Chapter 12
The Giancoli onLine Tutor

Sound

We deal here with various aspects of sound. In particular, the major items we deal with are aspects of frequency, intensity and motion. Sound has two aspects, physiological and physical. There is a rough correspondence between the terms both of these fields use in discussing sound:

Physiological	Physical
pitch	frequency
loudness	intensity

Throughout this chapter we will need to know the speed of sound. When the Bulk Modulus for air is factored into the speed of sound formula the result is

v_sound = (331+.6*T) m/s,
where T is the air temperature in Celsius.

I.Intensity and Loudness

The intensity of any wave is proportional to the square of the wave amplitude, and it has the units of watts/m². the human ear has a tremendous range of sensitivity to sound, varying from the threshold of hearing at 10^-12w/m² to the maximum tolerable, 1 w/m².

The loudness the ear/brain hears is not proportional to the intensity of the sound. As a result we define a new physics concept called intensity level.

Working with Logarithms

Intensity level is defined in terms of logarithms, specifically the log to the base 10 usually denoted by Log. There are other logarithms specifically the "natural log" or the log to the base "e" (e=2.718...) usually denoted by ln. There should be no mystery to the log function.

If 10^a=x then Log(x)=a

If e^a (or 2.718^a) =x then ln(x)=a

My birthday is October 3 that is to say 10/3, if I wanted to I could define a log to the base carl by the following

Interactive Example
If 103^a=x then log_carl(x)=a.

This log will work just as well as any of the others. See there really is nothing to it! Just to check - What is the Log to the base 2 of 16? Enter your answer and click outside.

All Logarithms, log₁₀, ln or madeup ones have the following properties:

log(1)= 0

log(ab)= log(a)+log(b)

log(1/a)= -log(a)

log(aⁿ)= nlog(a)

Intensity Level

Intensity Level is denoted by lower case greek letter beta, ß and is given by

ß = 10Log(I/I_o),

where I_o is the threshold of hearing and beta is given in decibels (db).

Interactivity Activity
Enter a value for I in watts/m² from 1e-12 (this is how you write the number 1x10^-12 in JavaScript ) to 100 here:

and press this button:

and get the result in decibels here:

Verify with your calculator that you can get these results.

II. Standing Waves

Just as standing waves in strings serve as the basis for music of string instruments, standing waves in air columns, such as, organ pipes and flutes, form the basis for the music produced by these instruments. There are two separate situations wherein this occurs:

Pipes open at both ends The standing waves in this situation are the same as those in a string.

f₁= (v/2L),
where v is the speed of sound and L is the length of the pipe, and,
f_n=nf₁ (n=1,2,3,...).
Pipes open at one end. Here the situation is different, The fundamental frequency or first harmonic is given by

f₁= (v/4L)
and
f_n= nf₁ where (n=1,3,5,... ).
Notice that the first harmonic (fundamental frequency) is half that as for the open pipe and that only the odd harmonics are present.

Interactive Examples
The temperature is 15 ^oC, what is the speed of sound? Enter your results below and click outside.

What will be the fundamental frequency of a 50 cm open ended organ pipe at this temperature? Enter your results below and click outside.

What will be the fundamental frequency of a 50 cm close ended organ pipe at this temperature? Enter your results below and click outside.

What will be the first overtone of a 50 cm open ended organ pipe at this temperature? Enter your results below and click outside.

What will be the first overtone of a 50 cm close ended organ pipe at this temperature? Enter your results below and click outside.

III. Doppler Effect

The final topic covered here is the Doppler effect. The frequency of a sound is affected by both the motion of the source (speaker) and the motion of the observer (listener). In the following, imagine that a source (of sound) and an observer (listening to the sound) are moving along the line connecting them. Either may be moving at or away from the other.

Definition of Terms
f_o	The actual frequency being emitted by the source if it is not moving.
f	The frequency of the sound heard by the observer.
v	The speed of sound
v_s	The speed of the source
v_o	The speed of the observer

The following formula will always give the correct answer under these circumstances:

f_o

v_o/v

v_s/v

If the observer (v_o) is moving towards the source use the top sign in the numerator.
If the source (v_s) is moving towards the observer use the top sign in the denominator.

Interactive Examples
A source of sound 1000 Hz is moving towards you at 34 m/s and you are moving away from it at the same speed. If the temperature is 15 ^oC what frequency to the nearest hertz do you hear? Enter your results below and click outside.

A source of sound 1000 Hz is moving away from you at 34 m/s and you are moving away from it at the same speed. If the temperature is 15 ^oC what frequency to the nearest hertz do you hear? Enter your results below and click outside.

A source of sound 1000 Hz is moving away from you at 34 m/s and you are moving towards it at the same speed. If the temperature is 15 ^oC what frequency to the nearest hertz do you hear? Enter your results below and click outside.

A source of sound 1000 Hz is moving towards you at 34 m/s and you are moving towards it at the same speed. If the temperature is 15 ^oC what frequency to the nearest hertz do you hear? Enter your results below and click outside.