ABSTRACT. The velocity of sound varies with the medium where it travels. The velocity of the sound in a metal rod was obtained in the experiment by the application of the principles of the resonance. A metal rod that is clamped at its midpoint is stroked in a lengthwise manner to produce friction and energy in a form of longitudinal wave. A glass tube, containing lycopodium powder and is closed at its one end, is connected to the metal rod through a disk which is not touching the tube. When the wave enters the tube, it will agitate the dust inside and forms visible shapes of a wave. At the same time, the wave resonates and results on producing sound as well as generating constant frequency. Measurements of the length of the visible wave segments along with the length of the rod are done. The velocity of sound in air is also measured by recording the temperature of the room. Using the data, the experimental value of vm is obtained. Afterwards, a comparison with the theoretical value found in the book and with the value obtained from the elastic property of the metal is accomplished. For each comparison, we had obtained a percentage error of 0.429% and 1.35%, respectively, but still, lies within the velocity uncertainty range of 3460.09m /s±58.31m /s.

INTRODUCTION

During your childhood, have you ever been tried to make a telephone toy using Styrofoam cup and a very long string? You noticed that at a certain distance, you still hear each other. Unlike without it, you still need to shout just to hear your friend. Generally, the speed of the sound is faster in solids than in liquids and gases. You also experienced that when you are in a baseball game or sat far away from the stage during a concert, you may have noticed something odd. You saw the batter hit the ball, but did not hear the crack of the impact until a few seconds later. Or, you saw the drummer strike the drum, but it took an extra moment before you heard it. It is due to light is faster than sound which we are used to see.

Sound surrounds us everyday. We may not notice it sometimes. Sound has great importance in our daily life.  We have learned that sound is a form of energy. It is produced by vibrations. Sound waves are propagated through longitudinal waves. They are also elastic waves hence they need a material medium for their transmission. They can not be transmitted in vacuum.

Sound travels in solids, liquids and gases. Their velocity is maximum in solids and least in gases. We hear various kinds of sound in our daily life, pleasant sounds called the musical sounds, unpleasant sound called the noise, loud sound, high pitched sound etc. In this laboratory, we can be able to measure speed of sound using the traditional method. The sound is allowed to travel at solid and gas, particularly in a metal rod and an air column. By that, frequency and the speed of the sound can be obtained. This method is called the Kundt’s tube method. A diagram of the Kundt’s tube is drawn on the next column for visualization.

Figure 1. The complete diagram of a Kundt’s Tube

The Kundt’s tube is a long narrow tube made of glass that is close by a stopper at one end. It is mounted in a certain metal bar frame so that it can be adjusted vertically/longitudinally with respect to the frame. A metal rod is clamped to the support frame, exactly at the center of the rod. The rod is clamped at the center to produce a fundamental mode of vibration. This rod has a disk at its one end. That is inserted into the glass tube but it does not touch the tube. In addition, the glass tube contains some dust inside which is distributed evenly in the entire length of the tube.

While the rod is stroked lengthwise with a cloth, longitudinal standing waves are observed to be formed from the dust inside. Those waves are set up in it with a minimum vibration (node) at the clamped part (at the center) and maximum vibration (antinode) at each end. Since the distance between two consecutive nodes or antinodes in a standing wave is exactly half of its wavelength, then, the wavelength of the tone in the rod is twice the length of the rod.

The sound waves are produced in the metal rod as the powder in the cloth is being displaced as if its molecules are vibrating successively. Furthermore, the frequency of the vibrations in a given metal rod depends on the length of the rod and the position of the clamp. The vibrations are transmitted to the disk, which in turn transmits them into the air column at the same or equal frequency. The wavelength and velocity change

as the wave chain goes from one medium to another, but the frequency is still consistent.

As the longitudinal waves leave the rod at the end containing the disk, it will now proceed inside the tube. These waves act as a close pipe. In that particular scenario, the wave is reflected on the close end of the tube so that the air in the tube is acted upon by two similar sets of waves travelling in opposite directions. Consequently, if the distance between the disk and the close end is such to produce resonance, that is when the length of the tube is exactly a multiple of the half of the wavelength, the cork dust will be agitated at the antinodes positions and remain relatively still at the nodes positions as to produce standing waves. For that, the length of one dust loop is one-half of the wavelengths in the air.

