CH351L Wet Lab 1/ page 1
CH351L Wet Lab I.
Waves on an Elastic String
Reminder: Bring your answers to the pre-lab questions to lab for credit.
I. Introduction & Concepts:
Many of the labs this semester will be concerned with changes in the quantum
state of a molecule. It is from these changes that molecular properties originate.
Classically we can determine when something changes by watching for a change
in its position or velocity. In quantum mechanics, however, the concepts of
position and velocity become fuzzy. Instead of describing a molecule by these
parameters, therefore, we describe it by its quantum state and the energy of this
state.
The quantum states of a molecule are described by something we call a State
Function. While this function is unlike anything we have previously studied in
classical mechanics, it shares many properties with waves in elastic materials.
This lab will be concerned with developing a more fundamental understanding of
these classical waves and also of the differences between these and quantum
mechanical State Functions.
Concepts to be learned:
•
Waves are characterized by an amplitude that varies with spatial
position and time.
For classical waves on a string, this amplitude is the height deviation of a
point on the string relative to its position at rest. It is a real number, and can
be either positive or negative. (See the experimental setup in Fig. 1 below.)
Pre-lab Question: Are there waves in nature whose value are never
negative? Can you name a few?
One of the strange facts of quantum waves is that their amplitudes are
complex numbers, not just real numbers. These amplitudes are related to
probability not displacement!
Question: Can you try to imagine a wave which moves not only in the up (+)
and down (-) directions, but somehow in an imaginary (I) and negative
imaginary (-I) directions? Can you think of something that can be described
by a probability amplitude?
•
Waves can be superimposed. This is called superposition.
CH351L Wet Lab 1/ page 2
In the same location in space, we might observe two different waves at
different times. If so, it is possible to observe a wave whose amplitude is the
sum of the amplitudes of the two other waves. These two waves are "added"
together to form the wave superposition.
Pre-lab Question: Can you think of some examples in your experience where
waves added together to form a wave that is the sum of the original two?
h(x,t) = height deviation from rest
Computer-controlled
mechanical vibration
drives waves
Height of string
at rest
Elastic string
travels over
pulley
Choice of fixed weight
controls string tension
Fig 1. Experimental Setup: Tension-controlled vibrating string
This superposition principle is one of the most important features of waves.
Notice that it does not seem to our senses that matter is wavelike in the
superposition sense: if a rock can be found at a particular spot on the ground,
and so can a tree, it's hard to imagine what it means for a "sum" of a rock and
a tree to be at that spot!
!
CH351L Wet Lab 1/ page 3
Pre-lab Question: Think of a State Function of a rock not as a thing but as the
probability of finding a rock at a point. Think of the State Function of a tree as
the probability of finding a tree at a point. What expression combining these
state functions would describe the probability of the rock and tree being at the
same point in space?
•
The superposition property can be deduced from the wave equation.
The classical wave equation, which we are discussing in lecture, describes
the possible wave motions observable on the elastic string. The equation
results from nothing more than applying Newton's law to each element of the
string, and assuming the restoring force is proportional to the stretch (Hooke's
law). It is a partial differential equation that is second-order in time and in
space:
2
∂ 2h
2 ∂ h
=v
∂t 2
∂x 2
(1)
where h(x,t) is the amplitude as a function of x, the position along the string,
and t , the time. For an elastic string, the wave velocity,v, is a function of the
linear mass density, r ,(mass per unit length) and the tension, T, on the
string:
T
v=
r
(2)
Pre-lab Question: Show that if h1(x,t) and h2(x,t) satisfy the wave equation,
then so does their sum h(x,t) = h1(x,t) + h2(x,t).
•
The set of possible wave motions is restricted by boundary conditions.
While the wave equation reflects Newton's law at each point on the string, the
constraints of clamping the string down at the ends is expressed
mathematically in the form of boundary conditions. For instance, the
boundary conditions h(0,t) = 0 or h(L,t) = 0 says that the ends of the string at
x = 0 and x = L are prevented from moving up or down.
Pre-lab Question: If boundary conditions on a classical wave restrict the
amplitude of a wave at its boundary, how do boundary conditions restrict state
functions?
CH351L Wet Lab 1/ page 4
•
The classical wave equation is linear.
Linearity is very similar to the concept of superposition, but is somewhat
stronger. Superposition requires the sum of two possible waves to be a
possible wave (solution of the equation). Linearity is defined to hold when
1) any linear combination of possible waves, i.e. h(x,t) = c1 h1(x,t) + c2 h2(x,t),
where c1 and c2 are any real numbers, is also a possible wave, and
2) h(x,t) = 0 is a possible wave.
Pre-lab Question: From what you know about probability, can you explain
why state functions must be linear?
Linear systems (those described by equations obeying linearity) have many
useful properties. Many of these properties are analogous to the properties of
vectors (a linear system). The following concept of "normal modes" is one
such example.
•
An elastic string has normal modes that are characteristic of the string
properties, as well as the particular boundary conditions.
A normal mode is a particular vibration or motion of the wave that oscillates in
time, and whose overall spatial shape is unchanged. More precisely, a wave
h(x,t) is a normal mode if it happens to be a solution of the wave equation
satisfying the boundary conditions that has the separable (factorizable) form:
h( x, t) = f (x)g(t)
(3)
where g(t) = sin(2pvt + j ) modulates the spatial shape function f(x) by an
oscillation of frequency n and phase shift j .
CH351L Wet Lab 1/ page 5
L
Integer multiples of halfwavelengths between boundaries:
nl/2=L
i.e., l = 2 L/ n
n=1:
l=2L
n=2:
l=L
n=3:
l=2L / 3
n=4:
l=L/2
l
h(0,t) = f(0) = 0
h(L,t) = f(L) = 0
Fig 2. Normal modes on an elastic string with fixed ends.
For a finite length of elastic string with fixed end boundary conditions
h(0,t) = 0 and h(L,t) = 0, the spatial shape functions are also sine functions,
f ( x) = sin(2px / l ) , where l is the wavelength. (See Fig. 2 above.)
Note that to satisfy the boundary conditions here, an integer multiple of halfwavelengths need to fit exactly into the length L.
The normal modes thus form a discrete hierarchy, characterized by an
increasing number of "nodes". A node is a point where the amplitude is
always zero as a function of time, and can be seen from above to be the
zeroes of f(x). An "anti-node" is the point where the envelope shape f(x) is an
extremum.
CH351L Wet Lab 1/ page 6
As the number of nodes increases, the wavelength decreases, and the
frequency of oscillation associated with that node also increases. (Shorter
wavelengths oscillate faster.)
Preview comment: Other important properties of normal modes which we
mention here, but will discuss later, is that mathematically, they are
eigenfunctions of the wave equation, and form a basis for all possible waves.
Pre-lab Question: For a state function to be analogous to a normal mode
what conditions must be satisfied? Hint: Think about the time dependence of
the probability.
•
Travelling waves can be formed from a superposition of normal modes.
Other wave motions are possible on a string besides the normal modes
(which are the special oscillating motions). These other motions will typically
involve the familiar "travelling" of a waveform along the string. Nonetheless,
these travelling waves (and any allowed wave) can still be mathematically
decomposed into a linear combination of the normal modes.
Try generating travelling waves in the lab room with the bungee cord
stretched across the room. Do wave pulses appear to superpose and pass
through each other when travelling in opposite directions?
•
The classical wave equation describes many wave phenomena in
nature.
Since the derivation of the classical wave equation involved only Newton's
laws and a harmonic (Hookean) restoring force, most waves that are
relatively small disturbances in an equilibrium system obey the classical wave
equation. Thus water waves, sound waves, mechanical waves (as long as
they are not too strong) as well as electromagnetic (e.g. light) waves all obey
the classical wave equation.
