Lab Exercise 0: Water Waves
Contents
0-6 P RE - LAB A SSIGNMENT . . . . . . . .
0-7 F ORMAT AND COMMENTS . . . . . . .
0-8 I NTRODUCTION . . . . . . . . . . . . .
0-8.1 Traveling Waves . . . . . . . . .
0-8.2 Standing Waves . . . . . . . . . .
0-8.3 Spreading Factor: Circular Waves
0-9 E QUIPMENT . . . . . . . . . . . . . . .
0-10 E XPERIMENT . . . . . . . . . . . . . .
0-10.1 Traveling Waves . . . . . . . . .
0-10.2 Standing Waves . . . . . . . . . .
0-11 L AB WRITE - UP . . . . . . . . . . . . .
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. xix
Objective
The objective of this lab is to examine experimentally the characteristics
of waves through the use of water waves.
General concepts to be covered:
• Amplitude
• Circular waves
• Frequency
• Linear waves
• Phase
• Standing waves
• Traveling waves
• Wavelength
vi
0-6
PRE-LAB ASSIGNMENT
0-6
P RE -L AB A SSIGNMENT
0-6.1 Read Section 1-3 of the text.
0-6.2 To be entered into your lab notebook prior to coming to lab:
Summarize the experimental procedure (1 paragraph per section) of:
(a) Section 0-10.1: Traveling Waves
(b) Section 0-10.2: Standing Waves
0-7
F ORMAT A ND C OMMENTS
The format of this document has been chosen to make the lab as easy to use as possible.
Each experiment is subdivided into 5 sections. These sections are:
• Setup
This section contains the instructions and required information for setting up the
experiment.
• Procedure
This section gives detailed instructions on what steps to perform and how to
accomplish them.
• Measured Data
This section gives a summary of what data should have been collected. In addition, a
layout for reporting the data is also given.
• Analysis
This section contains the required steps to analyze the collected data.
• Questions
This final section asks a few questions that test your understanding of the general
concepts covered in the experiment.
This laboratory manual contains four appendices which describe the instruments and
components used in this and forthcoming labs. These appendices are:
• Appendix A: HP 8712C Network Analyzer
This appendix includes a summary of all of the instrument procedures used in this
and other labs. In addition, a picture of the front panel of the instrument is included
with the various types of keys and inputs/outputs annotated on the picture for easy
reference.
• Appendix B: HP 54645A 100 MHz Digital Oscilloscope
This appendix includes information on the operation of the oscilloscope used in this
and other labs.
• Appendix C: HP 8648B RF Signal Source
This appendix includes information on the operation of the signal source used in this
and other labs.
vii
viii
LAB EXERCISE 0: WATER WAVES
• Appendix D: Adapters, Cables, Connectors, and Components
This appendix is a pictorial reference guide for all of the adapters, cables, connectors,
and components used in Lab Exercises 1-5. When you are told in a setup or procedure
to use a certain component and you are unsure of what it is, you can use the picture
in the appendix to identify that component.
When you perform a measurement with either the oscilloscope or the network analyzer,
you measure the desired signal plus some additive noise. If the noise level is appreciable in
comparison with the signal, it will make the measurement jumpy. When you are asked to
record the measured value of a signal that is jumpy, watch the value for a few seconds and
then record the “average” value. For example, if the measured value over an interval of a
few seconds is:
2.14, 2.89, 1.9, 3.1, 2.4,
record the measured value to be around 2.5. The recorded value should have one more
decimal place than the most stable part of the reading. For example, if the measured value
is consistent in the first decimal place but is jumping around in the second decimal place,
record the signal to only two decimal places.
0-8
I NTRODUCTION
Waves carry energy from one place to another. This property makes waves useful not only
as a means of delivering power over long distances, but also as a means of transporting
information over long distances. The 60 Hz power grid delivers power to homes and
industry which can be easily transformed by a resistive element into heat for ovens or
furnaces, or by an electric motor into mechanical power which drives fans, refrigerators,
etc. Examples of waves carrying information include electrical signals on telephone wires
or sound waves in air carrying music.
