Investigating the Properties of Sound

Demonstrating the temperature dependence of the speed of sound in air

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

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.

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.

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?

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?

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

Display on oscilloscope showing delay between 2 signals

Apparatus Large Styrofoam cooler Liquid nitrogen Heating lamps (60W) Digital thermometer Two-channel digital oscilloscope Speaker Microphone Battery (9V) with switch

Entire experimental setup (outside view)

Inside the cooler

Boiling liquid nitrogen inside cooler

Halogen heating lamp

Fan to promote air circulation

Digital thermometer

Battery (inside box) with switch and signal splitter

Speaker

Microphone

Digital oscilloscope

Entire experimental setup again

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.

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 travelling compressions and rarefactions in a medium.

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

Number of particles is on the order of 1023 – impossible to calculate the movement of every single particle!

Real method: Same idea, but no need to keep track of every

particle individually. Use probability and statistics to “guess” the

collective behaviour of particles. The branch of physics that uses statistics to

model very large systems is called thermodynamics, or statistical mechanics.

Sound is a statistical mechanical phenomenon.

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

PVK

xkF

V

VKP

Just as a high spring constant corresponds to a stiffer spring, a high bulk modulus corresponds to a less compressible gas – a “stiffer” gas.

PK For diatomic gases

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 gas pressure.

γ ≈ 1.4 for diatomic gases.

The speed of sound in theory A rigorous derivation of the speed of sound from first

principles 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?

Recall from chemistry class the ideal gas law:

TNkPV B where P is pressure, V is volume, N is the number of particles, kB is the Boltzmann constant, and T is temperature in Kelvin.

V

TkNc B

Substituting for P in our previous expression:

Now realize:

MV

mN

V

Therefore where m is the mass of a

single molecule.

m

Tkc B

Substituting in m gives us:

15.273115.273

m

kB

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

15.273 m

kc 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?

The speed of sound vs. temperature in theory

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

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.

The speed of sound vs. temperature in theory

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

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

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

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 beginning

of the microphone signal. Acoustic noise from sounds inside room. Electronic noise from battery, microphone, etc.

The approximations made in the derivation of the speed of sound:

PK and 4.1

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:

Tm

kKc B