Download A simple experiment on diffraction of light by interfering liquid

A simple experiment on diffraction of light by interfering liquid
surface waves
Tarun Kr. Barik, Anushree Roy, and Sayan Kara)
Department of Physics, Indian Institute of Technology, Kharagpur 721 302, WB, India
共Received 30 September 2004; accepted 24 January 2005兲
We describe a simple experiment on the diffraction of monochromatic light by interfering liquid
surface waves. The surface wave profile, which acts as a reflection phase grating for incident light,
is generated by electrically driven vertical oscillations of two or more slightly immersed exciters.
The theoretical intensity of the diffracted light agrees well with observations. In particular, we show
the relation between the intensity and the amplitude 共height兲 of the surface wave. Although invisible
to the naked eye, the interference of liquid surface waves can be optically detected from the
characteristic features of the diffraction pattern. In addition, we can measure the amplitude,
wavelength, and phase velocity of the surface wave that propagates along the line joining the
exciters. © 2005 American Association of Physics Teachers.
关DOI: 10.1119/1.1870032兴
I. INTRODUCTION
Imagine dropping a pebble in a lake. Everyone is familiar
with the circular ripples that propagate radially outward.
Dropping two such pebbles simultaneously at two nearby
locations would result in interfering waves on the water surface. Most textbooks1 on waves and optics include photographs of such interfering water waves while introducing the
notion of superposition of waves.
However, if the wavelength of the surface wave is small
共say a fraction of a centimeter兲, it is not easy to see such
waves with the naked eye. Sophisticated modern-day cameras can detect these small-wavelength waves, but such cameras are expensive and not readily available to undergraduates. To detect these waves, we need to find other ways,
which also could provide valuable physical insight. Monochromatic laser light falling on the vibrating liquid surface
will be diffracted and result in an intensity distribution with
important characteristic features. In simple single slit or grating experiments with a He–Ne laser we can infer the nature
of the diffracting object from the intensity variations. Similarly, we also can learn about the surface wave profile 共which
is the diffracting object兲. The aim of this article is to show
how we can learn about interfering waves on the liquid surface from the diffraction of incident monochromatic light.
Before describing the experiment, we briefly discuss the
importance of surface waves, which also are known as Rayleigh waves. These waves can be generated on solid as well
as liquid surfaces, resulting in diverse phenomena with several useful applications. In particular, it is well established
that information on the properties of surface acoustic waves
is crucial for device related applications in solid-state
technology.2 For example, the acousto-optic interaction in a
layered structure controls the guided optical wave. In slab
waveguides, this interaction has been studied extensively because of its importance in applications related to optical
correlations.3–5 The acousto-optic figures of merit and the
effective photoelastic constants of GaN layer have recently
been measured by diffraction of light.6 A general overview of
surface acoustic waves is available in Ref. 7. Details on the
diffraction of light by surface waves as a diagnostic tool in
surface wave technology can be found in Refs. 8 and 9.
There have been fewer investigations of surface acoustic
725
Am. J. Phys. 73 共8兲, August 2005
http://aapt.org/ajp
waves in liquids. The traditional ripple tank method of measuring the surface tension of a liquid is one of the earliest
studies.10 Recent advances in related areas 共for example, the
measurement of the amplitude and wavelength of circular
ripples, the study of thermally activated liquid surfaces and
spatial damping of liquid surface acoustic waves, light
scattering/photon correlation spectroscopy兲 have been
performed.11–16 The basic principle of light diffraction by
liquid surface acoustic waves remains the same as for solids,
although the quantitative features and the generation mechanism differ. As for solids, the features of the intensity distribution of the diffracted light provide information on the liquid surface acoustic waves as well as of properties of the
liquid itself. Because liquid properties 共for example, surface
tension and viscosity兲 can be measured by other methods, we
are more interested here in exploring the nature of surface
acoustic waves using external optical probes. In Secs. II and
III we describe the experimental setup and the background
theory. Section IV discusses the results and Sec. V summarizes our results.
II. THE EXPERIMENTAL SETUP
A schematic diagram of the experimental setup is shown
in Fig. 1. Water is kept in a petri dish of diameter 18.5 cm.
The depth of water in the dish is 1.0 cm. A long metallic pin
is glued to the center of the diaphragm of a small loudspeaker, which is driven by a low-frequency signal generator.
