Ch 12: Sound

Medium vs velocity

• Sound is a compression wave (longitudinal) which needs a

medium to compress. A wave has regions of high and low

pressure.

• Speed- different for various materials. Depends on the

elastic modulus, B and the density, ρ of the material.

• Speed varies with temperature by:

• Here we assume t = 20°C so v=343m/s

v B /

v (331 0.60T)m /s

Sound perceptions

• A listener is sensitive to 2 different aspects of sound:

Loudness and Pitch. Each is subjective yet measureable.

• Loudness is related to Energy of the wave

• Pitch (highness or lowness of sound) is determined by the

frequency of the wave.

• The Audible Range of human hearing is from 20Hz to

20,000Hz with higher frequencies disappearing as we age.

Beyond hearing

• Sound frequencies above the audible range

are called ultrasonic (above 20,000 Hz).

Dogs can detect sounds as high as 50,000

Hz and bats as high as 100,000 Hz. Many

medicinal applications use ultrasonic

sounds.

• Sound frequencies below 20 Hz are called

infrasonic. Sources include earthquakes,

thunder, volcanoes, & waves made by

heavy machinery.

Graphical analysis of

sound • We can look at a compression (pressure) wave from a

perspective of displacement or of pressure variations. The

waves produced by each are ¼ λ out of phase with each

other.

Sound Intensity

• Loudness is a sensation measuring the intensity of a wave.

• Intensity = energy transported per unit time across a unit

area. I≈A2

• Intensity (I) has units power/area = watt/m2

• Human ear can detect I from 10-12 w/m2 to 1w/m2 which is

a wide range!

• To make a sound twice as loud requires 10x the intensity.

• In the open sound intensity follows inverse square law as

you get farther from the source.

Decibels, dB

• We measure sound intensity levels, β using a logarithmic scale having a unit of bels.

• 1 decibel, dB is 0.1 bel (10dB =1bel)

• I0 is the minimum intensity audible (1x10-12 W/m2) and the log is base 10, so we define intensity level as

• The threshold of hearing would be 0dB

• Each 10dB increase is 10x the intensity. (20dB = 100x intensity of 0dB; 30dB = 1000x >0dB)

10logI

I0

Vibrating

Strings • Recall how standing waves are made on a string…

• Fundamental- pitch determined by lowest resonant frequency

where nodes are only at the ends. Here λ=2L so v=f*2L on the

string.

• Strings on some instruments are the same length, but sound

different b/c each has a different m/L which effects velocity:

• Adjusting tension is how we tune the instrument

• Velocity on a heavy string is less and f will be less for the same λ.

v FT /(m /L)

Ex 12-6 Piano strings

• The highest key on a piano corresponds to a frequency

about 150 times that of the lowest key. If the string of

the highest note is 5.0cm long, how long would the

string for the lowest note be if it had the same mass per

length and was under the same tension?

• Solution: The velocity would be the same on each

string so the frequency is inversely proportional to the

length L of the string (f=v/λ= v/2L)

Solution cont’d

• Thus where the subscripts L and H refer

to low and high notes. It follows that

• This would be too long for a piano so the longer lower

strings are made heavier to avoid this.

LL

LH

f H

fL

LL LH ( fH / fL ) (5.0cm)(150) 750cm 7.5m

Amplification

• Stringed instruments need a ‘sounding board’ to help

amplify the sound by putting a greater surface area in

contact with the air. This produces a stronger wave.

• Other instruments use a column of air to set up a

standing wave.

• The fundamental frequency on a string has antinodes

at both ends and a single anti node. Harmonics or

overtones are whole # multiples of the fundamental.

pipes

• In a pipe the air is vibrating forming the standing wave

• We can discuss the waves by the air displacement OR

by the pressure in the air. (Recall these are separated

by ¼ a λ)

• We can have pipes that are open at both ends (called

open tubes) or that are closed at one end (closed tube).

• Read last paragraph on pg 358- first 2 on page 359

about air displacement on open tubes like a flute.

Pipe diagrams

Ex 12-8 open & closed

organ pipes

• What will be the fundamental frequency and first three

overtones for a 26cm long organ pipe at 20°C if it is

(a) open and (b) closed?

• Solution: At 20°C, the speed of sound in air is

343m/s.

• (a) For an open pipe, the fundamental frequency is

• The next 3 overtones would be 1320 Hz, 1980 Hz,

2640 Hz and so on…

• (b) For a closed pipe, the fundamental is

• The next 3 overtones will only be heard at odd

multiples so we have 990Hz, 1650 Hz, and 2310 Hz

f1 v

2L343m /s

2(0.26m) 660Hz

f1 v

4L343m /s

4(0.26m) 330Hz

Interference- beats

• As two or more waves superimpose, their amplitudes add

algebraically.

• We have constructive interference when waves are in phase

and destructive when out of phase.

• Beats occur when sound waves have frequencies close but

different. As the waves interact a listener perceives sound

levels that alternately rise and fall.

• Beat frequency is found as the difference between the 2 waves.

• A 400 Hz tone and a 396 Hz tone will have a fbeat=4Hz

Doppler effect

• Recall wave speed depends only on medium.

• If a source or listener move relative to a wave, the wave

frequency (and therefore wavelength) can be perceived

to change resulting in a change in pitch.

Perceived frequency

• The shift in wavelength is directly proportional to the speed of the source, vs.

• The new frequency, f ’ will then be:

• for a source moving toward you

• this is called blue shift

• for a source moving away (red shift)

f f

1vs

v

f f

1vs

v

Doppler effect in

space

Your turn to prac

• Please do Ch 12 Review pgs 375-378

• Problems # 4, 29, 36, 48, 55

• 1998 FRQ B5