If we assign Vm as the velocity of sound in the metal and λm as the wavelength, the frequency f is given by

f m=vmλm

Equation 1

Again, when the rod is stroked lengthwise, standing waves are set up in the vibration rod. The vibration contains a node at the clamped part (at the center) and antinodes at both ends. On the other hand, if we let va and λa to be the velocity and the wavelength, respectively in the air column at a specific temperature, then the frequency is given by

f a=vaλa

Equation 2

For both metal rod and in air, the frequency is the same because for sound waves, the frequency is just dependent on its source. By equating both frequencies, we can be able to derive the relationship of the velocity and wavelength of the sound for two different mediums.

f m=f a

vmλm

=v aλa

Equation 3

Since for fundamental mode of vibration,

λ=2 L Equation 4

Then,

vm2Lm

=va2La

Equation 5

Finally, we can get the velocity of sound in rod in terms of velocity of sound in air and the length of the rod and the air column.

vm=va LmLa

Equation 6

The velocity of sound in air varies with the temperature. Velocity is directly proportional to the square root of the absolute temperature.

v1v2

=√T 1T 2 Equation 7

By manipulation, we can simplify this equation into a linear equation. This equation is the best fit linear equation with the original equation. The revised equation is given below.

va=332ms+0.6 (T ) Equation 8

The best fit linear equation is exactly portrayed in the graph below.

0 10 20 30 40 50 60 70 80310

320

330

340

350

360

370

380

f(x) = 0.570865790870683 x + 332.416619059234

T, o C

velo

city,

m/s

Graph 1. Best fit linear graph of Equation 7

The velocity v is the velocity of air at T, temperature in oC. On the other hand, the factor 0.6, is the increase in velocity per second at T, temperature.

The velocity of a compression wave in a metal depends on the elastic properties and the density of the metal and its value in meters per second is given by

vm=√Yρ Equation 6

Where Y is the Young’s modulus (coefficient of elasticity) and ρ is the density. Y must be expressed in force unit per unit are, N/m2. In the experiment, we were using brass as the metal. The value of the Y and ρ are shown below.

Property Brass (Drawn and Compact)

Young’ Modulus Constant

1x1011 N/m2

Density 8580 kg/m3

The aims of this experiment are to determine the velocity of sound in a metal rod and also to determine the speed of sound in the tube applying the principles of resonance.

METHODOLOGY

In this experiment we will be using the Kundt’s Tube Apparatus, a meter stick, a piece of cloth, a thermometer, rosin and lycopodium powder.

The Kundt’s tube consists of a long, narrow glass tube mounted in a metal frame case. A metal rod (any desired material) is clamped in such a way that its end containing the disk is inside the tube. The rod can be clamped at any distance. However, it is better to clamped it at the center to make the experiment not complicated. The Kundt’s tube is closed at one end by a stopper. The wave produced in this column follows the wave behavior of the close type case. In vibrating the rod, energy comes from the friction produced by stroking cloth at the rod. To produce friction, rosin is rubbed in the cloth. The waves produced inside after vibration is visibly seen through agitation of lycopodium powder. All the lengths needed in this experiment is measured using the meter stick. A low accuracy instrument is just fit with the experiment because it is not important to measure accurate lengths. Finally, thermometer is the instrument used in measuring the temperature of the room.

Normally, we have to put the powder inside the glass tube. But in the experiment, it is already

prepared by the laboratory assistants to prevent waste of materials. The powder is evenly distributed throughout the tube. It is done to make the wave visible later, that is in similar shapes and sizes. The kind of material where the rod is made is to be recorded. The value of the constants, Y and ρ, for the specific material used are obtained using any form of resources. Furthermore, the length of the tube is to be measured using a meter stick. One must be careful to see if the rod is clamped horizontally at its center. This allows the experiment performer to easily calculate the value of velocity of the rod. The rod has a disk at its one end inside the tube. This disk has not to touch the walls of the glass tube. It must be leave free to vibrate. Also, it should be necessary to measure and record the temperature of the room, inside the tube or near the apparatus itself.

