Experiment 108: Transverse Waves: Frequency of Vibration

Analysis

In this experiment, we will be dealing with mechanical waves, to be

specific, transverse waves and its relationship with a medium. However,
before we delve into the discussion of transverse waves, we must first
understand what a wave is. By understanding the ideas and concepts that
surrounds it, we will be able to fully grasp the functions and theory of
transverse waves and its importance later on in the field of engineering.

As defined, a wave is a propagation of disturbances from place to

place in a regular and organized way. One of the most familiar ones that
we always see on water are surface waves which travels on water, the
things we hear on through our ears are longitudinal waves such as sound
that travel through air, the things we barely see nor feel are
electromagnetic waves such as light which can travel through a vacuum of
space, and also the motion of all subatomic particles that exhibit wavelike
properties.

Waves are classified into two main categories, electromagnetic waves

which does not require any medium to propagate, and mechanical waves
which
requires a medium in order to transport their energy from one location to
another. Electromagnetic waves consists of radio waves, microwaves,
infrared, light, ultraviolet rays, x-rays, and gamma rays, whereas
mechanical waves consists only of Longitudinal and Transverse waves.
Refer to figure 1 for the classification of waves and their types. Although in
this experiment, we will exclusively focus on mechanical waves, to be
specific, transverse waves.

Figure 1. Classification of Waves

A mechanical wave is a disturbance that travels through some

material or substance called the medium for the wave to transfer its energy
from one location to another. As the wave travels through the medium, the
particles that make up the medium undergo displacements of various
kinds, depending on the nature of the wave.

In the figure 2 the three varieties of mechanical waves is shown. In

Figure 2A, the medium is a string or rope under tension. If we give the leg
ten a small upward shake or wiggle, the wiggle travels along the length of
the string. Successive sections of string go through the same motion that
we gave to the end, but at successively later times because the
displacements of the medium are perpendicular or transverse to the
direction of travel of the wave along the medium, this is called a transverse
wave. In Figure 2B, the medium is a liquid or gas in a tube with a rigid wall
at the right end and a movable piston at the left end. If we give the piston a
single back-and-fort motion, displacement and pressure fluctuations travel
down the length of the medium. This time the motions of the particles of
the medium are back and forth along the same direction that the wave
travels. We call this a longitudinal wave. In Figure 2C, the medium is a
liquid in a channel, such as water in an irrigation ditch or canal. When we
move the flat board at the left end forward and back once, a wave
disturbance travels down the length of the channel. In this case the
displacements of the water have both longitudinal and transverse
components.

Figure 2. Three Ways to Make a Wave that Moves to the Right

Each of these systems has an equilibrium state. For the stretched

string it is the state in which the system is at rest, stretched out along a
straight line. For the fluid in a tube it is a state in which the fluid is at rest
with uniform pressure. And for the liquid in a trough it is a smooth, level
water surface. In each case the wave motion is a disturbance from
equilibrium state that travels from one region of the medium to another.
And in each case, there are forces that tend to restore the system to its
equilibrium position when it is displaced, just as the force of gravity tends
to pull a pendulum toward its straight-down equilibrium position when it
is displaced.

These examples have three things in common. First, in each case the
disturbance travels or propagates with a definite speed through the
medium. This speed is called the speed of propagation, or simple the wave
speed. Its value is determined in each case by the mechanical properties of
the medium. Second, the medium itself does not travel through space; its
individual particles undergo back- and-forth or up-and-down motions
around their equilibrium positions. The overall pattern of the wave
disturbance is what travels. Third, to set any of these systems into motion,
we have to put in energy by doing mechanical work on the system. The
wave motion transports this energy from one region of the medium to
another. Wave transport energy, but not matter, from one region to
another.

