CYMATICS Measuring Surface Tension of Water using Chladni Patterns
St. Xavier’s College Kolkata
RAHUL CHAKRABORTY 3 r d Ye a r . P H S A . 1 4 8 Guided By : Dr. Dibakar Dutta
B.SC FINAL YEAR PROJECT WORK
CONTENTS 1. Abstract
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2. Introduction
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3. The Science and Art of Cymatics
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4. Chladni Patterns
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5. Chladni’s Law
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6. Surface Waves on Water
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6.1. Finding a Governing Equation
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6.2. The Experiment
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6.3. Experimental Conditions & Results
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6.4. Error Analysis
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6.5. Discussions
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7. Conclusion
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8. Acknowledgment
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9. Bibliography
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ABSTRACT
C
ymatics is the study sound vibrations by visualizing it. Chladni Patterns are patterns generated on thin metal sheets using grains of sand and frequency generator. Here, we stretched this same concept to Surface waves. We measured the variation of wavelength with frequency of surface aves in water. This relation helped us find the surface tension of water. The final value of surface tension that we obtain is : T = 77.66 Âą 6.44 milliN/m
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INTRODUCTION
E
very one of us in this digital age are very familiar with Music Players/Organizers present in our personal computers/mobiles/etc. Many of these players, especially those present in the PCs has a visualization function which allows the user to listen to music while some sort of visualizations/designs corresponding to music’s loudness and frequency is displayed. These functions are an artistic interpretation of various sound frequencies. In reality sound frequency has a specific graphical function that can display a visual representation of the mathematical interpretation of sound. This study of sound is known as Cymatics. Also if we look in Nature, we see widespread evidence of periodic systems. These systems show a continuously repeated change from one set of conditions to another, opposite set. This repetition of polar phases occurs alike in systematized and patterned elements and in processes and series of events. If we turn our eyes to the great natural domains, the whole vegetable kingdom, for instance, is a gigantic example of recurrent elements, an endless formation of tissues on a macroscopic, microscopic and election microscopic scale. Much the same applies to the active muscle system which is actually in a state of constant vibration. Again, periodic rhythms are a dominant feature of the animal kingdom. This underlines Cymatics doesn’t deals with vibratory phenomena of sound in the narrow sense, but rather with the effects of vibrations and periodicity in general. This was an opportunity to research more about Cymatics and get a better understanding of Cymatics. One can use various mediums or objects to study Cymatics. We will be specifically taking the help of The Chladni Patterns in this project work. One can also take the help of Interactive Applets that can simulate the Chladni Patterns and show the interesting characteristic of sound , thereby helping to get a better grip on the topic of Cymatics. The applet simulates the Chladni’s Pattern corresponding to the different sound frequencies, just like the visualization function of Media players.
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THE SCIENCE AND ART OF CYMATICS
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he Science and art of Cymatics is a process for making sound waves visible. Typically the surface of a plate, diaphragm, or membrane is vibrated, and regions of maximum and minimum displacement are made visible in a thin coating of particles, paste, or liquid. Different patterns emerge in the excitatory medium depending on the geometry of the plate and the driving frequency.
The most typical representation of Cymatics experiment is Chladni’s Patterns using Chladni’s plates. This plate is a metal plate with edges that are free to vibrate while the center of the plate is fixed with a rod. It can be vibrated by running a bow on the edge of the plate or by vibration of the rod that is fixed in the center of the plate.
Cymatics (from Greek: “wave”) is the study of periodic effects that sound and vibration have on matter. The word Cymatics was coined by Hans Jenny, a Swiss medical doctor and a pioneer in this field.
A Circular Chladni Plate driven by a Electro-mechanical Vibrator
Each frequency has its own unique shape, and this visualization of sound can be explained mathematically. Sound can be visualized through a medium when the sound frequency vibrates through a medium. Cymatics basically deals with these periodic effects of visualization of sound and vibrations.
As the frequency increases, the complexity of the pattern increases. Also, the pattern of sound changes with the different shape of the plate. Some typical plates include plates that are Circular, Rectangular, Violin shaped, and Square shaped. However instead of solid particles such as salt/sand, we shall be using a liquid medium (such as water) on a Circular disc, to study the patterns generated according to Chladni’s Law.
The patterns are created because of the presence of nodes of the sound wave.
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CHLADNI PATTERNS
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rigid plate will have a set of natural resonance frequencies just like a string, and when the plate is excited at one of these frequencies, it will form a standing wave with fixed nodes. These nodes will form lines on the plate, in contrast to points on the string. This was what German physicist Ernst Chladni (1756-1827) demonstrated using the clever technique that now bears his name.
Chladni excited these resonant vibrations by drawing a violin bow across its edge. The plate was bowed until it reached resonance, when the vibration causes the sand to move and concentrate along the nodal lines where the surface is still, outlining the nodal lines. The patterns formed by these lines are what are now called Chladni Patterns. Today, however, we can mechanically drive the plate from below with a mechanical driver. The patterns that result are beautiful, and increasingly complicated as the frequency is increased.
Chladni realized that sand sprinkled on the top of the plate would be pushed away from the vibrating regions and settle into these nodes, allowing the node patterns to be seen.
The diagrams of Ernst Chladni are the scientific, artistic, and even the sociological birthplace of the modern field of wave physics and quantum chaos. The plates he used were made to vibrate by stoking them with a violin bow and these metal plates vibrated at pure, audible pitches, and each pitch has a unique nodal pattern. Chladni took the trouble to carefully diagram the patterns, which helped to popularize his work. The following is a set of drawings from Chladni’s original publication.
Chladni’s method of creating Chladni’s Patterns. He used a violin bow to resonate the plate.
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Patterns sketched by Chladni himself, in his 1809 book TraitÊ d’Acoustique
There are an infinite number of Chladni patterns, corresponding to the infinite number of vibrational modes/frequencies. Chladni patterns can be observed on plates of a wide range of materials such as glass, metals, ceramics, wood, composite materials, laminated board, corrugated board and rocks. For most frequencies, nothing at all happens, but when certain special frequencies are hit, standing waves appear on the plate, driving the sand/solid particles away from the points of large vibration to the points of no vibration. For liquid excitation particles, it is just the opposite. By varying the frequency of oscillation, we can find a
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large number of the Resonance Frequencies and their corresponding patterns, which become increasingly complex and beautiful as we increase the rate of oscillation/vibration. The patterns become more complicated at higher frequencies, even providing inspiration for some artists.
at which the object “wants� to move. A pendulum or swing is the most simple example of a system that can be driven at resonance; most objects have multiple resonance frequencies, indeed an infinite number of them. The next example in order of complexity is a vibrating string which is fixed at both end, like those on a guitar or a clothesline. If we were to mechanically drive this string, we would find that it has a lowest (fundamental) frequency, which we will call f, and then higher-order frequencies (harmonics) at 2f, 3f, 4f, and so on to infinity. The vibrations on the string (which we call modes) look different for each harmonic, as illustrated below.
Thus, Chladni Patterns formed due to multiple modes vibrations are directly dependent on the following factors : 1) The dimensions of the plane. 2) Mass per area of the plane. 3) Excitation frequencies. 4) Locations of excitation. Chladni Patterns are lovely examples of resonance, an important concept in almost all branches of physics, including vibration. Rigid and Semi-rigid bodies possess an (in principle) infinite number of natural frequencies of vibration
Strings of different lengths, thicknesses and tensions will have different fundamental frequencies, but they will all follow the same pattern: higher-order
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resonant frequencies will all be integer multiples of the fundamental. The illustration itself tells us the origin of the patterns. The fundamental and harmonic modes are those for which one round trip of waves on the string is a single wavelength. A single cycle of a wave, the size of a wavelength, is one for which the wave has completed one complete up/down motion, as shown below.
of the string at these points, which are called Nodes.
The fundamental mode in the illustration above is only a half-wavelength. A full round trip of the wave back and forth across the string constitutes a complete wavelength. When we drive the string at one of these fundamental frequencies, we are always adding to the vibration of the string in the same way at the same point in the string’s cycle, much like a child on a swing always pumps it at the same point in the motion to move it higher.
This sort of mechanical resonance has had unexpected and devastating consequences in the past. Suspension bridges have been brought down when soldiers marched across them in formation, inadvertently driving the bridges at the resonance frequency until they collapsed completely.
We could in principle put a finger on the string at those points, blocking its motion there, and not affect the harmonic mode at all; this technique is used in guitar playing and is known as a Pinch Harmonic. The waves themselves don’t move or change shape at all, except to move up or down; they essentially “stand in place.” Because of this, they are known as Standing Waves. We shall be making use of the standing waves/surface waves of liquid to study the resonant patterns thus formed.
A string can only vibrate along its length. When we consider objects that have length and width, we can get correspondingly more complicated resonance phenomena. This was what Ernst Chladni (1756-1827) discovered and named them Chladni Patterns.
Looking again at the mode illustration, it can be seen that the higher harmonics each have points where the string amplitude is zero, i.e, there is no motion
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Next, we shall study some of these patterns which were created by using an Aluminum plate 0.125 inch thick. Six shapes have been used, viz. Round, Square, Hexagonal, Triangular, Stadium-shaped & Violin-shaped, all (except one) driven by an electro-mechanical vibrator at few specific resonant frequencies. Finally, these patterns were compared to the sketches of Chladni Patterns by Ernst Chladni himself.
Round plate (70 cm across, held at center & bowed) (a) 10 spoke pattern (b) 14 spoke pattern
Square plate (70 cm on a side, driven from center with electro-mechanical vibrator) (a) 142.2 Hz (b) 225.0 Hz (c) 1450.2 Hz (d) 3139.7 Hz (e) 3678.1 Hz (f) 5875.5 Hz
Now, we can compare these patterns with those sketched by Chladni himself, in his 1809 book TraitÊ d’Acoustique. Similarities between them can be readily seen.
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Next, we compare the patterns generated by using the Hexagonal, Triangular, Stadium-shaped & Violin-shaped Plates.
Hexagonal plate (driven from center with electro-mechanical vibrator) (a) 402 Hz (b) 1.013 kHz (c) 1.314 kHz (d) 1.364 kHz (e) 1.622 kHz (f) 1.959 kHz
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Triangular plate (driven from center with electro-mechanical vibrator) (a) 406 Hz (b) 1.342 kHz (c) 1.381 kHz
Stadium Shaped plate (70 cm on a side, driven from center with electro-mechanical vibrator) (a) 387.8 Hz (b) 519.1 Hz (c) 649.6 Hz (d) 2667.3 Hz (e) 2845.0 Hz (f) 3215.0 Hz
Violin Shaped plate (120 cm on a side, driven from center with electro-mechanical vibrator) (a) 145.2 Hz (b) 268.0 Hz (c) 762.4 Hz (d) 954.1 Hz (e) 1452.3 Hz (f) 1743.5 Hz
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Few more sketches of different plate patterns by Chladni that were published in his 1809 book TraitÊ d’Acoustique
So, we can see that the vibration patterns were almost similar to the original patterns drawn by Chladni. We shall next study the mathematical equation behind the formation of these patterns. However, we constrict ourselves to the 2-D plate equation of Square & Round plates only.