After the preliminary assessment of the apparatus, one may now proceed on vibrating the rod. The rosin is initially rubbed on the cloth. The rosin allows the cloth to produce friction with the tube. The energy due to friction will serves as a wave. Strokes on the rod are done afterwards. It is ideal to do smooth, high-pitch tone stroke in a lengthwise manner. It is important not to let the hand slip off at end of the rod. This is because, it causes both ends of the rod to vibrate transversely, and the vibrating disk may break the glass tube.

When the dust inside the tube does not form visible waves, it is advised to adjust the air column by moving it towards the tube in a minimal distance. Continual adjustment can be made until best resonance condition is achieved. This happens when dust agitated formed perfect waves which are measurable and looks exactly the same from one another. When the rod gets warmed greatly, we could cease the stroking and let is cool for a while.

Another problem encountered in this experiment is when one observed that majority of the dust is

concentrating on one side of the node. This can be due to the apparatus is not oriented

Figure 3.Figure 3. Vibrating the Vibrating the metal rodmetal rod

Thermometer

Kundt’s Tube w/ Lycopodium Powder

Meter Stick

Cloth w/ rosin Brass Rod

horizontally. We can minimize this problem by removing some dust.

Once satisfied with the visible waves formed, one may now proceed on measuring the length of the waves (wavelength). On measuring, the first dust loop nearest to the disk of the rod is neglected. It is an option to measure one, two, three or any number of waves desired. However, it is more accurate to measure many waves. From the measured distance, one is to determine the average half wavelength of the sound in air column, La, by dividing it to the total number of loops or segments measured.

Using equation 8, calculate for the velocity of sound in air at the temperature recorded earlier. Once done, the value of the vm, or the velocity of the sound in rod can now be obtained using equation 6.

From the table of velocity of sound in solid in the textbook, compare the obtained experimental value with the theoretical value. In addition, the experimental value can also be compared with the value obtained in equation 9. The percentage error gives us the numerical digit of the error done in the experiment. Notice that the value obtained in equation 9 is much smaller with the

theoretical value. To be intact with the theoretical value, it is better to used internet for the more accurate value of Y and ρ.

When time permits, it is an option to repeat all the measurements for another trial and record the results.

DISCUSSION OF RESULTS

The determined velocities of sound in both rod and in air are accompanied by the data obtained in the experiment. The preliminary and the post data are presented in the table below.

Table 1. KUNDT'S TUBE VELOCILTY OF SOUND IN SOLID

length of metal rod Lr 91.5 cm

average length powder segments, La

9.2 cm

temperature of air t 26.5 °C

velocity of sound in air Vair 347.9 m/s

While doing strokes on the rod, friction is produced between the cloth and the rod. As a result, energy or disturbance will occur in a form of longitudinal wave. The vibrations of the rod are transmitted by the disk to the air in the glass tube closed at one end. The waves set up in the air in the glass tube have the same frequency as those in the rod. Hence, a resonance will be formed and produces sound.

The waves are reflected at the closed end of the tube and the air in the tube is thus acted upon by two similar sets of waves traveling in opposite directions. Since the length of the air column is some multiple of half wavelengths, the two oppositely traveling waves produce standing waves.

The standing waves are characterized by alternate points of maximum and minimum disturbance called respectively nodes and antinodes. These waves become visible after the dust inside the tube agitated in the antinodes and remain still at the nodes.

Since the wave loops are visible, we can easily measure its wavelength using the meter stick. As we all know, all waves segments are in the same sizes. Thus, we can measure the length of one segment by measuring two or more segment and get its average.

In that method, error can be minimized because the uncertainty in measuring will be distributed evenly along the whole segments. For example, if we have a constant uncertainty of 0.3 cm. That is for increasing no. of segments being measured, we can see a trend of the decrease in uncertainty produced.