The transverse wave on a stretched string in Figure 2A is an example

of a wave pulse. The hand, shakes the string up and down just once,
exerting a transverse force on it as it does so. The result is a single
“wiggle”, or pulse, that travels along the length of the string. The tension in
the string restores it straight-line shape once the pulse has passed. A more
interesting situation develops when we give the free end of the string a
repetitive, or periodic motion. Then each particle in the string also
undergoes periodic motion as the wave propagates, and we have a periodic
wave.

Figure 3. A Block of Mass m Attached to a Spring Undergoes

Simple Harmonic Motion, producing a Sinusoidal Wave that Travels to
the Right of the String
In particular, suppose we move the string up and down with simple
harmonic motion with amplitude A, frequency f, angular frequency ω =
2πf, and period T = 1/f = 2π/ω. Figure 3 shows one way to do this. The
wave that results is a symmetrical sequence of crests and troughs. As we
will see, periodic waves with simple harmonic motion are particularly easy
to analyze; we call them sinusoidal waves. It also turns out that any
periodic wave can be represented as a combination of sinusoidal waves. So,
this particular wave motion is worth special attention. In Figure 3, the
wave that advances along the string is a continuous succession of
transverse sinusoidal disturbances. Figure 4 shows the shape of a part of
the string near the left end at time intervals of 1/8 of a period, for a total
time of one period. The wave shape advances steadily toward the right, as
indicated by the highlighted area. As the wave moves, any point on the
string oscillates up and down about its equilibrium position with simple
harmonic motion. When a sinusoidal wave passes through a medium,
every particle in the medium undergoes simple harmonic motion with the
same frequency.

Figure 4. A Sinusoidal
Transverse Wave Travelling to the Right Along a String
For a periodic wave, the shape of the string at any instant is a
repeating pattern. The length of one complete wave pattern is the distance
from one crest to the next, or
form one trough to the next, or from any point to the corresponding point
on the next repetition of the wave shape. We call this distance the
wavelength of the wave, denoted by λ. The wave pattern travels with
constant speed v and advances of one wavelength λ in a time interval of
one period T. So, the wave speed is given by v = λ/T or v = λf, where f =
1/T, as shown in Equation 1. The speed of propagation equals the product
of wavelength and frequency. The frequency is a property of the entire
periodic wave because all points on the string oscillate with the same
frequency f.

Waves on a string propagate in just one dimension. But the ideas of

frequency, wavelength, and amplitude apply equally well to waves that
propagate in two or three dimensions. As with waves on a string,
wavelength is the distance from one crest to the next, and the amplitude is
the height of a crest above the equilibrium level. In many important
situations including waves on a string, the wave speed v is determined
entirely by the mechanical properties of the medium. In this case,
increasing causes λ to decrease so that the product wave speed remains the
same, and waves of all frequencies propagate with the same wave speed.
Aside from periodic transverse waves, there is also a periodic longitudinal
wave that won’t be discussed anymore since it will be out of the topic of
transverse waves. In addition, the discussion of the wave function and
mathematical description of a wave will be left untouched because it will
be discussed during the lecture proper of the co- requisite course of this
laboratory course. However, the wave function is one of the most
important equations in all of physics. It provides important foundation for
the study of Quantum Mechanics especially the idea of Time Dilation and
proving the idea of Schrodinger’s Equation, and other important ideas in
Quantum Mechanics.

One of the key properties of any wave is the wave speed. Light
waves in air have a much greater speed of propagation than do sound
waves in air versus that’s why you see the flash from a bolt of lightning
before you hear the clap of thunder. In this section we’ll see what
determines the speed of propagation of one particular kind of wave:
transverse waves on a string. The speed of these waves is important to
understand because it is an essential part of analyzing stringed musical
instruments, as we’ll discuss later in this chapter. Furthermore, the speeds
of many kinds of mechanical waves turn out to have the same basic
mathematical expression as does the speed of waves on a string. The
physical quantities that determine the speed of transverse waves on a
string are the tension in the string and its mass per unit length (also called
linear mass density). We might guess that increasing the tension should
increase the restoring forces that tend to straighten the string when it is
disturbed, thus increasing the wave speed. We might also guess that
increasing the mass should make the motion more sluggish and decrease
the speed. Both these guesses turn out to be right. We’ll develop the exact
relationship among wave speed, tension, and mass per unit length by two
different methods. The first is simple in concept and considers a specific
wave shape; thus, it will be the one used in proving the equation for the
wave speed.

Figure 5. Propagation of a Transverse Wave on a String

We consider a perfectly flexible string (Figure 5). In the equilibrium

position the tension is F and the linear mass density (mass per unit length)
is (When portions of the string are displaced from equilibrium, the mass
per unit length decreases a little, and the tension increases a little.) We
ignore the weight of the string so that when the string is at rest in the
equilibrium position, the string forms a perfectly straight line as in Figure
5A. Starting at time we apply a constant upward force at the left end of the
string. We might expect that the end would move with constant
acceleration; that would happen if the force were applied to a point mass.
But here the effect of the force is to set successively more and more mass in
motion. The wave travels with constant speed so the division point P
between moving and nonmoving portions moves with the same constant
speed (Figure 5B). Figure 5B shows that all particles in the moving portion
of the string move upward with constant velocity not constant acceleration.
To see why this is so, we note that the impulse of the force up to time t is
According to the impulse–momentum theorem, the impulse is equal to the
change in the total transverse component of momentum of the moving part
of the string. Because the system started with no transverse momentum,
this is equal to the total momentum at time t, as shown in Equation 2.

𝐹𝑦 𝑡 = 𝑚𝑣𝑦 (2)
The total momentum thus must increase proportionately with time.
But since the division point P moves with constant speed, the length of
string that is in motion and hence the total mass m in motion are also
proportional to the time t that the force has been acting. So, the change of
momentum must be associated entirely with the increasing amount of mass
in motion, not with an increasing velocity of an individual mass element.
That is, changes because m, not changes. At time t, the left end of the string
has moved up a distance and the boundary point P has advanced a
distance the total force at the left end of the string has components F and
Why F? There is no motion in the direction along the length of the string, so
there is no unbalanced horizontal force. Therefore F, the magnitude of the
horizontal component, does not change when the string is displaced. In the
displaced position the tension is (greater than F), and the string stretches
somewhat. To derive an expression for the wave speed we again apply the
impulse–momentum theorem to the portion of the string in motion at time
t—that is, the portion to the left of P in Figure 5B. The transverse impulse
(transverse force times time) is equal to the change of transverse
momentum of the moving portion (mass times transverse component of
velocity). The impulse of the transverse force in time t is In Fig. 15.11b the
right triangle whose vertex is at P, with sides and is similar to the right
triangle whose vertex is at the position of the hand, with sides and F.
Hence; the concept of transverse impulse and transverse momentum, as
shown in Equation 3 and 4, respectively, shall be use in formulating the
equation for the wave speed shown in Equation 5.

Note that
does not appear
in Equation 5; thus, the wave speed doesn’t depend on Our calculation
considered only a very special kind of pulse, but we can consider any
shape of wave disturbance as a series of pulses with different values of So
even though we derived Equation 5 for a special case, it is valid for any
transverse wave motion on a string, including the sinusoidal and other
periodic waves. Also, the wave speed doesn’t depend on the amplitude or
frequency of the wave, in contrast to what we have stated earlier. Equation
5 gives the wave speed for only the special case of mechanical waves on a
stretched string or rope. Remarkably, it turns out that for many types of
mechanical waves, including waves on a string, the expression for wave
speed has the same general form, as shown in Equation 6. To interpret this
expression, let’s look at the now-familiar case of waves on a string. The
tension F in the string plays the role of the restoring force; it tends to bring
the string back to its undisturbed, equilibrium configuration. The mass of
the string—or, more properly, the linear mass density —provides the
inertia that prevents the string from returning instantaneously to
equilibrium. Hence, we have for the speed of waves on a string.

