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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
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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 Microphone – converts sound to electronic signal Battery – outputs electronic signal Speaker – converts electronic signal to sound to channel 2 to channel 1 Oscilloscope – graphs electronic signal against time
Apparatus • Large Styrofoam cooler • Liquid nitrogen • Heating lamps (60W) • Digital thermometer • Two-channel digital oscilloscope • Speaker • Microphone • Battery (9V) with switch
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 Just as a high spring constant corresponds to a stiffer spring, a high bulk modulus corresponds to a less compressible gas – a “stiffer” gas. 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. 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: where P is pressure, V is volume, N is the number of particles, kB is the Boltzmann constant, and T is temperature in Kelvin. Substituting for P in our previous expression: Now realize: Therefore where m is the mass of a single molecule.
Substituting in m gives us: Now realize T = + 273.15, where is temperature in Celsius. Therefore 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: ms-1 Why do we want the expression specifically for nitrogen gas?
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.
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: and
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:
