Physics of Sound

Logarithms

• Do you know how to use your calculator?

Find the following functions

+ , - , x , / , ^ , log

The log is the exponent to which 10 is raised, representing a number

Antilogarithm

The antilog is 10 raised to the x power, or 10x

Logarithms

• Solve the following together:– Log (20) = ___________

– Log (400) = ____________

• Solve on your own:– Log (0.5) = ____________

– Log 2 = _____________

Antilogarithms

• Solve the following together:– X = 2 => _______________

– X = 4.3 => _______________

• Solve on your own:– X = 8.5 => _____________

– X = 9.0 => _____________

What is sound?

• Any change in air pressure– The molecules in the air exerts a pressure of

over 1 ton per square foot on our ears– Must be a rapid change in sound pressure to

be heard, a small rapid change will create noise

• Travels as sound waves

Frequency of Sound

• Rate at which complete high and low pressure regions are produced by the sound source.

• 1000 cycles per second is 1000 high and low pressure regions passing a point in one second. This is called 1000 Hertz (Hz) or 1 kHz.

Speech frequencies

• Speech frequencies: generally regarded to be 500 to 3000 hertz

• Frequency range of perceivable sound: 20 Hz to 15,000 to 20,000 Hertz.

• Tuning forks

Tone and Noise

• Tuning fork – pure tone and related frequencies– We cannot see the tines

moving back and forth because they are moving back and for the too quickly. Two-hundred cycles a second is too fast to see.

• Noise – random frequencies

Noise travels through a medium

• A vibrating object creates a disturbance that travels through a medium– A train’s noise can travel through the steel

tracks by creating sound waves– The vibrations of a speaker creates sound

waves

• Frequency is the number of complete back and forth vibrations per second

Noise travel

• Vibrational motion of the medium is set up by the object.– The vibrations set the molecule of the medium

into motion.– The motion of the molecule in the medium

sets the molecule next to it, in motion.– The transfer of energy continues as the

vibration of one molecule sets the next molecule into motion.

Sound wave is a pressure wave

• Thus an instrument can be used to measure the oscillations of high and low pressure variations in the pressure.

• These oscillations are shown as the typical sine wave that you may have seen

Wavelength• Distance which a disturbance travels along

the medium in one complete wave cycle.– Measured from one wave trough or crest to

the next wave trough or crest, in a transverse wave. It is from one wave compression to the next wave compression in a longitudinal wave

• With a pressure wave it is from one high pressure region to the next high pressure region

Speed of Sound

• Sound waves are pressure disturbances traveling through a medium by means of particle interaction– How fast the disturbance is passed from

particle to particle determines the speed of sound.

– How easily the medium transfers the disturbance determines the speed, which is measured in feet per second (ft/sec)

Speed of Sound

• Speed is equal to distance traveled per unit time.– Speed = distance/time

• If a sound waves travels 2,300 feet in 2 seconds the sound is traveling at 1,150 ft/sec

• Examples of sound travel would be the time it takes for thunder to reach the observer and an echo

Speed of Sound

• The speed depends on the properties of the medium, the elastic properties are much greater than are the inertial properties.– Thus longitudinal sound waves will travel

faster in solids than in liquids, and longitudinal sound waves will travel faster in liquids than in gases

Speed of Sound in air

• The speed of sound in air will depend on temperature and pressure of the air. The relationship is:

C = 1054 f/s + (1.07 f/s/oF)xT

Speed of Sound• At 72 oF, the speed of sound is 1,130 f/s

• The delay between lightning and thunder.– Light travels 980,000,000 f/s or reaches the

observer in almost no time.

• The time delay of an echo is the same phenomenon, the distance of a reflecting surface can be determined by the time it takes for the echo to return

Speed of Sound

• The speed of sound in different mediums at standard conditions:– In air : the velocity is 1,130 f/s– In water: the velocity is 4,700 f/s– In wood: the velocity is 13,000 f/s– In steel: the velocity is 16,500 f/s

Speed, frequency and wavelength

• The mathematical relationship between the three is:

C = λ x f

The speed is a constant based on the properties of the medium. The length of a wave will vary with the frequency.

Speed, frequency and wavelength

At standard conditions, the speed of sound is 1,130 f/s. Say we have a 440 hertz frequency pure tone sound, what is the wavelength of the sound?

C = λ x f

λ = C / f

λ = 1,130 / 440

λ = 2.57 feet

Speed, frequency and wavelength

• How about air at 1,000 oF as part of an exhaust stream?

• Find the speed of sound traveling through the exhaust?

C = 1,054 f/s + (1.07 f/s)x oF

C = 1,054 + 1.07*1,000

C = 1,054 + 1070 = 2,124 f/s

Speed, frequency and wavelength

• Engine rotating at 3,000 rpm and has 4 cylinders

3,000 * 4 = 12,000 rounds per minute

Which equals 200 rounds per second (Hz)

What is the frequency of this sound?

λ = C / f

λ = 2,124 / 200

λ = 10.62 ft

Speed, frequency and wavelength

• What is the wavelength of the following frequencies? (at standard conditions)– λ 20 hz = ________ remember: λ = C/f

– λ 1000 hz = ________

– λ 16000 hz = ________

Period

• The period is the time for one complete cycle of pressure transition. It is the reciprocal of the frequency.

T (sec) = 1/f

The period of a 1000 Hz sound wave is:

T = 1/f = 1/1000 = 0.001 seconds

Period

• What is the period of a 20 Hz and a 16,000 Hz wave?

