**Unit 4: Sound and Music**

**Problem Set B**

**Overview:**

Problem Set B targets your ability to analyze physical situations involving the transmission of sound waves through strings, cables, ropes, etc. The first five problems target your ability to relate the speed of a wave to the tension and the mass density of the medium. The last 15 problems target your understanding of the relationships between the length of the medium, the frequency of the harmonics and the speed of the waves (or the properties of the medium upon which wave speed depends).

**Speed of Waves in Strings, Wires, Ropes
and Cables**

The speed of a wave depends upon the properties of the medium through which it is transmitted, and NOT upon the properties of the wave itself. For sound waves being transmitted through strings, wires, ropes and cables, the primary properties effecting wave speed are the tension of the medium and the mass density of the medium. Tension pertains to the force with which the two ends of the medium are pulled tight; being a force, it is expressed in units of Newtons. Mass density pertains to the mass per unit length of the string, wire, rope or cable and is expressed in standard units of kilogram / meter. The equation expressing the relationship between these variables is

where **v** represents the wave
speed and
represents the mass density. When using these equations in problems
#1-5 and subsequent problems, it is important to pay attention to the
units with which the given quantities are expressed and to make
appropriate conversions where necessary. It is recommended that
substitutions be made into the equation using standard metric units.
For speed, use m/s; for tension, use Newtons (abbreviated N); and for
mass density, use kg/m.

**Analyzing Problems Involving the
Wavelength-Speed-Frequency Relationship**

The last 15 questions of the set target your ability to analyze physical situations involving the wavelength-frequency-speed relationship for standing wave patterns in strings, wires, ropes and cables. Any wave, whether a standing wave or a traveling wave, will have a wavelength-frequency-speed relationship which follows the wave equation:

where **v** represents the speed
(or velocity) of the wave, **f**
represents the frequency of the wave, and
represents the wavelength of the wave. As mentioned above, the speed
of a wave is dependent upon the properties of the medium. For
strings, wires, ropes and cables, the properties of importance are
the tension and the mass density. As such, the properties of the
medium are also related to the frequency and the wavelength of the
wave.

Waves introduced into a string, wire, rope or cable will typically
travel the length of the medium and reflect back upon reaching its
end. At certain frequencies, the reflected portion of the wave meets
up with the original wave to create a pattern known as a standing
wave pattern. In a standing wave pattern there are points along the
medium which appear as if they are always standing still. These
points are known as **nodes** and are
easily remembered as the points of
**no**
**dis**placement. Separating the
nodes are **anti-nodes**: points of
maximum positive and negative displacement. In such standing wave
patterns, there is a unique half-number relationship between the
length of the medium and the wavelength of the waves which have
established the pattern seen. These relationships are shown below for
the standing wave patterns having one anti-node (first harmonic), two
anti- nodes (second harmonic) and three anti- nodes (third
harmonic).

It is clear from the above graphic that the length of the string, wire, rope or cable is related to the wavelength of the standing wave which is established within it. As such, the length of the medium is related mathematically to the frequency of the wave and the speed of the waver (or the properties of the medium upon which wave speed depends).

A standing wave pattern such as those shown in the graphic above
is established within a medium only when it is being disturbed at
specific frequencies. Not any frequency will result in a standing
wave pattern; only the discrete frequencies which lead to wavelength
values which are mathematically related to the length of the medium
as illustrated by the equations in the graphic above. Such
frequencies are known as the harmonics of the string, wire, rope or
cable. As observed in the graphic above, the pattern associated with
the second harmonic has a wavelength which is one-half the wavelength
of the pattern associated with the first harmonic. And the pattern
associated with the third harmonic has a wavelength which is
one-third the wavelength of the pattern associated with the first
harmonic. Continuing this same logic, one would reason, that the
pattern associated with the fifth harmonic has a wavelength which is
one-fifth the wavelength of the pattern associated with the first
harmonic. And in general, the pattern associated with the *nth*
harmonic has a wavelength which is 1 / *n* the wavelength of the
pattern associated with the first harmonic. Each harmonic pattern -
whether the second, third, fifth or nth - is characterized by a
wavelength which is smaller than the wavelength of the first by a
factor of **n**. In this case,
**n** is known as the
**harmonic number**. The speeds of
the waves for each of these harmonics is the same. Thus, the decrease
in frequency which results from a progression from the first to the
third (and higher) harmonic must correspond to an increase in the
frequency by the same factor **n**.
That is, the frequency of the second harmonic is two times the
frequency of the first harmonic; the frequency of the third harmonic
is three times the frequency of the first harmonic; and the frequency
of the *nth* harmonic is **n**
times the frequency of the first harmonic. Put in equation form, one
could state that

where **f _{n}** is the
frequency of any harmonic pattern,

**An Effective Problem-Solving
Strategy**

As mentioned earlier, the last 15 questions will probe your ability to relate the frequency, wavelength and speed of waves to properties of the medium and to the length of the medium. The graphic below depicts the relationships between the various quantities in the problems. As is the usual case, when approaching a problem, first identify what you know and what you are trying to find. Locate the knowns and unknowns on the graphic below and plot out a strategy which allows you to determine the unknown quantity. The strategy for solving for the unknown will be centered around the relationships depicted in the graphic. The stated equations provide the mathematical expression of those relationships.

**Additional Readings/Study
Aids:**

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in accomplishing the above tasks.

View Sample Problem Set.

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