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CHLADNI’S LAW
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hladni also derived a law, known as the Chladni’s Law, which relates the frequency of modes of vibration for flat surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes. It is stated as the equation :
f = C (m + 2n) p where C and p are coefficients which depend on the properties of the plate. For flat circular plates, p is roughly 2, but Chladni’s law can also be used to describe the vibrations of cymbals, hand-bells, and church bells in which case p can vary from 1.4 to 2.4. In fact, p can even vary for a single object, depending on which family of nodes is being examined. We find experimentally and theoretically that thin plates or membranes resonate at certain “nodes.” This means due to initial conditions imposed upon the plate (i.e. fixed edges) the plate can vibrate only at certain allowable frequencies and will demonstrate predictable “node” patterns. Nodes are points on the plate that vibrate with zero amplitude, while other surrounding points have non-zero amplitude. This concept can be seen with a vibrating string: tie one end of a string to a fixed object and smoothly vibrate the other end of the string. If vibrated fast enough, there will be a point or points in the middle that seem to be still while the rest of the string vibrates wildly. These points are the nodes. On a 2-Dimensional vibrating plate, the nodes are not points, but curves. With the circular plate, we most commonly observe concentric circular nodes and diametric modes, while with the rectangular plate, we commonly observe nodes parallel with the boundaries. The square plate equation for the 2-Dimensional image is,
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rmy rny rnx l cos b l - cos b rmx l=0 cos b L l cos b L L L n = radial (circular) node m = diametric (linear) node L = length of the sides of the plate The Chladni’s pattern is obtained from graphing the contour lines of this function. We note that the solution is uninteresting for n = m. In the case of 3D solutions,
cos (c 1 x) + cos (c2 y) + cos (c3 z) = 0 with 0 < x < Π , 0 < y < Π , 0 < z < Π . And for the circular plate,
J m (h m,n r) (c 1 cos (mi) + c2 sin (mi)) = 0 where hm,n is the nth zero of the Bessel function of order m ; Jm is the Bessel function of the first kind and the radius is equal to 1. As the vertical moment of the plate is rather small, it is hard to make direct observation with naked eyes, so Chladni invented the clever experiment by showing the vibration of the plate by revealing the sand/water pattern. From a intuitive guess, for the same plate, different frequencies of vibration give different pattern, and Chladni has made drawing of those patterns in a systematic way as seen before. Later on, Chladni postulated the law (known as Chladni’s Law), that there is a relationship between the frequency and nodal numbers, by saying f=(m+2n)2 ,where the integers m and n are the number of node lines by counting the diametric node number a and side node number n, and m + n = a. For example, in the pattern illustrated in the next page, we have m + n = 7 and n = 4, so m = 3. Therefore, according to Chladni’s hypothesis, the frequency is f=112. Thus we get the desired resonant frequency at which the plate vibrates.
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A case from Chladni pattern, to show the number of m and n, Here, m = 4 and n = 3.
Below is the schematic representation of Chladni Patterns, from where we can determine the number of m and n respectively.
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SURFACE WAVES ON WATER
A
time. This velocity is also known as the Phase Velocity vph .
wave is a disturbance that transfers energy progressively from point to point in a medium. The disturbance may take the form of elastic deformation or variation of physical quantities such as pressure, electric or magnetic intensity, or temperature.
Surface Wave is an everyday phenomenon that is very different from most other waves in that its velocity is strongly dependent on water depth and wavelength. A surface wave is a mechanical wave that propagates along the interface between differing media, usually as a gravity wave between two fluids with different densities. A surface wave can also be an Elastic (or Seismic) Wave. It can also be an electromagnetic wave guided by a refractive index gradient. In radio transmission, a ground wave is a surface wave that propagates close to the surface of the Earth.
Waves are usually analyzed in terms of sinusoidal waves. There are several physical quantities used to describe sinusoidal waves. (1) The amplitude A is the maximum displacement of the wave disturbance. (2) The wavelength l is the distance between peaks or troughs.
The term surface wave can describe waves over an ocean, even when they are approximated by Airy functions and are more properly called Creeping Waves. Examples are the waves at the surface of water and air (ocean surface waves), or ripples in the sand at the interface with water or air. Another example is internal waves, which can be transmitted along the interface of two water masses of different densities.
