with which a wave travels over the surface of a liquid depends upon the surface tension of the liquid as well as on a gravity force due to the displacement of the liquid from its undisturbed position. For waves of large wavelength the effect of surface tension on the velocity of the wave is small; however for small wavelengths the gravity force is very small and the velocity of the wave is determined almost entirely by the surface tension. We shall therefore consider waves of very small wavelength

Oelke, W. C.

Yamakawa, K. A.

English

Name not available

The Determination of Surface Tension by Standing Waves

University of Northern Iowa

Proceedings of the Iowa Academy of ScienceVolume 52 | Annual Issue Article 241945The Determination of Surface Tension by StandingWavesK. A. YamakawaGrinnell CollegeW. C. OelkeGrinnell CollegeCopyright © Copyright 1945 by the Iowa Academy of Science, Inc.Follow this and additional works at: https://scholarworks.uni.edu/piasThis Research is brought to you for free and open access by UNI ScholarWorks. It has been accepted for inclusion in Proceedings of the Iowa Academyof Science by an authorized editor of UNI ScholarWorks. For more information, please contact scholarworks@uni.edu.Recommended CitationYamakawa, K. A. and Oelke, W. C. (1945) "The Determination of Surface Tension by Standing Waves," Proceedings of the Iowa Academyof Science: Vol. 52: No. 1 , Article 24.Available at: https://scholarworks.uni.edu/pias/vol52/iss1/24THE DETERMINATION OF SURFACE TENSION BY STANDING WAVES K. A. YAMAKAWA A'.'>D W. C. OELKE The velocity with which a wave travels over the surface of a li-quid depends upon the surface tension of the liquid as well as on a gravity force due to the displacement of the liquid from its undis-turbed position. For waves of large wavelength the effect of surface tension on the velocity of the wave is small; however for small wave-lengths the gravity force is very small and the velocity of the wave is determined almost entirely by the surface tension. We shall there-fore consider waves of very small wavelength. J. H. Poynting and J. J. Thomson in their "Text Book of Physics" derive an expression showing this relation for the case of a wave whose amplitude is small compared to its wavelength neglecting the effect of the depth of the liquid. Their expression is : T-surft>cet • .,,,.,A -wo.ve '""S11. whe.-e t-de11~ 1t~ 9-a.ocelcr-1: .. t1011of3ra~·t_y n -fre111e,,cy h..-"~v;::~·:v·:i!t "°!;,....1 Lord Kelvin deduced an equation for the case of an infinitely. small wave and included the effect of the depth of the liquid. This expres-sion agrees with the above for depths of the order of a centimeter and wavelengths of the order of a half centimeter. Lord Rayleigh was the first (Phil. Mag. xxx page 386) to success-fillly apply this method to the determination of the surface tension and his method was used by Dorsey (Phil. Mag. xliv page 369) to determine the surface tension of a large number of solutions. An electrically driven tuning fork generated the waves and care was taken to keep the amplitude small compared to· the wavelength (to the extent that they were invisible to the unaided eye). In the present case standing waves were produced in a glass tube containing the liquid by vibrating the tube. In this case the amplitude of the waves was no longer small compared to the wavelength. Fur-thermore the wave shape changed and peaked waves were produced as shown py the photograph in Figure 1. An expression for the form of the wave in such a case is derived in A. G. Webster's "The Dy-namics of Particles and of Rigid, Elastic, and Fluid Bodies"; how-ever it is very general and appears difficult to handle. An attempt is made in this paper to derive and experimentally check a relation for this type of wave so that the surface tension of liquids can be de-termined in this way. It can be seen from figure 1 that the shape of the wave approximates an inverted cycloid. This was assumed and an expression derived by following the general method of Poynting and Thomson in their "'l'ext Book of Physics." The force of gravity causing the wave to travel is due to the verti-cal pressure produced by the weight of the liquid above the undis-1Yamakawa and Oelke: The Determination of Surface Tension by Standing WavesPublished by UNI ScholarWorks, 1945200 IOWA ACADEMY 01<' SCIENCE [VOL. 52 turbed level of the liquid. The surface tension will give rise to an additional vertical force. The addition of the surface tension force produces an effect equivalent to that of increasing the normal gravity force. The velocity of a gravity wave, neglecting the effect of the depth of the liquid, is the velocity of a body would acquire under gravity by falling vertically through a distance ~4 .,,. A. where A. is the wave~ length. An expression for this equivalent "g" is obtained and in~ Figure 1 serted in the expression v = \I 2 gh relating velocity and distance in free fall. This results in the expression as given on the first page for the case of a wave whose amplitude is small compared to the wavelength. The wave was generated by a point fixed close to the center of a circle rolling in a straight line. In the following derivation the point is fixed at a point on the circumference of the circle result-ing in a cycloid. The pressure (p) due to surface tension (T) is given by the rela-tion p=T/ R where 'R' is the radius of curvature at that point. The pressure due to surface tension can therefore be obtained from the general expression f'or the radius of curvature of any curve: .,.i .1...- ""~fa ;t''" R -[i.+("¥.uJ"]r.,_ by substituting the equations of the cycloid: .,3' X= 4 (~- Sin-6-) ~. a. (l- cos-9-) 2Proceedings of the Iowa Academy of Science, Vol. 52 [1945], No. 1, Art. 24https://scholarworks.uni.edu/pias/vol52/iss1/241945] DETERMINATION OF SURFACE TENSION 201 It can be shown that this results in the following expression for the radius of curvature: The pressure due to surface tension acting normal to the surface of the liquid, is therefore: It can be seen that the vertical component of this force is active in propagating the wave and that the horizontal component cancels. Therefore the effective pressure is: •s' P,.=Pc.osf whe .. +•a.rc1"-.11"J'o1. The relation between o and </> can be obtained by obtaining an ex-pression for dy/dx from the parametric equations of the cycloid and equating it to that obtained in terms of </>· d!I. - -~ r •• ~ ey •• f,,"' 'f~cr·'''" Y<l.li-to. .. +- l-COS9-F .. o,.,, wli1cJ.. .t C'-2'1 Ir s~o~,, fJ,p,.f" cos cf> :z S•n ! Therefore: T Pv = 4 a. It is interesting to note that the vertical component of the pressure is independent of (}. The pressure due to gravity is age pressure) P,,.., ... "'3 J..., ( due to ~..,.,,,t~ The total average pressure is therefore: This can be changed to the form: This results in an equivalent "g·•. that is the aver-The average height h is taken as the average height of the curve av 3Yamakawa and Oelke: The Determination of Surface Tension by Standing WavesPublished by UNI ScholarWorks, 1945202 IOWA ACADEMY OF SCIENCE [VOL. 52 above .the normal undisturbed level of the water. The normal un-disturbed level of the water is taken as the line shown as mn in Fig-ure 2 drawn so that A is equal to B. Upon integration and averaging over the portion of the curve above the normal undisturbed level there results in 0.53a for the average height. 'l'he velocity of a gravity Figure 2 ~1 1---1 '~ I I I I oq: I ~' wave is that acquired by a freely falling body falling through a dis-tance equal to 1471" of the wavelength, and is therefore given by: ~-.,. Y.., .. ,,1 ·,/A s., .... of Q.j,. ... ,~, w ...... c. Rigure 3 shows a diagram of the apparatus. The surface tension of water at 28°C was determined. Six determinations were made. The wavelength was determined by photographing the tube contain-ing the liquid along with a scale mounted along the. tube. 4Proceedings of the Iowa Academy of Science, Vol. 52 [1945], No. 1, Art. 24https://scholarworks.uni.edu/pias/vol52/iss1/241945] DETERMINATION OF SURFACE TENSION 203 A I I I I I I ['""'f"'·r .. "·l"'""l"'"·i'"'·"\'""''I E L~-.:.:-_-_ A Test Tube containing liquid B Vibrator C Amplifier D Variable frequency oscillator E Scale Frequency cy/sec 230 163 194 84 194 163 Figure 3 EXPERIMEN'rAL RESULTS Wavelength ems. 0.1613 0.202 0.181 0.198 0.181 0.213 Surface Tension dynes/cm. Calculated fromEq.1 34.0 34.2 34.5 8.6 39.3 40.0 CONCLUSION c I l I 0 Surface Tension dynes/cm. Calculated from Eq. 2 72.2 74.6 74.9 57.4 84.9 85.6 The paper is not complete in that more experimental data is neces-sary. Also, we realize that many assumptions were made in the deri-vation of the equation. In particular, the assumption that the velocity of the wave is that of free fall through i,;;,,,. of the wavelength may not be a valid one. However the method is very interesting and sug-gests many possibilities. We hope to continue the work. DEPTS. OF PHYSICS AND CHEMISTRY GRINNELL COLLEGE GRINNELL, row A 5Yamakawa and Oelke: The Determination of Surface Tension by Standing WavesPublished by UNI ScholarWorks, 1945

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The Determination of Surface Tension by Standing Waves

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