Figure 4.Figure 4. Measuring the Measuring the temperature of the roomtemperature of the room

0 1 2 3 4 5 6 7 80

0.050.1

0.150.2

0.250.3

0.35

No. of segment measured

ARel

ative

unc

erta

inty

in

leng

th, c

m

Graph 2. Graph on relative uncertainty produced versus the no of segment measured

In our data, we use 3 segments being averaged. For three segments, we measured the length of 27.5 cm, which is 9.2 cm in average per segment. Since one segment is half of the length of one wavelength (two successive wave loops), then, we can easily solve the frequency of the sound in air by determining the velocity of sound in air. It is done by applying its linear relationship with the temperature.

As we all know, the apparatus is set in such a way that the sound will be propagated on its fundamental mode, where H = 1. After completing all the necessary data, we can use equation 2 for the numerical value of the frequency. In the experiment, the sound wave produced in rod and in air is in resonance, so their corresponding frequencies are the same. For that, we can equate each frequency and relate the velocity of sound in rod in terms of the frequency of the sound in air and the wavelength of the sound wave in the metal rod. The metal rod is clamped at the middle, so node will occur at that point while antinodes will occur at both ends, thus producing one segment of a wave. Again, one segment is actually half of one wavelength. Since the relationship is completed, we can now solve for the velocity of the sound in rod. Using our data, we had obtained the value of the velocity. It is shown below.

Table 3. VELOCITY OF SOUND IN ROD USING EQUATION 3

velocity of sound in the rod Vr equation 3

3460.092 m/s

We can also calculate for the value of the velocity of sound in metal rod using the elasticity property of the metal used through the used of equation 9. The values for constants are not that accurate because of few digits of significant figures. After

obtaining the value, it is then compared with the value obtained earlier. Since the constants are not that accurate, we can introduce an uncertainty value of about 3% of the original value. By doing that, we can observe that the velocity we had obtained in the first equation is still within the range.

Table 2. PERCENTAGE ERROR COMPARED WITH THE VALUE OBTAINED USING EQN 4

velocity of sound in the rod Vr equation 4

3413.944 m/s

Percentage error 1.35177 %

The error in this part is quite small. By considering the nature of the apparatus, we can prove that uncertainty measurement is a small factor in the accumulation of the error; Since the Kundt’s tube is a special apparatus made just for this type of experiment, there is a great chance that the data obtained has a great accuracy when we are using it. The possibility of having an uncertainty is also diminished since the over all set up is already prepared by the professional laboratory assistants. Probable source of error for the determination of velocity of the sound in metal rod are as follows:

[1] The tube is not horizontally placed in the table[2] The tube is somehow open at the end where it should be closed[3] The uncertainty in measuring the length of the segments by around ±0.15cm

Using the theoretical value, using the textbook (more accurate), we can also calculate for the amount of error produced after committing uncertainties in measuring. The table is shown above.

It is glad to see that we got a smaller percentage error upon comparing it with theoretical value in the textbook. It only means that our obtained value is somewhat closer with the more accurate basis.

Table 4. PERCENTAGE ERROR COMPARED WITH THE VALUE OBTAINED USING THE THEORETICAL VALUE IN THE TEXTBOOK

velocity of sound in the rod Vr textbook'

3475 m/s

Percentage error 0.42899 %

As we observed, the main passage of the longitudinal wave from the rod to the tube is through the disk. We also notice that when the dust is agitated inside, the antinode starts from the disk (and not a node) and a node is seen at the end of the tube. It signifies that the tube is a close type air column. It is open at the side where the disk is located while closed at the other end.

By making an analysis on temperature, if the experiment is done in a hot summer day, the velocity of sound in air will be affected, so as to the length of the segments formed. Since

va∝ La

Then, we can say that when the temperature goes higher, the length of the segments formed will be much longer. More variation can be discussed in this report. This allows us to study the flexibility of the experimentation and the apparatus itself, if still; the same result will be obtained.

If the rod were clamped not at the center, for instance, two clamps are placed at each one-fourth distance from the end, we can observe that the length of each segments will be shorter by half. This can be proven by considering the frequency of the sound in rod.

f m=V mLm

=va2La

At the same conditions given in the experiment (except for the clamping), we could say that,

La=vaLm2vm

La=12Laoriginally

On the other hand, if the stopper is removed from the other end of the tube, open air column type of sound wave behavior will be produced. So, we can observe that antinodes are on both ends. Also, if the same conditions is applied, then,

vm2Lm

=f a=va4 La

La=2 Laoriginally

So, we can also notice that the length of each segment formed inside the tube is longer by a factor of 2.