(3) The period T is the time elapsed for two consecutive crests to pass over a given point. (4) Frequency f is the number of complete wave cycles produced or traveling through in one second. (5) The velocity v is the distance traveled by the wave crest in a given unit of
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It is fascinating that the same physical impetus (an object striking the water surface) can generate qualitatively opposite behavior. This particular wave motion is an example of Dispersion, where wave speed depends on wavelength. The waves resulting from an impact on a surface are dispersive, spreading away from each other over time. In general, wave theory also includes the study of non-dispersive waves such as waves on a vibrating string or sound waves in air or water.
the fluid is taken to be finite and constant below the undisturbed surface, and the free surface of the water is described as a Perturbation about z = 0. In this context, the word â&#x20AC;&#x153;perturbationâ&#x20AC;? means that the unknown free surface may be described by a function
z = h (x, y, t) assumed to be as smooth as necessary and that both h and its first spatial derivatives are small. The motion of every point inside the water is described by the velocity field,
In order to observe this interesting phenomenon, we have made use of resonant patterns formed on a shallow circular plate, filled with water, when driven by an acoustic speaker according to the input of specific frequencies to the speaker.
uv (x, y, z, t) = (u 1, u 2, u 3) where u1, u2 and u3 are all functions of x, y, z and t.
Finding a governing equation The variables in which we develop the model are sketched in the figure aside, showing the coordinate x and z axes, positioned so that the z axis points in the vertical direction with the undisturbed water level at z = 0. We assume that the y axis points into the page. We also assume that the fluid lies below an infinite layer of still air. The depth h of
There are three primary assumptions about the water and its motion : (1) The water is inviscid. (2) The flow is irrotational.
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(3) Water is neither created nor destroyed within the fluid. The second of our primary assumptions can be expressed mathematically in terms of the velocity field by the equation,
d # uv = 0 where d = (2 x, 2 y, 2 z) is the vector of spatial derivatives. The third assumption is the physical statement that the vector field is incompressible.
d $ uv = 0 By a standard theorem from vector calculus, if d # uv = 0 , then there exists some scalar function z (x, y, z, t) such that uv = d (z) . Combining this definition of with the equation above yields d $ dz = 0 , so that our second and third assumptions result in the Laplace equation,
d2 z = 0 Using z = h (r, i, t) we can say,
2z 2h z = u z = 2z 2t We consider water does not flow in the solid bottom surface at z = -h and r = R (rod of petri-dish).
2z u r = 2r = 0
at z = - h
2z u z = 2z = 0
at r = R
The Dynamic Boundary Condition realizes the physical fact that there is a pressure jump across the free water surface z = h (x, t) that depends on surface tension.
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As developed previously, we begin with a form of Eulerâ&#x20AC;&#x2122;s equation of motion for an irrotational flow of velocity uv = dz and pressure p of a fluid with density t under the influence of gravity g.
2uv = 2 (dz) = - d ( p + 1 u 2 + gz) t 2 2t 2t v ` t 22tu =-dP - tge z Let us split the total pressure into static and dynamic parts, P = p 0 + p p0 = static pressure = -tgz p0 satisfies 0 =-dp o - tge z
v 2dz ` t 22tu = t 2t =-dp
so that p =-t 2z
2t
This relates the dynamic pressure to the velocity potential. Let us assume that the air above the petri-dish is stagnant essentially. Because of its very small density, we ignore the dynamic effect of air and assume the air pressure to be constant, which can be taken as 0, without loss of generality. To calculate Surface Tension we consider a thin film covering the water surface with tension T per unit length. We consider a horizontal rectangle dx dy on surface,
2h (T 2x
2h x + dx - T 2x
2h x ) dy + (T 2y
2h y + dy - T 2y
22 h 22 h + 2y ) dx dy y ) dx = T ( 2x 2
By continuity of vertical force on unit area of surface, we get,
22 h 22 h p o + p + T ( 2 + 2y ) = 0 2x 2z 22 h 22 h ` - tgh - t 2t + T ( 2 + 2y ) = 0 2x
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at z = 0
22 z 2z 2 2z & 2 + g 2z - T (d t 2z ) = 0 2t
at z = 0
Let us take z = f (z) J 0 (kr) sin ~t . This is because the second order wave equation has solution of this form. 2 22 f 2 2 1 ` f (z) sin ~t [ 2 J 0 (kr) + r 2r J 0] + J 0 (kr) sin ~t 2 = 0 2r 2z
& f m - k2 f = 0
--- (A)
also,
-~2 tf (z) J 0 (kr) sin ~t - f l (z) gt J 0 (kr) sin ~t - Tf l (z) sin ~t d2 J 0 (kr) = 0 & - ~2 t + f l tg + k2 Tf l = 0 f l = 0 at z = - h
--- (B) --- (C)
Thus, f satisfies the following :
(A) f m - k2 f = 0 (B) - ~2 t + f l tg + k 2 Tf l = 0 (C) f l = 0 at z = - h Now,
f (z) = B cosh k (z + h) 3 & ~2 = (gk + T k t ) tanh (kh)
If we get the nth zero at L, then,
k = ZR1n
J 1 (kR) = 0
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( R = radius of petri-dish )
THE EXPERIMENT
is r. The light falling on the petri-dish
Water is taken as the medium for which
produces a projection of the image of concentric nodes and anti nodes formed
surface tension is to be determined. A
on the surface of the liquid on the table.