While waves of an electromagnetic nature are perhaps most interesting due to their
ability to be easily transformed into other forms of energy (or information), all waves exhibit
the same basic characteristics. Waves in water will be studied in this lab because most
electromagnetic waves are not directly observable with the senses humans are endowed
with (sight is an exception, but optical wavelengths are a bit too small to directly observe),
and they tend to travel at exceedingly fast speeds (which, except for trying to watch them
move, is a very good thing!).
In this lab, you will familiarize yourself with the basic characteristics of waves,
including the relationships between the frequency, wavelength and velocity for traveling
waves, and the generation of standing waves when two traveling waves interfere.
0-8.1 Traveling Waves
Any single wave traveling in one direction is a traveling wave. The traveling wave may
be composed of many frequencies, or of a single frequency component. If composed of
many frequency components, the rules of superposition apply and the wave shape seen is
the result of constructive interference and destructive interference. In dispersionless media,
a multi-frequency wave maintains its shape as it moves along. For example, vibrations on a
wire strung along the x-axis may appear like the waves in Fig. 0-1.
0-8
INTRODUCTION
ix
5.
4.
3.
t = 0 ms t = 2 ms t = 4 ms t = 6 ms
amplitude (cm)
2.
1.
0.
-1.
-2.
λ
-3.
-4.
-5.
5.
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4.
3.
t = 6 ms t = 4 ms t = 2 ms t = 0 ms
amplitude (cm)
2.
1.
0.
-1.
-2.
-3.
-4.
-5.
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Distance x (m)
Figure 0-1: A pair of traveling waves. Increasing time is marked by increasing darkness (as
if we were looking at an oscilloscope screen with persistence). The upper figure represents
a forward traveling wave and the lower figure represents a backward traveling wave.
The text explains how to determine the wave parameters (direction, wavelength, period,
frequency, reference phase, etc.) from a wave expression, but how do we determine the
parameters from the wave itself, or a figure of a wave? This is important because in this lab
and ones to follow, we will be looking at real waves, not expressions for them.
First, a wave must be recognized as a cosine function, and secondly, if there is a change
in amplitude due to attenuation. If it doesn’t look cosinusoidal at all, then the wave must
be broken down into Fourier components, but we’ll leave that exercise to a later class.
These waves certainly look close enough to cosinusoidal to be considered cosinusoidal
waves. If there is attenuation, the wave is traveling in the direction of decreasing amplitude,
otherwise, some other means must be used to determine the direction.
From the figure, there is no attenuation, but the time progression clearly reveals that the
upper figure represents a wave traveling in the +x direction and the lower figure represents
x
LAB EXERCISE 0: WATER WAVES
a wave traveling in the −x direction. Thus the form of the wave expressions is
¶
µ µ
¶
t
x
upper figure: ff (x,t) = Af cos 2π
−
+ φf
Tf λf
µ µ
¶
¶
t
x
lower figure: fb (x,t) = Ab cos 2π
+
+ φb
Tb λb
(0.1)
(0.2)
where the subscript “f” refers to a forward wave and “b” refers to a backward wave.
Wavelength λ
The easiest parameter to extract is the wavelength, λ. This is done by measuring one cycle of
the wave, from crest to crest or trough to trough or some other phase point to the same phase
point on the next cycle. For the forward wave in the figure, the t = 6 ms “snapshot” has
one trough at x1 = 2.0 m and the next one occurs at x2 = 7.0 m. Therefore, the wavelength
is λf = |x2 − x1 | = 5.0 m. It turns out waves in the lower figure have the same wavelength:
λb = λf .
Phase Velocity u
The next easiest parameter to get is the velocity, u. The forward wave, at t2 = 6 ms, has a
crest at x20 = 4.5 m. But at t1 = 2 ms, this same crest appears to have been at x10 = 2.0 m.
Therefore, the velocity is uf = (x20 − x10 )/(t2 −t1 ) = 2.5 m/4 ms = 625 m/s. Similar analysis
reveals that the backward wave has a velocity of ub = −625 m/s. Strictly speaking, the
backward velocity is negative, but giving the absolute value of the velocity (known as the
speed) is ok, because we are separately specifying the wave direction.
Frequency f
With the velocity and wavelength known, the frequency can be derived from u = f λ. For
both the forward and backward waves, the frequency is f = 125 Hz.
Attenuation Coefficient α and Amplitude A
The attenuation coefficient, α, and amplitude at the origin, A, can be measured by finding
the height and location of a pair of wave crests or troughs. For example, the forward
00
traveling wave at t = 2.0 ms has a wave crest with a height A1 = Ae−αx1 = 2.2 cm at
location x100 = 2 m. The next crest of the same wave occurs at x200 = 7.0 m with a height
00
00
00
of A2 = Ae−αx2 = 2.2 cm. Taking the ratio of these two crests, we get A2 /A1 = e−α(x2 −x1 ) .
Solving this for α, we have
α=
− ln (A2 /A1 )
x200 − x100
− ln (2.2 cm/2.2 cm)
7.0 m − 2.0 m
= 0 Np/m
=
Of course, we already decided the attenuation was zero from the fact the wave was not
decaying, so this is a happy confirmation. The amplitude at the origin can then be solved
0-8
INTRODUCTION
xi
from one or the other of the two crest equations: A = A1 eαx1 or A = A2 eαx2 . In this case,
A = 2.2 cm. (If troughs are used instead of crests, use −A1 and −A2 in place of A1 and A2 .
Also, if the wave is traveling backwards, then the sign in front of the expression for α is +,
not −).
00
00
Reference Phase φ
The reference phase is trickiest of all to get. Again, picking a crest or a trough is the easiest
place to start, as the phase at that location is known. Crests correspond to the phase being
0◦ : φ(x,t) = 0, and troughs correspond to the phase being 180◦ : φ(x,t) = π. For example,
the backward traveling wave has a trough at x = 3.0 m and t = 6 ms. The phase at this
trough is therefore
φ(x,t) = 2π f t + 2πx/λ + φb = π
which can be solved for the reference phase:
φb = π − 2π(125 Hz)(6 ms) − 2π(3.0 m)/(5.0 m)
= −1.7π = −306◦
The reference phase is usually reported as a value between −180◦ and +180◦ . Therefore,
φb = 54◦ , which is equivalent to −306◦ .
0-8.2
Standing Waves
Two traveling waves traveling in opposite directions create a standing wave. The standing
wave is caused by the constructive and destructive interference of the two waves as they pass
by each other. The standing wave is stationary in position, if the two traveling waves travel
at the same speed. But the interference pattern of the two waves does not have a fixed shape;
individual points of the superposition vary over time at the same frequency as that of the
traveling waves that compose it. Some places are always experiencing some constructive
interference, and at these points the superposition of the two traveling waves varies with
a larger amplitude than either of the two traveling waves individually. At other places,
destructive interference dominates, and at these points the superposition of the two traveling
waves varies with a smaller amplitude than either of the two traveling waves individually.
The envelope, that is, the maximum extent of the interference of all the waves present,
which varies from location to location depending on the nature of the interference there,
is the standing wave pattern. For a forward traveling wave by itself, the upper and lower
envelopes are constants. The same is true for a backward traveling wave by itself. Figure 0-1
shows the upper and lower envelopes for the traveling waves by themselves.
Mathematically, the standing wave for two traveling waves can be constructed from the
sum of the two traveling waves. For the relatively simple case of the forward and backward
traveling waves both having the same amplitude A, ie. A = Ab = Af , we get
fs (x,t) = ff (x,t) + fb (x,t)
³ ³t
´
³ ³t
´
x´
x´
= A cos 2π
−
+ φf + A cos 2π
+
+ φb
T λ
T λ
(0.3)
(0.4)
xii
LAB EXERCISE 0: WATER WAVES
From a trig identity, namely
cos α + cos β = 2 cos 12 (α − β) cos 12 (α + β)
(0.5)
¡
¢
with α ¡= 2π ¢Tt + λx + φb representing the phase of the backward traveling wave and
β = 2π Tt − λx + φf representing the phase of the forward traveling wave, the expression
can rewritten as
³ x
´
³ t
´
fs (x,t) = 2A cos 2π + 12 (φb − φf ) cos 2π + 12 (φb + φf )
(0.6)
λ
T
In this expression, the last cosine is the only part that varies with time, so we can view the
rest as an amplitude modulating the wave. This amplitude varies with space and forms an
envelope within which the standing wave oscillates. In other words, we can write Eq. 0.6 as
³ t
´
fs (x,t) = fe (x) cos 2π + 12 (φb + φf )
(0.7)
T
with
´
³ x
fe (x) = 2A cos 2π + 21 (φb − φf )
λ
(0.8)
The amplitude of the wave is bounded by the upper envelope, given by +| fe (x)|, and the
lower envelope, given by −| fe (x)|. For the waves shown in Fig. 0-1, their sum is shown in
Fig. 0-2 for various values of t. Also shown is the envelope which has a spatial pattern that
repeats itself twice within one wavelength λ.
λ
Upp
er E
nvel
ope
5.
4.
3.
2.
t = 3 ms
t = 4 ms
1.
0.
-1.
t = 5 ms
-2.
-3.
-4.
-5.
ope
nvel
er E
Low
amplitude (cm)
7.
6.
t = 6 ms
-6.
-7.
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Distance x (m)
Figure 0-2: A standing wave. The solid lines represent the superposition of the two traveling
waves of the previous figure. Increasing time is marked by increasing darkness (as if
we were looking at an oscilloscope screen with persistence). The upper envelope of the
superpositions is shown as the long dashed line and the lower envelope is the short dashed
line. The envelopes have a spatial periodicity of one half of a wavelength.
0-8
INTRODUCTION
We can show this mathematically from the trigonometric identity cos γ = − cos (γ + π),
which implies |cos γ| = |cos (γ + π)|, which in turn demonstrates that |cos γ| has a
fundamental period of 1π, not 2π. Thus, the spatial period of a standing wave is half the
wavelength, the wavelength being the spatial period of the traveling waves that compose
the standing wave.
Often, the rate of oscillation of a wave is much faster than can be directly observed, but
the envelope, which is stationary, can be easily observed. The wavelength of a very fast
traveling wave can be found by simply doubling the spatial periodicity of a standing wave
created by the traveling wave.
The derivation of the standing wave equation is more complicated if Af 6= Ab , but is not
so bad if phasors are used. This is done in Section 2-5.2 of the text.
0-8.3 Spreading Factor: Circular Waves
For waves on a wire, Eqs. (0.1) and (0.2) describe a one-dimensional wave. But these same
equations can describe multi-dimensional waves if we interpret all points on the plane x =
constant to have the same phase. These waves are known as plane waves because the phase
fronts (all points with the same phase) lie on a plane. Plane waves do not spread out. That
is, different points on a phase front do not become farther apart as the wave travels along;
they travel parallel to each other. This means that, in the absence of attenuation, the energy
carried by the wave does not spread out and the amplitude of the wave, which is proportional
to the square root of the wave energy, remains constant.
This is in contrast to two- or three-dimensional waves which have a point source. For
two-dimensional waves, like those on the surface of water, the phase fronts are circles if
the source of the wave is at a single point. The energy carried by the wave spreads out in
circles of ever increasing radius. In the absence of attenuation, the total energy in the wave
at different radii is the same. Therefore, the amplitudes must decrease as the square root
of the radius from the source, to compensate for the increasing circumference of the larger
circles. Thus, the form for a circular wave, such as a water wave of height h(r,t), with a
source at the origin is
³ ³t
´
r´
e−αr
hcirc (r,t) = Acirc √ cos 2π
−
+ φcirc
T λ
r
where the attenuation is included. The only difference between this expression and that
for the forward propagating
plane wave (Eq. 0.1) is that r is used instead of x, and the
√
spreading factor of 1/ r is included in the amplitude. The effect of this spreading factor
on the amplitude is shown in Fig. 0-3.