Figure 6. A series of images of a

wave pulse, equally spaced in time from top to bottom. The pulse starts
at the left in the top image, travels to the right, and is reflected from the
fixed end at the right
 Up to this point we’ve been discussing waves that propagate
continuously in the same direction. But when a wave strikes the
boundaries of its medium, all or part of the wave is reflected. When you
yell at a building wall or a cliff face some distance away, the sound wave is
reflected from the rigid surface and you hear an echo. When you flip the
end of a rope whose far end is tied to a rigid support, a pulse travels the
length of the rope and is reflected back to you. In both cases, the initial and
reflected waves overlap in the same region of the medium. This
overlapping of waves is called interference. (In general, the term
“interference” refers to what happens when two or more waves pass
through the same region at the same time.) As a simple example of wave
reflections and the role of the boundary of a wave medium, let’s look again
at transverse waves on a stretched string. What happens when a wave
pulse or a sinusoidal wave arrives at the end of the string?

Figure 7. Reflection of a
Wave Pulse
 If the end is fastened to a rigid support, it is a fixed end that cannot
move. The arriving wave exerts a force on the support; the reaction to this
force, exerted by the support on the string, “kicks back” on the string and
sets up a reflected pulse or wave traveling in the reverse direction. Figure 6
is a series of photographs showing the reflection of a pulse at the fixed end
of a long-coiled spring. The reflected pulse moves in the opposite direction
from the initial, or incident, pulse, and its displacement is also opposite.
Figure 7A illustrates this situation for a wave pulse on a string. The
opposite situation from an end that is held stationary is a free end, one that
is perfectly free to move in the direction perpendicular to the length of the
string.

For example, the string might be tied to a light ring that slides on a
frictionless rod perpendicular to the string, as in Figure 7B. The ring and
rod maintain the tension but exert no transverse force. When a wave
arrives at this free end, the ring slides along with the rod. The ring reaches
a maximum displacement, and both it and the string come momentarily to
rest, as in drawing 4 in Figure 7B. But the string is now stretched, giving
increased tension, so the free end of the string is pulled back down, and
again a reflected pulse is produced (drawing 7). As for a fixed end, the
reflected pulse moves in the opposite direction from the initial pulse, but
now the direction of the displacement is the same as for the initial pulse.
The conditions at the end of the string, such as a rigid support or the
complete absence of transverse force, are called boundary conditions.
 Figure 8. Overlap of Two Wave Pulse-One Right Side Up and One Inverted

The formation of the reflected pulse is similar to the overlap of two

pulses traveling in opposite directions. Figure 8 shows two pulses with the
same shape, one inverted with respect to the ther, traveling in opposite
directions. As the pulses overlap and pass each other, the total
displacement of the string is the algebraic sum of the displacements at that
point in the individual pulses. Because these two pulses have the same
shape, the total displacement at point O in the middle of the figure is zero
at all times. Thus, the motion of the left half of the string would be the same
if we cut the string at point O, threw away the right side, and held the end
at O fixed. The two pulses on the left side then correspond to the incident
and reflected pulses, combining so that the total displacement at is always
zero. For this to occur, the reflected pulse must be inverted relative to the
incident pulse.
Figure 9. Overlap of Two Wave Pulse-Both Right Side Up
 Figure 9 shows two pulses with the same shape, traveling in opposite
directions but not inverted relative to each other. The displacement at point

O in the middle of the figure is not zero, but the slope of the string at this
point is always zero. This corresponds to the absence of any transverse
force at this point. In this case the motion of the left half of the string would
be the same as if we cut the string at point O and attached the end to a
frictionless sliding ring (Figure 7B) that maintains tension without exerting
any transverse force. In other words, this situation corresponds to
reflection of a pulse at a free end of a string at point O. In this case the
reflected pulse is not inverted.

