Speed of Sound Presentation

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Investigating the

Properties of Sound

Demonstrating the temperature

dependence of the speed of sound in air

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Outline

Introduction to sound waves 

The experiment  – measuring the temperature

dependence of the speed of sound

The theory of sound propagation 

Data analysis and discussion of

experimental results

Conclusion 

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 What is sound in physics terms?

 A longitudinal travelling wave. Caused by an oscillation of pressure (the

compression and dilation of particles) in matter.

Other names for sound are pressure waves,

compression waves, and density waves.

Names derived from the motion of particles that carry

sound.

Sound wave animation:

http://paws.kettering.edu/~drussell/Demos/waves/wavemotion.html 

Notice that each individual particle merely oscillates.

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How do we perceive sound?

Pressure waves causes the eardrum to vibrate accordingly.

That vibration is transferred to the brain and then interpreted

as sound.

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Some properties of sound

Volume

 Amplitude of sound wave – how large are the

particle displacements?

Pitch Frequency of oscillations.

Speed of propagation

How fast does a sound wave travel? What factors affect the speed of sound?

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 The experiment

Purpose To determine how the speed of sound is

dependent on the temperature of the medium.

Motivation for this study

Musicians: try playing an (accurately tuned)

instrument in the freezing cold; the intonation will

be completely off.

Effect is most apparent with brass instruments. Then warm the instrument up again without

retuning, the intonation is fine again. Why?

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Schematic of experiment

Speaker   – converts

electronic signal

to sound

Microphone  – converts sound

to electronic

signal

Oscilloscope  – graphs electronic

signal against time

Battery  – outputs

electronic

signal

to channel 1

to channel 2

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Display on oscilloscope showing delay between 2 signals

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 Apparatus

Large Styrofoam cooler Liquid nitrogen

Heating lamps (60W)

Digital thermometer Two-channel digital oscilloscope

Speaker

Microphone Battery (9V) with switch

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Entire experimental setup (outside view)

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Inside the cooler

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Boiling liquid nitrogen inside cooler

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Halogen heating lamp

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Fan to promote air circulation

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Digital thermometer

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Battery (inside box) with switch and signal splitter

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Speaker

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Microphone

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Digital oscilloscope

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Entire experimental setup again

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Collecting the data

Cooler has already been cooled with liquid

nitrogen to approx. -60˚C.

We will periodically pause the lecture and

take a data point.

Turn on battery to send a voltage pulse.

This pulse triggers the oscilloscope to (1) start

reading and (2) freeze graph on screen (pre-set

oscilloscope functions).

Immediately record the temperature.

Use oscilloscope cursors to measure the time

delay between the signals on channels 1 and 2.

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How is sound modeled mathematically?

Sound is a somewhat abstract concept  A sound wave isn’t an object – it’s a type of

particle motion.

That motion can be understood as travellingcompressions and rarefactions in a medium.

Most straight-forward method to describesound is to keep track of the positions ofevery particle that mediates the sound wave.

Number of particles is on the order of 1023

  – impossible to calculate the movement ofevery single particle!

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Real method:

Same idea, but no need to keep track of everyparticle individually.

Use probability and statistics to “guess” the

collective behaviour of particles.

The branch of physics that uses statistics tomodel very large systems is called

thermodynamics, or statistical mechanics.

Sound is a statistical mechanicalphenomenon.

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Important Definitions

Bulk modulus (K)  A measure of the elasticity of a gas; ie. how easily is the

gas compressed?

 Analogous to the spring constant in Hooke’s law 

V 

 P V  K 

 xk  F   

V 

V  K  P  

Just as a high spring constant

corresponds to a stiffer spring, a

high bulk modulus corresponds to a

less compressible gas – a “stiffer”

gas.

 P  K     For diatomic gases

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 Adiabatic process

 A physical process in which heat does not enter or

leave the system. The compression and dilation of air to form a

sound wave is an adiabatic process.

 Adiabatic index (γ)

 A thermodynamic quantity related to the specific

heat capacities of substances.

Here γ accounts for the heat energy associated

with compression, which adds to the gaspressure.

γ ≈ 1.4 for diatomic gases. 

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 The speed of sound in theory

 A rigorous derivation of the speed of sound from firstprinciples in statistical mechanics is much too

complicated.

We need to start somewhere though, so lets begin

with a more easily accessible equation.

 K c  

    P 

The speed of sound is denoted as c by

convention; p is pressure and ρ is density.

So where’s the dependence on temperature? 

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Recall from chemistry class the ideal gas law:

T  Nk  PV   B where P is pressure, V is volume,

N is the number of particles, kB isthe Boltzmann constant, and T is

temperature in Kelvin.

V T k  N c   B

Substituting for P in our previous expression:

Now realize:

 M V    

m N 

V 

  Therefore where m is the mass of a

single molecule.

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m

T k c   B  

Substituting in m gives us:

15.273115.273       

mk  B

  is temperature in Celsius.Now realize T = + 273.15, where 

15.273     m

k c   BTherefore

15.2731353   nitrogenc ms-1

Notice that the first term is equal to the speed of sound at 0˚C.

Lastly, substitute in the correct numerical values and simplify to get:

Why do we want the expression specifically for nitrogen gas?

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The speed of sound vs. temperature in theory

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The (real!) theoretical speed of sound vs. temperature

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 Analyzing our data

Our raw data gives us, at each temperature,the travel time Δt of the sound wave.

To extract speed, divide the distance

between the speaker and microphone by Δt. Distance measured to be 73cm.

Now we can graph the speed of sound

against temperature.

See how closely our data matches up with

theoretical predictions.

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The speed of sound vs. temperature in theory

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Speed of sound vs. temperature in theory (experimentaltemperature range)

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Data set #1 plotted with theoretical speed of sound vs. temperature

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Data set #2 plotted with theoretical speed of sound vs. temperature

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Discussion of experimental errors

Many sources of measurement uncertainty. Distance between speaker and microphone.

Uneven temperature distribution inside cooler.

 Air leakage – escaping nitrogen replaced by normal air.

Oscilloscope screen does not clearly define the beginningof the microphone signal.

 Acoustic noise from sounds inside room.

Electronic noise from battery, microphone, etc.

The approximations made in the derivation of the speed ofsound:

 P  K      and 4.1 

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In summary

What we perceive as sound is actually oscillations

of air particles.

These oscillations are caused by pressure waves

travelling through the air.

Sound waves are mathematically described by

statistical mechanics.

The speed of sound is dependent on the

temperature of the medium carrying it, and obeys

the equation:

T m

k  K c   B