For circular waves, the expression for the amplitude at the origin approaches infinity.
There is no such thing as a true point sources, so the expressions must be considered
approximate. They are valid as long as the point of observation is “far” from the source. A
good rule of thumb to know if a point of observation is far enough from a point source is
if the distance to the source is larger than 2D2 /λ, where D is the largest dimension of the
source and λ is the wavelength.
xiii
xiv
LAB EXERCISE 0: WATER WAVES
Figure√0-3: A two-dimensional circular wave. The amplitude of the circular wave decreases
as 1/ r to make up for the larger circumference of the wave. The curve showing the
amplitude is not shown near the source at the center because the wave expression is not
valid there.
0-9
E QUIPMENT
Item
Advanced ripple tank system
Halogen light
Mechanical strobe
Plane wave generator
Power supply
0-10
E XPERIMENT
0-10.1
Traveling Waves
Part #
Pasco WA-9770
WA-9776
WA-9772
S52078-3A
HP E3620A
In this section of the lab, you will measure the characteristics of traveling waves in water.
The relationship
f λ = up
can be confirmed because all three variables can be measured. The velocity of the water
waves can be directly measured by timing the propagation of a wave crest over a set
distance.
With the use of a strobe light whose flash period matches that of the waves, the waves
can be made to appear to stand still. This is because our eyes observe the continuously
moving waves only part of the time: those moments when the wave crests (or troughs) have
moved to the locations of the previous wave crest (or trough) at the last flash of the strobe.
Since each cycle of the wave is identical to any other, the entire wave appears to stand still,
even though it has moved by an entire wavelength over the time the strobe light was out.
Because the waves appear to stand still, the wavelength can be measured directly. And
because the period of the strobe matches the period of the waves, the frequency readout on
the strobe reveals the frequency of the waves in the water.
Water waves can be made to propagate as linear waves or as circular waves. Part of this
experiment explores the effects of the shape of the wavefront on its characteristics.
Setup
This experiment uses the ripple tank, halogen light, mechanical strobe, plane wave
generator, and power supply.
0-10 EXPERIMENT
xv
Setup the experiment as shown in Fig. 0-4.
• Set up the ripple tank with the ripple generator and the strobe configuration.
• Fill the ripple tank with water to a depth of 5 to 10 mm.
• Place the thin, clear, plastic ruler in the ripple tank (make sure that it rests against the
glass and does not float).
Halogen Light Strobe
Motor
Connector
Ripple
Generator
Ripple Tank
Figure 0-4: Ripple Tank setup for Section 0-10.1.
Setup hints: The strobe will work best with the lights turned off and the readings made
on blank sheets of paper under the tank. You might want to make markings on the paper and
then turn the wave generator and strobe off to measure the distance between the markings
on the paper. Don’t measure the distances between the markings directly: Use the image
on the paper of the transparent ruler that sits in the tank.
Procedure
Low Frequency Case
1. Set the strobe frequency to a value between 9 Hz and 11 Hz. Record the frequency.
2. Turn on the halogen light and adjust the power supply voltage until the images of the
water waves appear to stand still on the floor. Record the power supply voltage.
3. Measure the wavelength of the water waves. Use whatever technique you feel
will give you the most accurate measurement, but do estimate the error in your
measurement (see Appendix E). Record your measurements, explain the technique
you used and why you chose to use it.
xvi
LAB EXERCISE 0: WATER WAVES
4. Turn off the strobe, but not the light. Rotate the strobe wheel such that the light
shines through the window onto the tank. Time a wave maximum as it travels from
the source at one end of the tank to the other end. Record this time. Don’t worry
about being too precise: you can make up for the imprecision by repetition. Repeat
this time measurement 9 times. Also, record the distance over which you timed the
wave, so you can convert these time measurements into a wave velocity.
High Frequency Case
5. Change the strobe frequency to a value between 15 Hz and 17 Hz. Record the
frequency.