A sinusoidal signal of frequency ⍀ 共220 Hz in our experiment兲 is used to drive the pin. This vibrating pin is the surface acoustic wave exciter. The circular ripples, which are
formed due to the vertically oscillating pin, act as a dynamical diffraction phase grating on the water surface. Best results are obtained when the pin is just below the liquid surface. If the pin is immersed more deeply, splashing and other
associated effects can occur.
To create superposed waves on the surface, we glued two
pins very closely at the center of the diaphragm of the same
loudspeaker. The separation between the pins is of the order
of 1 to 3 mm as measured by a microscope. The two pins are
made to oscillate in phase on the water surface. Each oscillating pin creates a circular wave about its location. These
waves interfere and produce the well-known two-source in© 2005 American Association of Physics Teachers
725
each diffraction spot is measured by a photodiode detector.
The linear relation between the photodiode current and the
intensity of diffracted light was checked.
III. THEORETICAL CONSIDERATIONS
Fig. 1. Schematic diagram of the experimental setup used to study surface
acoustic waves on a liquid. 共A兲 laser; 共B兲 frequency generator; 共C兲 screen;
共D兲 loudspeaker; 共E兲 microscope; 共F兲 exciter.
terference pattern on the surface of water 共Fig. 2兲. Notice that
the interference pattern along the line joining the sources 共the
solid black line in Fig. 2兲 results in a similar water wave with
a different 共increased兲 amplitude compared to the water wave
produced by a single pin. On the other hand, in the region
perpendicular to the line joining the sources, we can see a
lattice-like structure. Therefore, the diffraction pattern for
light incident on these two regions will be different, a feature
that we shall illustrate.
A 5 mW He–Ne laser of wavelength ␭⫽632.8 nm and
beam diameter 1.8 mm is directed to fall on the water surface
共as shown in Fig. 1兲, where the superposed surface acoustic
waves are formed. The angle of illumination, the angle at
which the laser beam falls on the liquid surface, ␪ i , in our
experiment is 77°. It is known that at frequencies of the
order of 220 Hz the decay length of surface acoustic waves is
much greater than the size of the illuminated spot.17 Due to
Fraunhofer diffraction of light by the phase grating due to the
surface acoustic waves, a pattern is observed on a screen
placed at a reasonably large distance, z 共3.15 m in our case兲
from the water surface. The images of the diffraction pattern
are taken using a digital camera. The spurious noise in the
images is removed by using PHOTOSHOP. The intensity of
The interference of circular waves generated on the surface of the liquid due to more than one source of disturbance
can be approximated as a linear superposition of individual
sinusoidal waves. We approximate the individual waves as
␺ 1 ⫽h sin(⍀t⫺kr1) and ␺ 2 ⫽h sin(⍀t⫺kr2),18 where we assume that the two oscillations have the same frequency ⍀,
wave number k, and amplitude h; r 1 and r 2 are the distances
of the oscillation sources from a point P on the liquid surface. We are interested in focusing light on a small region
along the line joining the two sources 共see Fig. 2兲. This configuration allows us to restrict ourselves to a onedimensional problem.
The sum of the two waves produces a resultant wave profile 共along the line joining the sources兲
␺ ⫽2h cos
kd
sin共 ⍀t⫺kr 兲 ,
2
共1兲
where we have assumed r 2 ⫺r 1 ⫽d, r 2 ⫹r 1 ⫽2r, and dⰆ1.
In general, if we assume two waves ␺ 1,2⫽h sin(⍀t
⫺k"r1,2), the superposed wave is given by ␺
⫽2h cos关kd/2 cos ␪兴sin(⍀t⫺k"r), where r1,2 are vectors
from the exciters to a point P, r is the vector from the center
of the line joining the sources to P, and ␪ the angle r makes
with the same line. In the following, we shall be concerned
only with the superposed wave along the line joining the
sources.
