**Unit 4: Sound and Music**

**Problem Set C**

**Overview:**

Problem Set C targets your ability to analyze physical situations involving resonating air columns - both open-end and closed-end. There are a variety of skills and conceptual understandings required to be successful in such analyses. Such skills and conceptual understandings include the following:

**Ability to Calculate the Speed of Sound from Air Temperature**The speed of a sound wave is dependent upon the properties of the medium through which it propagates. The predominant property effecting the speed of sound in air is the temperature of air. Increasing temperatures increase the speed at which sound travels. There are a few equations which quantitatively express the wave speed-temperature relationship. One simple equation which works well within the typical range of indoor and outdoor air temperatures is

**v = 331 m/s + 0.6 m/s/C • T**where

**v**is the speed of sound in air and**T**is the air temperature in degrees Celsius.

**Relating Wavelength of Wave and Length of Air Column: Open-End**The basis of all problems in this problem set is a forced vibration causing resonance of a column of air. The air column is either open at both ends (open-end air column) or open at one end and closed at the other (closed-end air column). A resonance situation in an open-end air column is characterized by the presence of an anti-node at each of the open ends, creating the standing wave patterns shown below. Each pattern is referred to as a harmonic and has its own unique wavelength and frequency. As shown in the graphic, there is a distinct relationship between the wavelength of the standing wave and the length of the air column. Knowing the pattern allows one to relate the length to the wavelength and ultimately to the frequency and the speed.

**Relating Wavelength of Wave and Length of Air Column: Closed-End**A closed-end air column is open to the surrounding air at one end and closed at the other end. A resonance situation in a closed-end air column is characterized by the presence of an anti-node at the open end and a node at the closed end, creating the standing wave patterns shown below. Again, there is a distinct relationship between the length of the air columns and the wavelength for each of the harmonics. Knowing this length-wavelength relationship allows one to relate the length of an air column to the speed and the frequency at which the air inside naturally vibrates.

**Using the Wave Equation**Any wave, whether a standing wave or a traveling wave, will have a wavelength-frequency-speed relationship which follows the wave equation:

wherevrepresents the speed (or velocity) of the wave,frepresents the frequency of the wave, and represents the wavelength of the wave. As mentioned above, the speed of a sound wave in air is dependent upon the air temperature and not upon the properties of the wave. Thus, alterations in the frequency for a wave in a non-changing medium result in inverse alterations in the wavelength with no change in wave speed.

**Relating the Frequency of Various Harmonics to the Fundamental Frequency**There is a clear mathematical relationship between the harmonic frequencies of an open- and a closed-end air column and the frequency of the fundamental. Each harmonic frequency is some whole number multiple of the fundamental frequency. Put in equation form, one could state that

wherefis the frequency of a harmonic,_{n}nis the harmonic number associated with that harmonic andfis the frequency of the first harmonic or the_{1}fundamental frequency.

**Understanding Beats and Beat Frequency**When two sound sources of very similar yet different frequencies meet at an observer's ears the phenomenon of beats is observed. This phenomenon is perceived by an observer as a sound which fluctuates in amplitude over the course of time. For instance, if two tuning forks - one with a frequency of 256 Hz and the other with a frequency of 254 Hz - produce sound waves, then an observer would hear a fluctuation in amplitude at a frequency of 2 Hz. This 2 Hz fluctuation in frequency is known as the beat frequency and is equivalent to the difference in frequency between the two sources. The fluctuations in amplitude which are observed is the result of the interference of the two waves. A low amplitude sound is observed when a compression from one source meets up with a rarefaction from the other source. A high amplitude sound is observed when a compression from one source meets up with a compression from the other source (or a rarefaction with a rarefaction).

**Implementing a Good Problem-Solving Strategy**Nearly all the problems in this problem set will probe your ability to relate the frequency, wavelength and speed of waves to properties of the medium (temperature) and to the length of the air column. 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 mathematical analysis of resonating air columns.

Open-end Air Columns | Closed-end Air Columns | Beats and Beat Frequency

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