standard petri-dish with radius r and
This is due to the anti nodes acting as
height H is taken. This petri-dish is at-
lenses for the strong beam and hence
tached to a 4â&#x201E;Ś, 20W speaker indirectly
the light through it is brighter than that
by a short plastic rod. The light source
through the nodes. We fix a scale with a
(laser source) is attached higher than
value L0 coinciding with the center C of
the level of the petri-dish at a distance d
the image formed. Suppose for a fre-
from the setup. The speaker is attached
quency f Hz, there are N nodes formed
to a woofer which is connected to a
on the surface. We count n nodes from
tone generator. The tone generator is
the center. Let R be the radius of the
used to produce frequency in the range
petri-dish projected on the table. In
30Hz -150Hz. The woofer amplifies the
the petri-dish let ln be the length for
output of the tone generator which is
n nodes. The distance between C and
fed to the speaker. Hence the petri-dish
projection of n nodes is Ln. Therefore
is vibrated at a particular frequency.
by similarity,
The petri-dish is filled with liquid (wa-
r ln = Ln R
ter) upto a height h. When a particular frequency is fed, ripples are formed on
The relation between no. of nodes, the dimension of petri-dish, and k as discussed before, involving Bessel zero Z1n is :
the surface of the liquid in the form of concentric circles. These are Surface Waves. There are nodes and anti nodes
k = Zl 1n n
formed on the surface. Therefore the distance between the center c of con-
This is used to calculate k. Using the value of k and ~2 we find Surface Ten-
centric nodes and edge of the petri-dish
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Experimental Setup for determining Surface Tension using Cymatics
Projected Image of Nodes and Anti-nodes generated
Tone Generated using Mathematica
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sion using the Dispersion Relation, 3 k ~2 = (gk + T t ) tanh (kh)
For our calculated values of k and h, we get tanh(kh) ~1 (minimum value of k calculated k=500, height of water h=1cm gives tanh(5)=0.999 ~1) Therefore, ~ = (gk + T k 3) t 2
(~2 - gk) t Hence, we get T = k3
Schematic Diagram of determining Surface Tension using Cymatics
Experimental conditions and Results Radius of the petri-dish r Position of center of image in the projection L0 Radius of the petri-dish in the projected image R Height of liquid in the petri-dish h Acceleration due to gravity g Density of the liquid Ď
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4.25 cm 11.8 cm 11.7 cm 1 cm 9.8 m/s2 1000 kg/m3
Table 1. Experimental Data
The Bessel Zeroes are taken from standard table. k=Zpm/R, where Zpm = mth zero of pth order Bessel function. Here we take 1st zero of nth order Bessel, i.e, Bessel (1,n).
