Lesson 5: Musical
Instruments
Guitar Strings
A guitar string has a number of
frequencies at which it will naturally vibrate. These
natural frequencies are known as the harmonics of the
guitar string. As
mentioned earlier, the natural
frequency at which an object vibrates at depends upon the
tension of the string, the linear density of the string
and the length of the string. Each of these natural
frequencies or harmonics is associated with a standing
wave pattern. The specifics of the patterns and their
formation were discussed in Lesson
4. For now, we will merely
summarize the results of that discussion. The graphic
below depicts the standing wave patterns for the lowest
three harmonics or frequencies of a guitar
string.
The wavelength of
the standing wave for any given harmonic is related to
the length of the string (and vice versa). If the length
of a guitar string is known, the wavelength associated
with each of the harmonic frequencies can be found. Thus,
the length-wavelength relationships and the
wave
equation (speed = frequency *
wavelength) can be combined to make perform calculations
predicting the length of string required to produce a
given natural frequency. And conversely, calculations can
be performed to predict the natural frequencies produced
by a known length of string. Each of these calculations
requires a knowledge of the speed of a wave in a string.
The graphic below depicts the relationships between the
key variables in such calculations. These relationships
will be used to assist in the solution to problems
involving standing waves in musical
instruments.
To demonstrate
the use of the above problem-solving scheme, consider the
following problem and its detailed solution.
Practice
Problem
The speed of waves in a particular
guitar string is found to be 425 m/s. Determine
the fundamental frequency (1st harmonic) of the
string if its length is 76.5 cm.
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The solution to the problem
begins by first identifying known information, listing
the desired quantity, and constructing a diagram of the
situation.
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Given:
v = 425 m/s
L = 76.5 cm = 0.765 m
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Find:
f1 = ??
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Diagram:
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The problem statement asks us to
determine the frequency (f) value. From the
graphic
above, the only means of
finding the frequency is to use the wave equation
(speed=frequency*wavelength) and knowledge of the speed
and wavelength. The speed is given, but wavelength is not
known. If the wavelength could be found then the
frequency could be easily calculated. In this problem
(and any problem), knowledge of the length and the
harmonic number allows one to determine the wavelength of
the wave. For the first harmonic, the wavelength is twice
the length. This relationship is derived from the diagram
of the standing wave pattern (and was explained
in detail in Lesson 4). The
relationship, which works only for the first harmonic of
a guitar string, is used to calculate the wavelength for
this standing wave.
Wavelength = 2 *
Length
Wavelength = 2 * 0.765
m
Wavelength = 1.53
m
Now that wavelength is known, it
can be combined with the given value of the speed to
calculate the frequency of the first harmonic for this
given string. This calculation is shown below.
speed = frequency *
wavelength
frequency =
speed/wavelength
frequency = (425 m/s)/(1.53
m)
frequency = 278
Hz
Most problems can be solved in a
similar manner. It is always essential to take the extra
time needed to set the problem up; take the time to write
down the given information and the requested information,
and to draw a meaningful diagram.
Seldom in physics are two
problems identical. The tendency to treat every problem
the same way is perhaps one of the quickest paths to
failure. It is much better to combine good
problem-solving skills (part of which involves the
discipline to set the problem up) with a solid grasp of
the relationships among variables, than to memorize
approaches to different types of problems. To further
your understanding of these relationships, examine the
following problem and its solution.
To demonstrate
the use of the above problem-solving scheme, consider the
following problem and its detailed solution.
Practice
Problem
Determine the length of guitar string
required to produce a fundamental frequency (1st
harmonic) of 256 Hz. The speed of waves in a
particular guitar string is known to be 405
m/s.
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The solution to the problem
begins by first identifying known information, listing
the desired quantity, and constructing a diagram of the
situation.
|
Given:
v = 405 m/s
f1 = 256 Hz
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Find:
L = ??
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Diagram:
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The problem statement asks us to
determine the length of the guitar string. From the
graphic
above, the only means of
finding the length of the string is from knowledge of the
wavelength. But the wavelength is not known. However, the
frequency and speed are given, so one can use the wave
equation (speed = frequency*wavelength) and knowledge of
the speed and frequency to determine the wavelength. This
calculation is shown below.
speed = frequency *
wavelength
wavelength =
speed/frequency
wavelength = (405 m/s)/(256
Hz)
wavelength = 1.58
m
Now that the wavelength is
found, the length of the guitar string can be calculated.
For the first harmonic, the length is one-half the
wavelength . This relationship is derived from the
diagram of the standing wave pattern (and was
explained in
detail in Lesson 4); it may
also be evident to you by looking at the standing wave
diagram drawn above. This relationship between wavelength
and length, which works only for the first harmonic of a
guitar string, is used to calculate the wavelength for
this standing wave.
Length = (1/2) *
Wavelength
Length = (1/2) *
Wavelength
Length = 0.791
m
If you have successfully managed
the above two problems, take a try at the following
practice problems. As you proceed, be sure to be mindful
of the numerical relationships involved in such problems.
And if necessary, refer to the graphic
above.
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