When the metal rod, is cut into half, and use only one half of it, (still the clamp is placed at its original position), and also the temperature inside the tube is maintained at 0oC with only hydrogen inside (vsound=1270m/s), then we may observe that the waves are longer than before by a certain factor.

vm

4 (0.5 L¿¿m)=1270

ms

2 La¿

La=1270Lmvm

= 1270vabefore

Laoriginally

If we use the velocity of sound in rod obtained from the experiment which is 3460 m/s (remember that the velocity of sound in a particular metal is generally constant), then we can say that

La=0.36Laoriginally

In our observations, we saw that the dust is accumulating on the node parts. It is because these parts are still and does not move at all. When agitation happens, the movement of the dust is not only sideward but also forward and backward. Thus, the dust may be crowded on those points, especially when the apparatus used is not horizontal. The movement of the dust particle is more on a forward or a backward. As a result, we can’t be able to measure the length of each segment accurately. Good thing that we may diminish this effect by making the apparatus horizontal or by removing excess dust particles.

In analyzing the relationship of elasticity of solid material and its density with the velocity of sound in a metal rod, we can use the general formula, where the velocity of sound in metal is directly proportional to the square root of Y (Young’s Modulus) and inversely proportional to the square root of its density.

vm=√YρFrom that, we can say that when the material is denser, the speed of the sound as it flows through the material is slower, while on the other

hand, when the Young’s modulus constant of the material is greater, then its velocity is faster.

The uncertainties in the result are from the measuring of the segment and the measuring of the length of the rod. If we put an uncertainty value for the ∆ La=±0.15cm and for the length of

the rod of ∆ Lm=±0.05cm, (but not much error on temperature so its effect is negligible)

vm=F (va , Lm , La)

vm=va LmLa

dvm=va [( d vmd Lm )Lad Lm+( d vmd La )Lmd La]∆ vm¿va[|∆ LmLa |+|−Lm∆ LaLa

2 |]¿(347.9ms )[|0.05cm9.2cm |+|− (91.5cm ) (0.15 cm )

(9.2cm )2 |]∆ vm=±58.31m /s

In our theoretical value of 3475 m/s, the obtained value of vm=3460.09m /s±58.31m /s is still within its range.CONCLUSION

This report has discussed the relationship of velocity of sound in both gas (air) and solid (metal rod) through manipulation of certain condition to produce resonance. The goals of this lab were to obtain the velocity of sound in metal rod as well as the velocity of sound in the tube with the application of resonance. Those objectives were met by obeying the conditions needed such as clamping the rod at its midpoint, closing the one end of the glass tube while the other end is free, and by making the disk in the rod not to be in contact with the glass tube. By that means, the relationship derived in the equation given in the laboratory manual is followed.

By constantly applying stroke on the rod, friction is produced. So, longitudinal wave is produced. It goes down the tube from the rod at equal

frequency, hence making a resonance, and produces sound.

In keeping track of the length of the visible wave pattern produced after the dust is being agitated inside the tube, the velocity of the sound in rod is obtained.

This lab introduces us an important topic on velocity of sound at a certain medium. Once the relationship of velocity, wavelength and frequency are established, we can be able to relate two different velocities at different medium. It is done by applying the principle of resonance where frequencies of each corresponding longitudinal wave are equal.

This lab experiment also tells us the changes in velocity of sound in solid after varying the position of the node (or the clamped part). When the position of the clamp divides the rod in a certain segment, as the division increases, its velocity changes in an inverse proportional manner.

The velocity of sound in an air column can be described as either open or close type. Its significant difference is the behavior of the wave produced, particularly on the position of the nodes and antinodes. For a close type, which is applied in the experiment, an antinode is seen on the open end while a node is seen at the close end. The velocity of sound in an air column of this type is proportional to the frequency and twice the length of each segment.