For a superposition of N such oscillations, the factor
2 cos kd/2 is replaced by sin(Nkd/2)/sin(kd/2) 共reminiscent
of the factor in the diffraction formula for N slits in the
theory of Fraunhofer diffraction兲.1
The phase modulation produced by a single circular surface wave is given by ␾ (x)⫽(2 ␲ /␭) 关 (2h cos ␪i)sin(⍀t
⫺ 关kx/cos ␪i兴)兴, where the factors of cos ␪i appear due to oblique incidence. This well-known formula is used extensively
to discuss sinusoidal phase gratings.19,20 The only change in
this phase modulation function for the reflected light from
interfering liquid surface waves is the replacement of the
h sin(⍀t⫺ 关kx/cos ␪i兴) factor by Eq. 共1兲. Thus, we obtain
␾共 x 兲⫽
2␲
␭
冋冉
4h cos
冊 冉
kd
kx
cos ␪ i sin ⍀t⫺
2
cos ␪ i
冊册
,
共2兲
where x is along the wave profile. We can easily rewrite Eq.
共2兲 to make it look like the phase modulation by a single
wave using the replacement h ⬘ ⫽2h cos(kd/2). For N
sources of oscillation we have
h ⬘ ⫽h
Fig. 2. Interference between two circular waves. The dotted and dashed
lines represent the positions for constructive and destructive interference,
respectively. Quantitative results are for experiments performed with light
incident in regions along the solid bold line.
726
Am. J. Phys., Vol. 73, No. 8, August 2005
sin共 Nkd/2兲
.
sin共 kd/2兲
共3兲
The light field strength E is the Fourier transform of the
object 共aperture兲 function given by exp(i␾).19 The intensity is
obtained by considering EE * and yields19
I共 x⬘兲⫽
兺n J 2n共 4 ␲ h ⬘ cos ␪ i /␭ 兲 ␦
冉
冊
x⬘
n
,
⫺
␭z ⌳ cos ␪ i
Barik, Roy, and Kar
共4兲
726
Fig. 3. The basic geometry of the measurement.
where z is the horizontal distance between the location of the
laser spot on the liquid surface and the screen, the wavelength of the surface acoustic waves is ⌳⫽2 ␲ /k, J n is the
Bessel function of order n, and ␦ denotes the Dirac delta
function. The coordinate x ⬘ measures the distance of the diffraction spots from a reference point 共central spot兲 on the
observation plane. The geometry of our measurement is
shown in Fig. 3.
The intensity distribution given in Eq. 共4兲 vanishes at the
location of the zeros of the Bessel function J n . The zeros of
J n for different n appear at different amplitudes 共values of
h ⬘ ). For instance, if n⫽0, x ⬘ ⫽0, the intensity will vary as
J 20 . Thus, the intensity will vanish at the values of h ⬘ for
which J 20 ⫽0 共the zeros of J 0 occur at values 2.405, 5.520,...
of its argument兲. Hence, for the superposed waves the value
of h ⬘ for which zero intensity will be observed will be a
factor 1/关2 cos(kd/2) 兴 共for N⫽2) smaller than the value h for
circular waves from a single excitation source.
We find that the intensity zero of the zeroth order appears
at nearly half the h value obtained for the single source case.
Because h is proportional to the voltage applied to the pin in
our experiment, we should be able to see this reduction in h
through the reduction in the voltage. For more than two
nearby sources of oscillation, we would expect a reduction
by a factor of N 共because sin关Nkd/2兴 /sin关kd/2兴 →N for kd
→0). In addition, any deviation from the factor N will contain information about the wavelength of the superposed liquid surface waves through the value of the factor
sin(N␲d/⌳)/sin(␲d/⌳).
Fig. 4. Theoretical plot of intensity versus the amplitude for zero-order
共solid line兲, first-order 共dashed line兲, and second-order 共dotted line兲 diffraction spots using Eq. 共4兲. Experimentally measured intensity for these spots
共for two exciters兲 are shown by the black-filled, gray-filled, and open circle,
respectively. The linear variation of the resultant amplitude in water and the
applied voltage in the signal generator 共the amplitude imparted by the exciter兲 is shown in the inset.
the plot and then compare these values to the experimentally
observed ratios for the same orders. The measured ratios of
the intensities for a given voltage correspond to a specific
value of h. We repeat this measurement for different applied
voltages to generate the voltage–amplitude curve shown in
the inset. We measured the amplitude of vibration of the
exciter 共in air兲 using a microscope for different values of the
applied voltage, and obtained the voltage–amplitude curve.
A linear voltage–amplitude relation is found, but amplitude
of the waves is approximately 100 times smaller than the
oscillation amplitude of the exciter in air.
