ChemPhys 173/273

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:

v = f •

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 displacement. 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

fn = n • f1

where fn is the frequency of any harmonic pattern, n is the harmonic number associated with that pattern and f1 is the frequency of the first harmonic. The frequency of the first harmonic (f1) is the fundamental frequency; it is the lowest possible frequency at which a standing wave could be established within the medium.

 

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.

 Speed of Sound | Standing Wave Patterns | Harmonics | Mathematics of Guitar Strings

 

View Sample Problem Set.

 

Problem

Description

Audio Link
1

Determination of tension from wave speed and mass density information.

2

Determination of tension from wave speed and mass density information.

3

Determination of tension from wave speed and mass density information.

4

Referring to the previous problem; determination of wave speed using a new tension value.

5

Determination of the mass of a string from wave speed and tension information.

6

Determination of the wave speed from the length, frequency and harmonic number.

7

Referring to the previous problem; determination of the wavelength value for a different harmonic of the same medium.

8

Determination of the tension in a string from the frequency, length and harmonic number.

9

Determination of the tension in a string from the frequency, length and harmonic number.

10

Determination of the frequency from the properties of the medium, the length and the harmonic number.

11

Referring to the previous problem; determination of the fundamental frequency from the frequency of the nth harmonic.

12

Determination of the frequency from the properties of the medium, the length of the string and the harmonic number.

13

Complex analysis relating length, frequency and tension for two different media; requires good thinking skills.

14

Determination of a harmonic frequency from another harmonic frequency.

15

Determination of the wavelength from the length and the harmonic number

16

Referring to the previous problem; determination of the fundamental frequency value from the frequency of a higher harmonic.

17

Determination of the tension in a string from the frequency, length and harmonic number.

18

Referring to the previous problem; determination of a new tension value required to vibrate the string at a higher frequency.

19

Complex analysis involving the comparison of two different strings with different lengths and different harmonic patterns; determine the frequency of a higher harmonic in the second string.

20

Complex analysis involving the effect of a variation in the tension upon the frequency of a string; requires good thinking skills.

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