II. Experimental
You will need the information you collect during the experiment to complete the
report at the end of this module. No separate report is required for this
experiment.
Safety and precautions:
CH351L Wet Lab 1/ page 7
1) Be sure the clamps are all tightened after moving a part of the
experimental assembly. The string will be under tension, and can yank the
mechanical oscillator off the table if not properly secured.
2) Securely fasten weights to the end of the string so that vibrations do not
shake the weight loose.
3) Do not leave the strobe running for more than a period of a few minutes.
Afterwards, allow the strobe to cool off for about ten minutes before turning on
again.
General:
The apparatus, as illustrated in Fig. 1 above, is set-up to excite normal mode
vibrations in the elastic string. The weight, mweight, off the end of the pulley is
equal to the string tension (T = mweight g).
Question: What have we assumed about the weight of the string in the above
statement? What have we assumed about frictional forces?
Since the pulley is not perfectly frictionless, there is some hysteresis in the
degree of stretching of the string for a given weight.
Estimate the amount of hysteresis by measuring the difference in length between
the maximum and minimum stable extensions for the same weight.
The motion of the mechanical oscillator is controlled by the computer-driven
waveform. For this experiment, we will always use a sinusoidal waveform
(controlled in the "Signal Generator" window).
The pulley acts to clamp the vertical string motion at that boundary. Without a
second clamp, the string is attached directly to the mechanical oscillator. This
end does not correspond to a fixed boundary since it is free to move up and
down. Therefore, without a fixed second boundary, the wavelength is not
constrained to fit a half-integer multiple in the length of vibrating string, and so
the wavelength can take continuous values.
Hint: The measurements are more easily made when the oscillations have the
highest amplitude. This can be found by tuning the frequency 1 Hz at a time.
CH351L Wet Lab 1/ page 8
Part I. Determination of the relation between wavelength and
frequency in normal mode oscillations
From the discussion above, we expect the frequency n to increase as the
wavelength l decreases for normal mode oscillations. In this first part of the
lab, we will determine the functional relationship between l and n by measuring
l with a meterstick while varying n on the computer. You will not likely need to
go to frequencies below 10 Hz or above 500 Hz, but you can try it (with caution!).
In this first part we will fix the tension by hanging a 500-g weight off the free end
of the string. To be consistent, use the maximum extension of the string at this
weight.
1.
Pick a frequency between 10 Hz and 500 Hz, and observe the formation
of a stable normal mode oscillation. What is the spatial envelope
function? Identify the nodes and antinodes, if any. View the results with a
strobe.. (See below for strobe hints).
2.
Increase or decrease the frequency 1 Hz at a time until you find a
maximum response (amplitude of the spatial wave). Notice that the
mechanical oscillator may be a short distance off from the nearest node.
Measure the wavelength. Record the frequency and wavelength.
Hint: If you have many wavelengths visible, is accuracy best achieved by
measuring a single wavelength?
3.
Scan the range from 10 Hz to 500 Hz, picking around 15 frequencies to
make measurements. They don't all need to be at maximum response,
though these are easier to measure.
4.
Plot (at home if you choose) your l vs. n data on a log-log plot. This is a
common method for detecting a power-law dependence: If l = c n n ,
then log( l ) = n log(n ) + log(c)…a linear relationship on a log-log plot.
The slope of the line is the exponent n. Do you find your data support a
power-law dependence of l vs. n ? If so, what is your best value of n, or
the nearest integer? Plot l vs. n n , for this value of n.
5.
Given Eq. (3) above for the functional form of the normal mode, with the
appropriate choices of f(x) and g(t), use the classical wave equation,
Eq. (1), to determine a theoretical relationship between l , n , and v (the
wave speed). How do your measurements compare with theory?