T20 Hz = 1/f =

T16000 Hz = 1/f =

Period

• Below 20 Hz – infrasound• Above 20,000 Hz – ultrasound

– Dogs – 50 Hz to 45,000 Hz– Cats – 45 Hz to 85,000 Hz– Bats – to 120,000 Hz– Dolphins – 200,000 Hz– Elephant – down to 5 Hz and up to 10,000 Hz

Sound Waves

• A pure tone (tuning fork) sound introduced into the room will create a change in the molecules in the room. – At 440 Hz the molecules will bunch up every 3

feet– Also there will be a net drift of molecules from

the bunched up section to the section where the molecules are further apart.

– The wave is moving toward me at 1,130 f/s

Sound Waves

• The sound wave is traveling toward me however, the molecules are not moving toward me.– An example would be a garden hose, when I

shake it a snaky wave travels away, however, the hose is not moving only the wave energy is moving along it.

– Other examples would include: sound, water, and football fans

Sound Waves

• Mechanical waves – they require a medium to transfer energy. So sound will not transfer through a vacuum.

• Slinky demo– Pulse and a wave – moves one coil at a time

• Medium is the slinky. In water it is the water, at a concert it is the air, at a football game it’s the fans in the stadium

Intensity

• The amount of energy which is transported past a given area of the medium per unit time– Intensity = energy / (time x area)

• Since power is energy per unit time, it can also be written as:– Intensity = power / area

• Typical units are Watts/meter2

Intensity

• Inverse square relationship– The mathematical relationship of intensity and

the distance from the source

– As you move away from the source (larger distance) the area gets larger and the intensity will decrease.

• If the distance from a source doubles the intensity will decrease by a factor of 4.

Threshold of Hearing

• Humans can detect sound of very low intensity. The faintest sound which the ear can detect has an intensity of 1x10-12 W/m2.

• At this level sound will displace particles of air by a mere one-billionth of a centimeter.

Loudness

• Loudness of a noise is a more subjective response. Factors that affect the perception of loudness includes age and frequency

Sound Intensity

• The average rate at which sound energy is flowing through a unit area

• Intensity can be measured by means of a twin microphone probe, with signal processing by a microprocessor controlled cross correlation spectrum analyzer

• Measurement of intensity is very useful in industrial noise situations

Decibels

• The decibel scale is a logarithmic scale. The logarithmic scale is based on multiples of 10.

• A sound which is 10 times more intense is assigned a sound level of 10 dB.

• A sound which is 100 times more intense is assigned a sound level of 20 dB.

• A sound which is 1000 times more intense is assigned a sound level of 30 dB.

Decibels

• Threshold of hearing 0 dB• Whisper 20 dB• Normal conversation 60 dB• Street traffic 70 dB• Vacuum cleaner 80 dB• Walkman at max setting 100 dB• Threshold of Pain 130 dB• Military Jet Takeoff 140 dB

Sound Pressure and Sound Pressure Level

• Sound pressure is the root mean square (rms) value of the pressure changes above and below atmospheric when used to measure steady state noise.

• The sound pressure level is the ratio expressed in decibels (dB) of the rms pressure to a reference rms pressure.

Sound Pressure Level

• Sound pressure level (Lp) is:

Lp = 10 log (P / Po)2

= 20 log (P / Po)

Where : Po= the reference sound pressure of 2 x 10-5 N/m2

Lp = sound pressure level in dB P = rms sound pressure in N/m2

Sound Pressure Level

For a sound source having a sound pressure of 1 N/m2, what is the sound pressure level in dB?

Lp = 20 log (P/Po)

= 20 log ((1 N/m2)/(2x10-5 N/m2)

= 20 log (0.5 x 105) = 94 dB

Sound Pressure Level

If the sound source has a sound pressure of 2x10-3 N/m2, what is the sound pressure level in dB?

Sound Pressure Level

If the sound source has a sound pressure of 2x10-3 N/m2, what is the sound pressure level in dB?

Lp = 20 log (P/Po)

= 20 log ((2x10-3 N/m2)/(2x10-5 N/m2)

= 20 log (1 x 102) = 40 dB

Sound Pressure Level

Weighted sound levels Fletcher-Munson curves.

Ear is most sensitive around 2 to 5 kHz

Decibel Addition

To add individual sound levels the equation to add these is:

LT = 10log(10L1/10+10L2/10+10L3/10+…+ 10Ln/10)

Example: Two machines each operate at 93 dB at a given location. What is the sound pressure level if both machines are on?

Decibel Addition

Add to sound pressure levels of 93 dB together

LT = 10log(10L1/10 + 10L2/10)

LT = 10log(1093/10 + 1093/10)

LT = 10log(2 x 109.3) = 96 dB

Decibel Addition

Exercise: Three machines have the following sound pressure levels at a given measurement location: 95, 96, 100 dB

What is the resulting sound pressure level if all three machines are on?

Decibel AdditionRule of thumb (can only be used when a limited

number of sources are added together)

0 dB difference add 3 dB to the higher value

1-1.5 dB difference add 2.5 dB

2-3 dB difference add 2 dB

3.5 to 4.5 dB add 1.5 dB

5 to 7 dB add 1 dB

7.5 to 13 dB add 0.5 dB

Decibel Addition

• Using the rule of thumb in the previous exercise, find the total sound pressure level:

Sound Pressure Level in decibels of common sources of noise

See page 38 and 39 in your manual

Examples: refrigerator 50 dB

rainfall 50 dB

doorbell 80 dB