Figure 5共a兲 shows the Fraunhofer diffraction patterns of
monochromatic light by superposed surface acoustic waves
generated by two exciters with light incident along the line
joining the sources. As the loudspeaker amplitude 共applied
voltage兲 is increased slowly, at a certain value the central
spot vanishes 关Fig. 5共b兲兴. We observe the reappearance 共disappearance兲 of the central spot 共first-order spot兲 when the
amplitude is increased further 关Fig. 5共c兲兴. This vanishing of
the central and first-order diffraction spots occurs for applied
voltages 2.4 and 3.9 V, respectively. Similar diffraction pat-
IV. COMPARISON OF THEORY AND EXPERIMENT
The solid, dashed, and dotted lines in Fig. 4 are plots of
the intensity of the zero-, first-, and second-order diffraction
spots as a function of the height 共amplitude兲 h, as obtained
from Eq. 共4兲. In the inset we show the relation between the
applied voltage in the function generator 共the amplitude of
oscillation imparted by the exciters兲 and the amplitude of the
resultant surface acoustic waves. We obtained this relation as
follows. First, we measure the ratio of the intensities for any
two or three orders 共say n⫽0, 1, 2兲, for a given voltage
applied to the pin. Note from Fig. 4 that a vertical line 共parallel to the y axis兲 will intersect the intensity curves for different orders and with a fixed h at certain points. Hence, we
estimate the ratio of these intensities of different orders from
727
Am. J. Phys., Vol. 73, No. 8, August 2005
Fig. 5. Diffraction patterns of two interfering waves on the surface of water.
Barik, Roy, and Kar
727
Table I. Summary of the results obtained for one, two, and three exciters.
No. of exciters
1
2
3
Remark
Central order
vanishes for
4.5V⫽V 01
2.4V⫽V 02
1.7V⫽V 03
V 02 /V 01⬃0.53
V 03 /V 01⬃0.38
First order
vanishes for
7.4V⫽V 11
3.9V⫽V 12
2.7V⫽V 13
V 12 /V 11⬃0.53
V 13 /V 11⬃0.37
terns appear for waves generated by a single exciter for applied voltages 4.5 and 7.4 V, respectively. In other words, for
two exciters, the zero-order diffraction spot vanishes for
nearly half of the applied voltage compared to that for a
single exciter. Similarly, we find that for three pins the zeroorder spot vanishes when the applied voltage is 1.7 V. The
results are summarized in Table I.
The displacement x ⬘ of the nth-order spot from the center
of the diffraction pattern is n␭z/⌳ cos ␪i . By measuring x ⬘
for a single exciter, we determined the surface acoustic
wavelength to be 2.1 mm. Moreover, with more than one
exciter two interesting possibilities emerge for certain special
values of kd/2, which can be used to verify interference
effects and determine the wavelength of the superposed surface wave.
Complete destructive interference. From Eq. 共3兲 we see
that if d⫽m⌳/N, where m is an integer not equal to or a
multiple of N, the factor sin(Nkd/2)/sin(kd/2) vanishes. If
N⫽2, this factor vanishes for d⫽ ⌳/2 ,3⌳/2 ,5⌳/2... . This
vanishing implies that for these values of d, the argument of
the Bessel function vanishes. But, J n (0) is nonzero only for
J 0 , which implies that we see only a central spot at maximum 共100%兲 diffraction efficiency. This special feature, essentially arising out of complete destructive interference can
be easily used to check the value of the wavelength of the
surface waves. If we remember that a change in the separation between the two pins does not change the wavelength,
we see that d⫽⌳/2⫽1.05 mm, d⫽3⌳/2⫽3.15 mm, and d
⫽5⌳/2⫽5.25 mm. Experimentally, when the separation between the two exciters is 1.0, 3.0, or 5.2 mm, we find that,
except for the central spots, the intensities of the other diffraction spots go to zero irrespective of the amplitude of
surface acoustic waves.