6. Turn on the halogen light and adjust the power supply voltage until the images of the
water waves appear to stand still on the floor. Record the power supply voltage.
7. Repeat steps 3 and 4 for the high frequency case.
8. Turn the power supply off. Attach a circular wave driver to one end of the plane wave
driver. Center the circular wave driver in the middle of the wave tank such that only
the circular wave driver makes contact with the water. Turn on the power supply and
adjust the voltage until the the water waves appear to stand still. Record the power
supply voltage.
9. Measure the wavelength of the circular waves near the source and far from the source.
Record the propagation distance and time as done in step 4. Record the wavelength
as measured far from the source.
Measured Data
Copy the following chart into your lab book and fill in the measured data. If you are
missing any data, please repeat the necessary parts of this experiment before proceeding to
the analysis section.
Wavefront
Frequency setting
Strobe frequency
Power supply voltage
Water wavelength
λ uncertainty
Propagation Distance d
d uncertainty
Propagation Time 1
Propagation Time 2
Propagation Time 3
Propagation Time 4
Propagation Time 5
Propagation Time 6
Propagation Time 7
Propagation Time 8
Propagation Time 9
Propagation Time 10
Linear
Low
Linear
High
Circular
High
0-10 EXPERIMENT
Analysis
1. For each case calculate the mean and standard deviation of the wave propagation
time. From these values calculate the mean wave velocity and the error around the
mean.
2. Using up = f λ, calculate the wave velocity from your measurements of λ and f , and
determine the error for this calculation.
Questions
1. Compare the velocities of the linear waves at the two frequencies. Do they agree to
within the uncertainty of your measurement?
2. Which method (timing propagation, or measuring wavelength) for finding the
velocity is more accurate? Why?
3. Compare the velocities of the circular waves to the linear waves at the same
frequency.
4. Compare the wavelengths of the circular waves to the linear waves at the same
frequency.
5. Based on your observations, which parameters ( f ,u p ,λ) remain constant in a material?
Explain any differences in the results.
0-10.2
Standing Waves
In this section of the lab, you will measure the characteristics of standing waves in water.
When a wave launched by the plane wave generator reflects from the aluminum barrier,
the reflected wave interferes with forward traveling waves created by the generator. The
result of this interference is a standing wave. By measuring the spacing of the nulls and/or
maximums, you can determine the wavelength.
Setup
This experiment uses the ripple tank, halogen light, mechanical strobe, plane wave
generator, aluminum barrier, and power supply.
Setup the experiment:
• Remove the circular wave driver from the plane wave generator. Adjust the blade of
the plane wave generator so it is level with respect to the surface of the water.
• Place the plane wave generator at the edge of the ripple tank.
Procedure
Low Frequency Case
1. Set the power supply voltage and the strobe frequency to the values recorded for the
low frequency case in Sec. 1-5.1.
xvii
xviii
LAB EXERCISE 0: WATER WAVES
2. Carefully adjust the power supply voltage until the traveling waves appear to stand
still.
3. Unplug the strobe and rotate the strobe disk to the clear position by hand (all of the
light should be shining through the strobe disk.)
4. Place the aluminum barrier in the tank 20 cm away from the plane wave generator.
Replace the paper underneath the ripple tank with clean sheets. Mark the location of
two consecutive minima and maxima. Make sure that the minima and maxima are
located adjacent to each other.
5. Measure and record the distance between the first minima and first maxima, second
minima and second maxima, first and second minima, and first and second maxima.
High Frequency Case
6. Remove the aluminum barrier from the ripple tank. Set the power supply voltage and
the strobe frequency to the values recorded for the linear high frequency case in Sec.
1-5.1.
7. Carefully adjust the power supply voltage until the traveling waves appear to stand
still.
8. Unplug the strobe, replace the aluminum barrier in the tank 20 cm away from the
plane wave actuator.
9. Replace the paper underneath the ripple tank with clean sheets. Mark the location of
two consecutive minima and maxima. Make sure that the minima and maxima are
located adjacent to each other.