CH351L Wet Lab 1/ page 9
Part II. Determination of the relation between wavelength,
tension and linear mass density in normal mode oscillations
In this second part, you will fix the frequency while varying the weight attached to
the free end of the string. You may want to choose a frequency that is low
enough that the wavelength is not too different from, L, the length between the
pulley and the mechanical oscillator, when there is a 500 g mass hanging. You
will vary the weight while measuring the wavelength.
The tricky part here is that while changing the weight alters the tension, it also
alters the linear mass density, since the same string (of constant mass, mstring) is
stretched over a longer or shorter distance. Thus the mass is distributed over a
total length Ltot, even though only the length L is vibrating.
1)
With the 500 g weight still on, measure the wavelength at your chosen
frequency. Also measure the total length of stretched string. Be sure you
are measuring the total stretched string--from the weight to the
mechanical oscillator. This can be done by measuring the length of the
string that is hanging vertically, and adding it to L, which is fixed. You will
need to estimate how much string is in the pulley, and record this value.
2)
Vary the hanging weight, mweight, (again using the maximum extension for
reproducibility against hysteresis) between 100g and 500g, and record the
wavelength, tension, and total linear extension of the stretched portion of
string. Make at least 5 different weight measurements.
3)
For the curious: You may wish to plot the total extension of the stretched
portion of the string, Ltot, versus the mass mweight. If the string were
perfectly Hookean, what would be the expected relationship between the
two? What do you observe and why?
4)
Given the relation between l , n , and v you derived from Part I, write an
mweightg
T
expression relating l to
. Note mstring and g are not
=
r
mstring / Ltot
being varied in this equation.
5)
Plot l vs. mweight Ltot . Consider how this compares with your theoretical
expression above?
6)
Since you know Ltot, calculate a value for mweight based on your
wavelength measurements. Weigh your string on a balance, and estimate
what fraction of your string was stretched (i.e. between the knots). From
this, you may estimate the mass of the string that was stretched and
determine how it compares to your calculated value of mweight?
CH351L Wet Lab 1/ page 10
Part III. Normal modes for a vibrating string with two fixed ends
In the previous parts, only one end (the pulley) was truly fixed. Here we will look
for the appearance of discrete normal modes when both ends are fixed.
Return to using a 500-g hanging weight. Using another clamp attached to the
clamp stand holding the mechanical oscillator, clamp down (pinch) the string
slightly away from the attachment point to the mechanical oscillator.
1)
Measure the distance between the two fixed ends of the vibrating string.
2)
Scanning in frequency, record when the amplitude of the vibration
between the two fixed ends reaches a local maximum. This should occur
roughly at the resonance frequencies of the normal modes.
3)
Plot n vs. l for observations of strong normal mode excitations.
Consider how your observations compare with those in Part I. In
particular, how does your wave speed compare to Part I. Do the data
from Part III fall on the same curve measured in Part I?
Part IV. Normal modes in two and three dimensions
Observe the normal modes for the circular drumhead and sphere (water balloon).
Make rough sketches of these modes identifying nodal surfaces. Can you find a
relationship between frequency and the number of nodal surfaces?
Notes on Strobe Observations
The strobe emits very short pulses of light at a precisely controlled frequency. If
that frequency matches the frequency of some oscillating motion, it will always
capture an image at the same point in its oscillatory cycle. This creates the
optical illusion of "freezing" the rapidly oscillating motion, and is often used to
time the frequency of rapidly oscillating or cycling industrial parts.
If the strobe frequency is slightly off from the oscillating motion frequency, the
snapshots captured by the strobe will slowly lag behind or get ahead of the last
snapshot. This creates the illusion of slow-motion viewing of the cyclic motion.
This can be effective for viewing the various phases of the cyclical motion, which
is ordinarily blurred out due to the high speeds.
In lab, the instructor may also present other vibrating or oscillating systems for
you to examine under the strobe, and to discuss in terms of normal mode motion.
For wave motion, it is particular interesting to observe the relative phases
(positive or negative amplitudes) of neighboring wave regions, separated by a
node.