No change in the diffraction pattern with changing N and
d. A second feature emerges when the diffraction pattern
remains unchanged even when we increase the number of
oscillation sources 共exciters兲. This phenomenon will happen
when sin(Nkd/2)/sin(kd/2) ⫽1, or when d⫽m⌳/(N⫹1),
where m is an integer that is not equal to or a multiple of
N⫹1. In particular, for N⫽2
we have d
⫽ ⌳/3 ,2⌳/3 ,4⌳/3... . Thus, for ⌳⫽2.1 mm, the values of
d⫽⌳/3⫽0.7 mm, d⫽2⌳/3⫽1.4 mm, d⫽4⌳/3⫽2.8 mm,
d⫽5⌳/3⫽3.5 mm should all produce the same diffraction
pattern. Experimentally, we find that when the separation between the two exciters is 1.4, 2.7, and 3.4 mm, we observe
diffraction patterns identical to those found in Figs. 5共a兲–共c兲,
depending on the applied voltage. We could not set the
separation between the exciters to values smaller than 1 mm,
which restricted us from checking our conclusions for
smaller values of d. This discussion is summarized in
Table II.
728
Am. J. Phys., Vol. 73, No. 8, August 2005
Table II. The separation between the two exciters, d, for complete destructive interference and unchanged diffraction pattern for ⌳⫽2.1 mm.
Separation between the two pins
d
Theory 共mm兲
Experiment 共mm兲
Complete destructive
interference
⌳/2
3⌳/2
5⌳/2
1.05
3.15
5.25
1.0
3.0
5.2
Unchanged diffraction
pattern
⌳/3
2⌳/3
4⌳/3
5⌳/3
0.7
1.4
2.8
3.5
¯
1.4
2.7
3.4
Nature of the pattern
The dispersion relation, ⍀ 2 ⫽ ␣ k 3 / ␳ , where ␣ and ␳ are
the surface tension and the density of the liquid, respectively,
provides a third theoretical check on the value of the wavelength of the surface acoustic waves. From this relation we
calculated the wavelength of surface acoustic waves in pure
water 共with ␣ ⫽72 dyne/cm and ␳ ⫽1 gm/cc) to be 2.1 mm.
Finally, from the values of the various quantities we can
estimate the phase velocity v p ⫽ 冑2 ␲ ␣ /⌳ ␳ of the surface
waves along the line joining the two sources.21 The result is
v p ⫽46 cm/s.
V. SUMMARY
We have seen that we can measure the amplitude and
wavelength of liquid surface waves using a simple optical
probe. We have noted how the intensity varies with the amplitude and experimentally locate the Bessel function zeros
from the vanishing of various diffraction orders. In addition,
we can see the effects of superposed surface acoustic waves
via optical diffraction.
We mention here an experimental result in Fig. 5共d兲. Recall that our quantitative results are for the region along the
line joining the sources. For light falling on other regions
where the liquid surface wave profile shows a lattice-like
structure 共see the region orthogonal to the line joining the
sources in Fig. 2兲, we have not been able to find the exact
analytical expression for the intensity distribution of diffracted light. Figure 5共d兲 shows the experimentally observed
diffraction intensity from the liquid surface when the light is
incident on a region near the dashed-dotted line in Fig. 2.
The explanation of Fig. 5共d兲 is left as an open question for
the curious reader.
We note that the experiment can easily be used to measure
the surface tension of the liquid 共the surface tension appears
in the dispersion relation兲. By using a single exciter, we can
measure the wavelength of the surface acoustic waves for
different frequencies of the exciter, and then plot ln ␻ versus
ln k. The intercept of this plot will be related to the surface
tension. A modification of the setup would let us measure the
viscosity by considering the spatial damping of the surface
waves.
ACKNOWLEDGMENTS
A.R. and T.K.B. thank the Department of Science and
Technology, Government of India for financial support. The
authors thank G. P. Sastry for useful discussions and C. S.
Kumar for his help with the photographs.
Barik, Roy, and Kar
728
a兲
Electronic mail: [email protected]
E. Hecht, Optics, 4th ed. 共Pearson Education, Singapore, 2003兲, pp. 460–
464.
2
D. P. Morgan, ‘‘A history of surface acoustic wave devices,’’ Int. J. High
Speed Electron. Syst. 10, 553– 602 共2000兲, and references therein.
3
A. Khan, R. Rimeika, D. Ciplys, R. Gaska, and M. S. Shur, ‘‘Optical
guided modes and surface acoustic waves in GaN grown on 共0001兲 sapphire substrates,’’ Phys. Status Solidi B 216, 477– 480 共1999兲.
4
C. S. Tsai, ‘‘Integrated acoustooptic circuits and applications,’’ IEEE
Trans. Ultrason. Ferroelectr. Freq. Control 39, 529–554 共1992兲.