10. Measure and record the distance between the first minima and first maxima, second
minima and second maxima, first and second minima, and first and second maxima.
Measured Data
Copy the following chart into your lab book and fill in the measured data. If you are
missing any data, please repeat the necessary parts of this experiment before proceeding to
the analysis section.
Frequency setting
Distance between first minima and first maxima
Distance between second minima and second maxima
Distance between first and second minima
Distance between first and second maxima
Low
High
Analysis
1. Using the distance between the first and second minima for the low frequency case,
compute the wavelength. Record the result as λl1 .
2. Using the distance between the first and second maxima for the low frequency case,
compute the wavelength. Record the result as λl2 .
3. Using the distance between the first minima and first maxima for the low frequency
case, compute the wavelength. Record the result as λl3 . Repeat for the second minima
and maxima, recording the result as λl4 .
4. Repeat steps 1 to 3 for the high frequency case, recording the results as λhn , where n
is either 1, 2, 3 or 4.
5. For the low frequency case, using the four computed wavelengths, determine the
mean wavelength. Use the standard deviation of the measurements to estimate the
uncertainty. Repeat for the high frequency case.
6. Using the wave velocity computed in the first experiment, determine the frequencies
of the waves for both the high and low frequency case.
Questions
1. Compare the computed frequencies for the low and high frequency cases to the
expected frequencies (from the first experiment). Are the frequencies close to the
expected?
2. Compare the spatial periodicity of the standing wave to the wavelength of the linear
propagating water wave at the same frequency.
3. Why didn’t you have to use the strobe light in this experiment?
0-11
L AB W RITE - UP
For each section of the lab, include the following items in your write-up:
1. Overview of the procedure and analysis.
2. Measured data where asked for.
3. Calculations (show your work!).
4. Any tables, plots and printouts.
5. Comparisons and comments on results.
6. A summary paragraph describing what you learned from this lab.
xix
#
Ã
no
bonus
points
for
typed
reports, but
deductions
will
be
made
for
illegibility
"
!
Chuck Divin
EECS 230-12 (Tues. 12:00)
Station. #3
GSI: Tsiawei Wu
Group Members:
Joe Smith
Jane Doe
Lab 0: Water Waves
Overview
In this lab we used a strobe light and a ripple tank to measure the frequency, wavelength,
and speed of water waves. A simple shadow-casting system was used to project the water
waves onto the floor below the tank. Velocity measurements were timed using a stopwatch
and the shadow system. We also measure the dependence of the wavelength on the shape of
the wave (planar vs. circular). We compared the experimental results with the theoretical
prediction, and found good agreement. The in-lab data and pre-lab are attached to the back
of this writeup.
Lab Part: 0-5.1
Data
Strobe Frequency [Hz]
∆f
Water Wavelength [cm]
∆λ
Supply Voltage [V]
Propagation Distance [cm]
∆d
Propagation time [s]
Propagation time t2
Propagation time t3
Propagation time t4
Propagation time t5
Propagation time t6
Propagation time t7
Propagation time t8
Propagation time t9
Propagation time t10
Linear (Low)
10.2
0.1
2.3
0.1
1.87
16.5
0.2
0.85
0.88
0.80
0.87
0.78
0.87
0.83
0.87
0.88
0.87
Linear (high)
15.4
0.1
1.5
0.1
2.23
16.5
0.2
0.81
0.91
0.86
0.77
0.69
0.90
0.89
0.82
0.87
0.78
Circular
15.4
0.1
1.5
0.1
2.24
12.75
0.2
0.64
0.53
0.58
0.66
0.64
0.62
0.66
0.71
0.69
0.82
Analysis
1. For each case, the mean and standard deviation were calculated using the formulas
listed in the Error Analysis Appendix. The results are listed below. The velocity and
velocity error formulas are:
xx
#
2. For each case, the velocity was calculated using u p = λ f , and the error was found
using the formulas listed in the Error Analysis Appendix. The results are listed below.
