Experiment VIII

Waves and Resonance:

The Velocity of Transverse Waves On a Vibrating
String

**Introduction and Objective**

The velocity of propagation of vibrational waves through a medium can
depend not only on the properties of the medium, but also on the external
conditions imposed on it. It is reasonable to suppose, for example, that
the velocity with which transverse vibrational waves propagate down a string
or flexible wire would depend both on the tension, T, in the string and
the mass per unit length (linear mass density), µ, of the string.
The tension supplies the restoring force when the string is given a transverse
displacement and the linear density determines the inertia of the string
and, therefore, its speed of response to the tension. On this basis it
would be expected that the velocity of propagation would increase with
increasing tension T and decrease with increasing mass density, µ.
A first guess might be V = KT/µ where K is a dimensionless constant,
but as you can and should show, this cannot be correct from dimensional
considerations. This being the case, a next logical step is to assume a
relationship of the form V = KT^{a}/m^{b}
and to take as your objective the determination of the coefficients K,
a and b experimentally.

**Equipment**

60 Hz electric vibrator, several strings of different linear density, pulley, swivel clamp, two table clamps, two 12" rods, a set of slotted weights, and a meter stick.

The string is held horizontally by the vibrator at one end and by weights hung over a pulley at the other end so that the tension may be varied. A sketch of the set-up is shown below.

**Theory of the measurements**

The velocity of propagation of waves on a string can be readily measured
by setting up a standing wave pattern using a simple harmonic vibrator
of known frequency. The vibrator sends down the string a wave whose mathematical
representation is y_{1} = A sin (2ft - 2p/lx)
where x is any location on the string between 0 and L (L being the length)
t is the time, f is the frequency and l is the
wavelength. The number of waves generated per second (f) multiplied by
the length of each (l) is just the distance
traveled per second by the wave, that is, the velocity V = fl.
The wave sent down the string by the vibrator will reflect at the end (x
= L) and return to the vibrator where it will again reflect. Leaving the
vibrator again it will now have the form Y_{2} = A sin [2ft - 2p/l(x
+ 2L)] as it has already traveled 2L along the string. Since the vibrator
also continues to create waves of the form of Y_{1}, the net effect
is obtained by adding Y_{2} to Y_{1}.

How the two waves combine depends on the relationship l
between and L; suppose 2L = ln where n is an
integer (1,2,3, etc). Then the length L is an integer multiple of one-half
wavelength or L = nl/2. In this case, 2p/l(x
+ 2L) = 2p/l(x + n) = 2p/l
x + 2pn. Now when an integer multiple of 2p
is added to the argument of the sine function nothing is changed, thus
when 2L = nl, Y_{1} = Y_{2}
and as the 2 waves add peak to peak or what is called *constructive interference*.
As waves Y_{2} continue one after another to reflect off the vibrator
they combine constructively with waves Y_{1} generated by the vibrator.
The net result is the build up of a large wave (much larger than the amplitude
of the vibrator) and the result is called resonance. The result is also
called a standing wave as it does not appear that waves are moving along
the string. To see why this is so, consider now the reflection off the
far end of the string. This wave can be represented by Y_{3} =
-A sin [2ft - 2p/l(2L - x)]. (The minus sign
in front corresponds to the fact that when the wave reflects off the fixed
end at x = L it flips over. Note that when the wave is at x = L the expression
shows that the wave has traveled a distance L and when it returns to x
= 0 the distance traveled is 2L as required). Now if the condition for
resonance, 2L = nl, is fulfilled, then 2p/l(2L
- x) = 2pn - 2xp/l
and Y_{3} = -A sin (2ft + 2p/l x). If
this wave is added to the wave Y_{1} = A sin (2ft - 2p/l
x) coming down from the vibrator, the sum Y_{1} + Y_{3}
is

Y_{1} + Y_{3} = 2A cos 2ft sin 2p/l
x. (8.1)

(This follows from the formulas for sin (a + b) and sin (a - b).)

Note that this sum is *always zero* at certain values of x, namely
when x = nl/2 for when sin 2p/l
x = sin 2pnl/l2 =
sin pn = 0. *These stationary points which
are separated by one-half wavelength are called nodes* as the string
does not move there. Hence at resonance one does not see any wave traveling
back and forth. The vibration pattern shows motion only in the Y direction.
The maximum Y or transverse displacement *(antinodes*) occur halfway
between the nodes as sin 2p/l x has its largest
values (±1) at these points. The net results is called a *standing
wave pattern*. The figure below shows some of the possible wave patterns
in order of increasing numbers of nodes.

**Procedure**

Set up standing wave patterns of 60 Hz frequency on your string. As the discussion above shows, by measuring the distance between nodes you can determine the wavelength and then, knowing the frequency, you can calculate the wave velocity. Since the length of string and frequency is fixed by the apparatus, the only quantity that may be adjusted to reach a resonance condition is the velocity which can be varied by changing the tension.

There will be nylon strings of several different linear densities available.
Your instructor will assign your group one of these. Vary the tension in
your string to produce at least 5 different standing wave patterns. It
is a good idea to start with enough tension to get 3 or 4 nodes not counting
the ends. Then you can increase or decrease T as needed for other patterns.
*Do not try to produce the fundamental or lst overtone as these patterns
have too few nodes to be useful and the required tension is excessive.*

The table below shows the approximate range of masses which must be hung
on nylon fishing lines of various strengths to achieve resonance over the
range of harmonics shown.

Table 8.1

Line
strength |
Range of
harmonics |
Range of hanging mass (kg) |

80 lb |
4 - 9 |
2.0 - 0.40 |

60 lb |
3 - 9 |
2.3 - 0.25 |

40 lb |
3 - 9 |
1.1 - 0.10 |

20 lb |
3 - 9 |
0.6 - 0.06 |

10 lb |
3 - 7 |
0.3 - 0.05 |

Note that because V = fl, if you wish to
decrease l so as to produce more nodes, V will
have to be decreased by reducing the tension. Hence, *the larger the
tension, the fewer the nodes* as the table above shows.

Certain combinations of vibrators, strings and tensions may result in vibration at 120 Hz rather then 60 Hz. Should you find that a particular tension produces a much shorter wavelength than the table above would indicate, it is likely that the vibration frequency is 120 Hz in that case. There is no need to discard such data. When you prepare your table of V and T for computer analysis, 120 Hz data will be recognizable.

Record the distances between nodes, the number of nodes not counting
the ends and the tension. (Remember that tension is a force). Note that
when you are near resonance you can lift up or pull down on the weights
a slight amount to see if more or less tension is needed. Try to get as
close to resonance (maximum amplitude) as possible. Note that the vibrator
and pulley ends are not quite fixed and are therefore not quite true nodes.
*Therefore do not include the distances from either end to the first
adjacent node in your measurements.*

Determine the wavelength and velocity of the waves traveling on your
string for each of your tensions. Since the assumption is that V = KT^{a}/m^{b}
a log-log plot of V versus T for a given string (constant µ) should
be a plot of the equation

log V = constant + a log T.

Similarly a log-log plot of V versus µ for constant T should be a plot of the equation

log V = constant - b log µ.

The data from your group will allow you to make a log-log plot and analysis of V versus T for the fixed density of your string. Use Quattro Pro and load the file "Waves.wb3". Fill in the values in the table. Then plot V versus T. Follow the instructions on the screen to convert the linear plot to a log-log plot.

Do a "Linear Regression Fit" of log V (dependent) against log T (independent). Do your data support the assumed relationship between V and T? What is your value for a, i.e., what is the dependence of V on T? Print your data and graph.

You will need to combine your data with those of other groups to make
the log-log plot of V versus µ for fixed T in order to determine
b. Each group will have a fit of the data for
their string to the equation,

log V = m log T + c.

Hence, by using each group's values for the y-intercept, c and slope,
m, a value for V can be calculated for *each* string for a *common
chosen tension* (say 4.0 N). These values of V can then be used to make
a log-log plot of V versus µ. (Your instructor will either give you
the values for the different string densities or show you how to measure
them. Use units of kg/m for these values).

Enter the combined values of V and µ in the second table. Then make a log-log plot of V versus µ. Do a "Linear Regression Fit" of log V (dependent) against log µ (independent). Do the data support the assumed relationship between V and µ? What value do you obtain for ? Print your data and graph.

**Discussion**

From your experimental results try to infer what values for a and b are likely to be obtained on a theoretical basis. Assume that simple results are the most likely. Do your experimental values for and agree within experimental error with your inferred values? Do your inferred values for and lead to a dimensionally correct result for V?

If time permits, you can determine the value for K in the relation V
= KT^{a}/m^{b} as follows. From
the above relation,

log V = constant + a log T - b log µ.

In comparison, the equation obtained by each group from their analysis is:

log V = c + a log T.

Hence,

c = log K - b log µ.

Thus, a value for K can be determined from the data for each string by first calculating log µ for each string and then combining it with the value of c for that string.

Average the values of K so obtained. What do you infer to be the theoretical value for K?

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© 1997 Dr. H. K. Ng.

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