Figure 10. Standing Nodes

We have talked about the reflection of a wave pulse on a string when
it arrives at a boundary point (either a fixed end or a free end). Now let’s
look at what happens when a sinusoidal wave is reflected by a fixed end of
a string. We’ll again approach the problem by considering the
superposition of two waves propagating through the string, one
representing the original or incident wave and the other representing the
wave reflected at the fixed end. Figure 10 shows a string that is fixed at its
left end. Its right end is moved up and down in simple harmonic motion to
produce a wave that travels to the left; the wave reflected from the fixed
end travels to the right. The resulting motion when the two waves combine
no longer looks like two waves traveling in opposite directions. The string
appears to be subdivided into a number of segments, as in the time-
exposure photographs of Figures 10A, 10B, 10C, and 10D.

Figure 10E shows two instantaneous shapes of the string in Figure

10B. Let’s compare this behavior with the waves we studied in the earlier
parts. In a wave that travels along the string, the amplitude is constant, and
the wave pattern moves with a speed equal to the wave speed. Here,
instead, the wave pattern remains in the same position along the string and
its amplitude fluctuates. There are particular points called nodes (labeled N
in Figure 10E) that never move at all. Midway between the nodes are
points called antinodes (labeled A in Figure 10E) where the amplitude of
motion is greatest. Because the wave pattern doesn’t appear to be moving
in either direction along the string, it
is called a standing wave.
(To emphasize the difference, a
wave that does move along the
string is called a traveling wave.)
The principle of superposition
explains how the incident and reflected waves combine to form a standing
wave.

Figure 11. Formation of Standing Nodes

In Figure 11 the red curves show a wave traveling to the left. The
blue curves show a wave traveling to the right with the same propagation
speed, wavelength, and amplitude. The waves are shown at nine instants,
of a period apart. At each point along the string, we add the displacements
(the values of y) for the two separate waves; the result is the total wave on
the string, shown in brown. At certain instants, such as the two wave
patterns are exactly in phase with each other, and the shape of the string is
a sine curve with twice the amplitude of either individual wave. At other
instants, such as the two waves are exactly out of phase with each other,
and the total wave at that instant is zero. The resultant displacement is
always zero at those places marked N at the bottom of Figure 11. These are
the nodes. At a node the displacements of the two waves in red and blue
are always equal and opposite and cancel each other out. This cancellation
is called destructive interference. Midway between the nodes are the points
of greatest amplitude, or the antinodes, marked A. At the antinodes the
displacements of the two waves in red and blue are always identical,
giving a large resultant displacement; this phenomenon is called
constructive interference. We can see from the figure that the distance
between successive nodes or between successive antinodes is one half-
wavelength, or λ/2

A standing wave, unlike a traveling wave, does not transfer energy

from one end to the other. The two waves that form it would individually
carry equal amounts of power in opposite directions. There is a local flow
of energy from each node to the adjacent antinodes and back, but the
average rate of energy transfer is zero at every point. As we have seen, the
fundamental frequency of a vibrating string is f1=v/2L. The speed of
waves on the string is determined by Equation 5. Combining these
equations will produce a general equation, as shown in Equation 7.

1 𝐹 (7)
𝑓1 = √
2𝐿 𝜇

This is also the fundamental frequency of the sound wave created in

the surrounding air by the vibrating string. Familiar musical instruments
show how depends on the properties of the string. The inverse dependence
of frequency on length L is illustrated by the long strings of the bass (low
frequency) section of the piano or the bass viol compared with the shorter
 strings of the treble section of the piano or the violin. The pitch of a violin
or guitar is usually varied by pressing a string against the fingerboard with
the fingers to change the length L of the vibrating portion of the string.
Increasing the tension F increases the wave speed and thus increases the
frequency (and the pitch). All string instruments are “tuned” to the correct
frequencies by varying the tension; you tighten the string to raise the pitch.
Finally, increasing the mass per unit length decreases the wave speed and
thus the frequency. The lower notes on a steel guitar are produced by
thicker strings, and one reason for winding the bass strings of a piano with
wire is to obtain the desired low frequency from a relatively short string.