I find it
easier (and
quicker) to
leave space
and write in
equations by
hand
"
#
Mean t[s]
Mean Velocity u p = d/t [cm/s]
∆u p [cm/s]
Mean Velocity u p = λ f [cm/s]
∆u p [cm/s]
Linear (Low)
0.85
19.4
1.2
23.4
1.2
Linear (high)
0.83
19.9
1.3
23.1
1.7
Circular
0.66
19.5
2.0
18.5
1.7
Questions
Every
answer
needs
to
show
the
equation
with
variables
at least once
"
Â
1. At both the low and high frequencies, the velocities of the linear waves agree to
within the uncertainty.
4. The circular and linear waves had similar wavelengths to within the uncertainty. We
can assume that the wavelength of a wave does not depend on its shape.
5. Based on our results, we can assume that the velocity, u p , is constant in a material.
Lab Part: 0-5.2
Data
Voltage [V]
Frequency [Hz]
Min1 [cm]
Max1
Min2
Max2
Low f
1.80
10.2
19.2
19.7
20.3
20.8
xxi
High f
2.32
15.4
18.0
18.4
18.8
19.1
!
Ã
!
¿
If they didn’t
agree, give a
possible reason why
2. We feel that the timing propagation method yields more accurate results. Although
each measurement was potentially less accurate, averaging many measurements Á
yielded a more accurate estimate of the mean velocity. The results imply that the
wave velocity is independent of its frequency
3. The circular and linear waves had similar velocities to within the uncertainty. We can
assume that the velocity of a wave does not depend on its shape.
Ã
À
Analysis
Steps 1-4:
®
λl1 = Min2 − Min1 = 20.3 − 19.2 =
©
1.1 cm
­
®
λl2 = Max2 − Max1 = 20.8 − 19.7 =
λl3 = Max1 − Min1 = 19.7 − 19.2 =
λl4 = Max2 − Min2 = 20.8 − 20.3 =
λh1 = Min2 − Min1 = 18.8 − 18.0 =
1.1 cm
­
®
0.5 cm
­
®
0.5 cm
λh4 = Max2 − Min2 = 19.1 − 18.8 =
ª
©
ª
®
©
0.8 cm
λh3 = Max1 − Min1 = 18.4 − 18.0 =
ª
©
­
­
®
λh2 = Max2 − Max1 = 19.1 − 18.4 =
ª
©
0.7 cm
­
®
0.4 cm
­
®
0.3 cm
­
ª
©
ª
©
ª
©
ª
Step 5: (since the period of a standing wave is 1/2 of the travelling wave)
®
λl−travelling =
2λl1 +2λl2 +4λl3 +4λl4
4
=
∆λl = Std.Dev. ([λl1 , λl2 , 2λl3 , 2λl4 ]) =
λh−travelling =
2λh1 +2λh2 +4λh3 +4λh4
4
=
∆λh = Std.Dev. ([λh1 , λh2 , 2λh3 , 2λh4 ]) =
Step 6:
fl =
v
λl
®
=
21.7
2.1
=
fh =
21.7
1.45
=
10.3 Hz
­
®
15.0 Hz
­
Questions
xxii
©
ª
©
ª
2.1 cm
­
®
0.06 cm
­
®
1.45 cm
­
®
0.09 cm
­
©
ª
©
ª
©
ª
©
ª
1. The difference in frequencies is almost negligible. They are definitely within the
uncertainty.
2. The spatial periodicity of the standing waves is 1/2 of the travelling wave.
3. We didn’t need the strobe light, because the waves peaks, troughs, and nulls are
stationary (although the peaks and troughs switch places twice every cycle).
Summary
In this lab we determined that the velocity of a wave is approximately independent of
the wave’s frequency, wavelength, or shape. We verified this by comparing the theoretical
velocity (λ f ) with the experimental velocity (d/t). In the second part, we verified that a
standing wave has one half the spatial periodicity of the travelling wave.
{att:
Prelab (1pg), Data (2pg)}
xxiii
Prelab
xxiv
Data
xxv
xxvi
Data
LAB EXERCISE 0: WATER WAVES