Besides from the application of the resonance, velocity of sound in rod can also be evaluated by considering the elasticity and density of the material where the sound wave travels. The velocity is directly proportional to the Young’s modulus of elasticity and inversely proportional to the density of the material being used.

Aside from the velocity of sound in metal, we learned that it is an important factor to consider the velocity of sound in air if we are to use the resonance principle in the Kundt’s tube. The said velocity is somewhat linearly dependent with temperature. So, for every rise in temperature per Celsius degree, the velocity of the sound is increased by 0.6 m/s.

The finding in the experiment for the value of velocity of sound in metal rod using the principle of resonance is 3460 m/s. When the uncertainties in measurements are considered, the velocity will

expand to a range of3460.09m /s±58.31m /s. Using the elasticity property of the material, brass, the velocity is 3414 m/s while using the book as a reference; it is 3475 m/s. Upon comparison with the experimental value, the error differences are 1.35% and 0.429%, respectively. However, the theoretical values lie within the range of uncertainty.

REFERENCES

[1] Young, H., Freedman, R., University Physics with Modern Physics, 11th Edition, 2004

[2] Bernard, C.H., Laboratory Experiment in College Physics, 7th Edition, 1995

[3] Wilson, Jerry D., Physics Laboratory Experiments, 6th edition, 2003

[4] http://hyperphysics.phy-astr.gsu.edu/hbase/ class/phscilab/kundt2.html

[5] http://www.nikhef.nl/~h73/kn1c/praktikum/phywe/LEP/Experim/1_5_06.pdf

[6] http://dev.physicslab.org/Document.aspx?doctype=2&filename=WavesSound_SpeedSoundCopper.xml

ACKNOWLEDGEMENT

First of all, I am heartily thankful to my group mates, Ninang Redden, Ms. Gimena, Ms. Ang and Mr. Deduyo, whose encouragement, guidance and support from the initial to the final level of the experiment enabled me to finish the work in the lab as well as develop my understanding of the topic. They are those who were doing the experiment with me and sharing their ideas. They were helpful that when we combined and discussed together our own respective thoughts, we had this experiment done successfully.

Secondly, I would like express my gratitude for giving me the strength and health to do this report until it done. Not forgotten to my family for providing everything, such as money, to buy anything that are related to this project work and their advise, which is the most needed for this project. They also provide me the internet, books, computers and all that as my source to complete this project. They supported me and encouraged me to complete this task so that I will not procrastinate in doing it.

Then I would like to thank my Professor, Mr. De Leon for guiding me and my friends throughout the experiment. We had some difficulties in doing this task, but he taught us patiently until we knew what to do. He tried and tried to teach us until we understand what we supposed to do in order to have a highly accurate data.

Lastly, I offer my regards and blessings to God who guides me to have an idea what to do during the completion of the report.

FREE SPACE

Our professor in Physics 3 Laboratory wonders why I am still quantifying the error in the experiment. He wonder why do I have still a time doing it, given that we are in Mapua, which has a busy and cramming environment.

The simple answer for that is because I have a simple reference. In this book, guidelines and rules on propagation of errors is instructed. This book is ideal on making a report, especially for quantitative analysis (while in chemistry, it’s usually qualitative). Without further a do, this is how it works.

Assume the quantity C depends on the measured quantities X and Y in some functional form expressed as:

C=F (X ,Y )

The total derivative of C is given by:

dC=( ∂C∂ X )Y

dX+( ∂C∂Y )X

dY

Assuming that errors are small so that we can neglect terms of higher order than the first, allows the use of errors in place of the differentials (usually for infinitesimal changes). Thus we have:

∆C= ∂C∂ X

∆ X+ ∂C∂Y

∆Y

∆X and ∆Y are intermediate errors and thus the largest possible error is required. This value is given by:

∆C=|∂C∂ X ∆ X|+|∂C∂Y ∆Y|Thus, the error propagated due to uncertainty in measuring can now be a component of the desired value. It is also applicable to use a more simple equation by assuming that each variable propagates error at the same magnitude.

∆ R=R√(∆ XX )2

+(∆YY )2

And that is simple it is!!!