Table 2
where,
(~2 - gk) t T= k3
Mean T= 77.66 miliN/m
Standard Deviation= 6.44
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ERROR ANALYSIS Maximum Proportional Error :
dk = dr + dR + dL r R L k dR + dL ) k + ` dk = ( dr r L R where, dL = 0.2cm , dR = 0.2cm , dr = 0.01cm
DISCUSSIONS (1) The mean calculated value of Surface Tension is found to be T = 77.66 miliN/m. The theoretical value for pure water is 71.97 miliN/m at 25o C. This can be account-
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ed for by the fact that temperature on the day of experiment was 29oC. Pure water was not used as the medium and it contained many surface impurities. (2) If a bigger petri-dish was taken, then more no. of nodes could be calculated for a given frequency. Hence error in k would have been less. (3) Frequencies taken are in the range 30Hz - 50Hz and then 95Hz - 150Hz. Readings could not be taken in the range 50Hz - 95Hz, as the gain of the amplifier was very low in that frequency range. (4) The projected length could be calculated with better precision. (5) The other methods to calculate surface tension are rather complicated and difficult. This is a simple and easy way to calculate Surface Tension. (6) We have taken projected image with laser light falling at an angle on the petri-dish. If a better method could be setup to make the light fall overhead on the pattern, then the image formed would be a perfect projected image with no loss of nodes at the edges. This would reduce the error considerably.
* * * * * * * * *
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CONCLUSION
W
e can learn from the existence of resonance vibrations, each occurring at a discrete isolated frequency. In fact, similar mathematics describes the energy levels of electrons in atoms and many other systems. The era of quantum mechanics really took off in 1913 when Niels Bohr speculated that the curious properties of light emitted by atoms was the result of electrons only being allowed to exist stably in certain specific, discrete, orbits, each with its own discrete energy. What Bohr could not explain is why electrons could only maintain these discrete orbits. This question was answered in 1924 when Louis de Broglie postulated that electrons themselves have wavelike properties. Just like waves on a string, then, an electron wave has certain natural modes of oscillation, and these are the stable states of the atom. Some of these probability waves are illustrated aside. Laser cavities also demonstrate special modes of resonance. In a rectangular laser cavity, a laser can be induced to output one of multiple so-called Hermite-Gauss modes (named after the mathematical functions used to describe them). These modes are a little different from those described earlier, as they are not distinguished by their energy (or frequency), but rather by their momentum. Nevertheless, the principle remains the same. If one uses a circularly symmetric cavity, one can produce the so-called Laguerre-Gauss laser modes, some of which are shown
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below. These patterns are strikingly similar to those generated by a Square and Circular Chladni plate.
The first few Hermite-Gauss modes
Some of the Laguerre-Gauss laser modes
So, by studying the vibrations of a simple metal plate, we can gain insight into everything from light to atoms !
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ACKNOWLEDGMENT
I
would like to thank my project supervisor Dr. Dibakar Dutta for his assistance and overall guidance in executing this project work. The project would not have been possible to complete without the support of my project collaborators Soumi Choudhury and Nirjhar Chakraborty. I would like to specially mention the immense help of Aditya Chowdhury for providing us with all the needed technical inputs.
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BIBLIOGRAPHY Insipration from http://nigelstanford.com/Cymatics/ Hans Jenny_Cymatics_A_Study_of_Wave_Phenomena_and_Vibration http://k3dsurf.s4.bizhat.com/k3dsurf-ftopic176.html 4http://www.tokenrock.com/explain-cymatics-16.html http://attunedvibrations.com/general/cymatics/ http://www.roelhollander.eu/en/tuning-frequency/cymatics/ https://en.wikipedia.org/wiki/Hans_Jenny_(cymatics) http://www.cymaticsource.com/ https://www.youtube.com/watch?v=2OeLMu1u5E4 http://blog.world-mysteries.com/science/making-sound-visible-through-cymatics/ http://www.ted.com/talks/evan_grant_cymatics?language=en#t-180866 http://en.wikipedia.org/wiki/Cymatics http://en.wikipedia.org/wiki/Ernst_Chladni http://demonstrations.wolfram.com/ChladniFigures/
http://www.physics.utoronto.ca/nonlinear/chladni.html
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