Now that all concepts are discussed, let us move on to the

experiment. The experiment consists of two main objectives, to determine
the relationship between tension on a string and the frequency of vibration,
and to determine the relationship between length of a string and the
frequency of vibration. The materials used for the experiment were a string
vibrator, a sine wave generator, two iron stands with clamp, a pulley, a set
of weights, a mass hanger, a meter stick, and a set of 5 guitar strings (diff.
sizes).

Table 1. Frequency of Vibration and Tension

Diameter of Wire = 0.010 in
Linear Mass Density of Wire; µ = 0.0039 g/cm
TRIAL Tension (T); Number Length Experiment Computed Percentage
(dynes) of of al Frequency of Error
Segment String, Frequency of of Vibration
s, n L Vibration
1 53 936.575 3 54.0 cm 100 Hz 103.30 Hz 3.19 %
 2 73 549.875 3 54.0 cm 105 Hz 120.63 Hz 12.96 %
3 102,969.825 2 54.0 cm 105 Hz 95.15 Hz 10.35 %
4 112,583.125 2 54.0 cm 110 Hz 103.82 Hz 5.95 %
5 142,196.425 2 54.0 cm 110 Hz 111.82 Hz 1.63 %
Table 2. Frequency of Vibration and Linear Mass Density
Tension = 49 000 dynes
TRIAL Linear Mass Number Length Experiment Computed Percentage
Density µ of of al Frequency of Error
(g/cm) Segment String, Frequency of of Vibration
s, n L Vibration

1 0.0039 2 54 cm 105.0 Hz 95.15 Hz 10.35 %

2 0.0078 4 67 cm 105.0 Hz 108.46 Hz 3.19 %
3 0.0112 2 56 cm 55.0 Hz 54.14 Hz 1.59 %
4 0.0150 2 53 cm 50.0 Hz 49.43 Hz 1.15 %
5 0.0184 3 56 cm 59.0 Hz 63.36 Hz 6.88 %

Conclusion

In the first part of the experiment which is shown on Table 1, shows

the relationship of frequency of vibration when the linear mass density is
constant. As the length of the string with complete number of segments
increases, the frequency of vibration of vibration decreases. It is due to the
fact that there was an inverse relationship between the two as stated on the
equation used to determine the frequency of vibration. The number of line
segments decreases as the tension is increased, because every time we
increase the weight with designated increments, the more the strings
wasn’t able to move freely. Thus, as it decreases its ability to move in a
wave form, few nodes are formed. In general, increasing tension to the
string will decrease the number of segments formed.

On the other hand, the second part of the experiment, which is

shown on Table 2, shows the relationship of frequency of vibration when
the tension is kept constant and used different sizes of strings. From the
experiment, it can be observed that as the linear mass density increases, the
computed frequency of vibration decreases. Moreover, as the table
presented, it was given in the table that the specific linear mass density of
the specific diameter of wire and considering the values, as the diameter of
wire increases, the linear mass density also increases due to the fact that
there is a direct relationship between the two parameters.

The experiment can be compared to a guitar. The pitch or frequency

of a guitar string also depends on its thickness and length. When the length
of a guitar is changed, it will vibrate with different frequency. Shorter
strings have higher frequency and therefore, higher pitch. On the other
hand, the longer the string, the lower the pitch is. So, when a guitarist
presses his/her finger on a string, he/she shorten its length. Thus, the
more finger he/she adds to the string, the higher the pitch will be made.
Meanwhile, the thickness of a string may also refer to its diameter. Thick
string with large diameter vibrates slower and have lower frequency than
thin ones which produces tones that are one octave higher the thicker ones.