5
C. Deger, E. Born, H. Angerer, O. Ambacher, M. Stutzmann, J. Hornsteine,
E. Riha, and G. Fischeraurer, ‘‘Sound velocity of Alx Ga1⫺x N thin films
obtained by surface acoustic wave measurements,’’ Appl. Phys. Lett. 72,
2400–2402 共1998兲.
6
R. Rimeika, D. Ciplys, R. Gaska, J. W. Yang, M. A. Khan, M. S. Shur, and
E. Towe, ‘‘Diffraction of optical waves by surface acoustic waves in
GaN,’’ Appl. Phys. Lett. 77, 480– 482 共2000兲.
7
P. A. Hess, ‘‘Surface acoustic waves in materials science,’’ Phys. Today
55„3…, 42– 47 共2002兲.
8
G. I. Stegeman, ‘‘Optical probing of surface waves and surface wave devices,’’ IEEE Trans. Sonics Ultrason. SU-23, 33– 63 共1976兲, and references therein.
9
J. P. Monchalin, ‘‘Optical detection of ultrasound,’’ IEEE Trans. Ultrason.
Ferroelectr. Freq. Control 33, 485– 498 共1986兲.
10
F. R. Watson, ‘‘Surface tension at the interface of two liquids determined
experimentally by the method of ripple waves,’’ Phys. Rev. 12, 257–278
共1901兲; F. R. Watson and W. A. Shewhart, ‘‘A study of ripple wave motion,’’ ibid. 7, 226 –231 共1916兲.
11
J. C. Earnshaw and R. C. McGivren, ‘‘Photon correlation spectroscopy of
thermal fluctuations of liquid surface,’’ J. Phys. D 20, 82–92 共1987兲; J. C.
Earnshaw and E. McCoo, ‘‘Mode mixing of liquid surface waves,’’ Phys.
Rev. Lett. 72, 84 – 87 共1994兲.
1
729
Am. J. Phys., Vol. 73, No. 8, August 2005
12
Light Scattering by Liquid Surfaces and Complementary Techniques, edited by D. Langevin 共Dekker, New York, 1992兲.
13
K. Y. Lee, T. Chou, D. S. Chung, and E. Mazur, ‘‘Direct measurement of
the spatial damping of capillary waves at liquid–vapor interfaces,’’ J.
Phys. Chem. 97, 12876 –12878 共1993兲.
14
V. Kolevzon and G. Gerbeth, ‘‘Light scattering spectroscopy of a liquid
gallium surface,’’ J. Phys. D 29, 2071–2081 共1996兲; V. Kolevzon, G. Gerbeth, and G. Pozdniakov, ‘‘Light-scattering study of the mercury liquid–
vapor interface,’’ Phys. Rev. E 55, 3134 –3142 共1997兲.
15
W. M. Klipstein, J. S. Radnich, and S. K. Lamoreaux, ‘‘Thermally excited
liquid surface waves and their study through the quasielastic scattering of
light,’’ Am. J. Phys. 64, 758 –765 共1996兲.
16
R. Miao, Z. Yang, J. Zhu, and C. Shen, ‘‘Visualization of low frequency
liquid surface acoustic waves by means of optical diffraction,’’ Appl. Phys.
Lett. 80, 3033–3035 共2002兲.
17
Spatial damping of the surface acoustic waves is measured by the decay
constant ␦ ⫽ 3 ␣ /4⍀, where ␩ is the viscosity of the liquid 共see Ref. 13兲.
For acoustic waves on the surface of water at 220 Hz, ␦ ⬵44 mm, which
is much larger than the spot size of the laser beam 共1.8 mm兲.
18
The wave function for two-dimensional surface waves is, in general, more
complicated. However, for our purposes here, the sinusoidal form is adequate and produces reasonable results.
19
J. W. Goodman, Introduction to Fourier Optics 共McGraw-Hill, San Francisco, 1968兲, p. 62.
20
B. D. Duncan, ‘‘Visualization of surface acoustic waves by means of synchronous amplitude-modulated illumination,’’ Appl. Opt. 39, 2888 –2895
共2000兲.
21
V. G. Levich, Physicochemical Hydrodynamics 共Prentice Hall, Englewood
Cliffs, NJ, 1962兲, p. 596.
Barik, Roy, and Kar
729