## Sound

#### Sound propagation in gases

Sound is a wave in pressure or density, not a transverse displacement like a wave on a string, but a compression and rarefaction that propagates with a well-defined speed. It is our best example of a longitudinal wave, and this distinguishes it from the vibrations that may originate or accompany it.

Sound waves have these characteristics

Frequency: $$f$$

Amplitude: $$p_0$$ (e.g. maximum pressure increase measured above the average pressure $$p_{av}$$ without the wave)

Speed: $$v_{sound}$$ (depending on the medium)

Direction: $$\hat{n}$$ a unit vector perpendicular to lines of constant amplitude at a fixed time

and they obey the wave equation

$$p = p_0 \sin( 2 \pi f t - 2 \pi x / \lambda)$$

where the wavelength is

$$v_{sound} = \lambda f$$

and $$p$$ is the pressure in the wave above or below the average pressure $$p_{av}$$ without the wave.  Usually the amplitude of the sound wave   $$p_{0} \ll p_{av}$$ and it is such a weak disturbance that the material responds linearly to the over- or under- pressure.  That condition makes the problem solvable with simple mathematics.

The amplitude of the sound wave is constant at a given time  at a position $$x$$ such that  the phase of the wave, that is the angle $$2 \pi f t - 2 \pi x / \lambda$$, is constant.  it seems from the way we have written this equation that  the $$x$$ is a coordinate in space and that the wave would be constant along a plane in which $$y$$ and $$z$$ vary.  That's the special case of a "plane wave",  More often, $$x$$ is just the distance from the source of the sound along the path the wave has taken, and the wave may be constant over a sphere or some other 3-dimensional shape.   Whatever that surface is, the perpendicular to the surface is the direction in which the phase and amplitude is changing most and is the "direction" of the wave's propagation.

The wave is a compression of the medium, and for air, would be recognized by the pressure of the gas.  The wave travels through the gas without moving the bulk of the gas, while displacing the constituent molecules to increase or decrease their density locally.  Since the wavelength is the speed divided by the frequency, for a typical sound wave a 1000 Hz (1 KHz) in air, moving at about 350 m/s,   this compression is on  a scale of under 10 cm (4 inches).  Of course higher frequencies have a smaller spatial scale, and lower frequencies a larger one.

The speed of a sound wave in air depends on the gas temperature and pressure, and is given by

$$v = \sqrt{ \gamma p / \rho }$$

where $$p$$ is the pressure, $$\rho$$ is the mass density, and $$\gamma$$ is  called the adiabatic  index.  Since it is an important factor we have not covered before, we pause to look at what it means.

A gas of atoms of molecules responds to the addition of energy in its heat by changing its temperature, pressure, or volume, and by storing the energy in the translation and internal motions of its molecules.  We treat this behavior by introducing the  specific heat, $$c$$ such that

$$\Delta Q = c m \Delta T$$

is the heat energy added to a mass $$m$$ of material producing a change in temperature $$\Delta T$$.  The value of $$c$$ depends on how the energy is added to the material.  For a gas, we may keep the volume constant, in which case this is $$c_v$$ the specific heat at constant volume.  This would be the case for piston which cannot move, and the heat increases the energy of the molecules in this confined constant volume. Or, we may keep the pressure constant, in which it is $$c_p$$, the specific heat at constant pressure.  That would be the case if we added heat to a gas in a piston, and allowed the piston to move so the gas expanded and did work against the piston. The two are not identical because of thermal expansion and the energy that goes into the work it does.  the ratio is

$$\gamma = c_p / c_v = 1 + \frac{2}{N}$$

where $$N$$ is the number of degrees of freedom the molecule has.  A gas of atoms only has 3 degrees of freedom in the spatial motions of the atoms (i.e. x, y, and z).  In that case $$\gamma = 1 + \frac{2}{3} = 5/3$$.  A gas of diatomic molecules, say pure N2 or O2, has 2 more degrees of freedom in the rotation of the molecules in space.  In that case, $$\gamma = 1 + \frac{2}{5} = 7/3$$.  The value for dry air is about 1.4.

#### Speed of sound in gases and Earth's atmosphere

In air, the speed of sound is approximately 343 meters per second at 20 °C when the humidity is very low.  At this speed and at low altitude, sound covers 767 miles in an hour, or a kilometer in 2.9 seconds.  However the speed depends on density, composition, and temperature. At higher altitude the speed falls until at 10 km it is 300 km/s at the top of the troposphere and where airliner fly long distances.  For mass density $$\rho$$ or the mass of a molecule $$m$$ and temperature $$T$$ in kelvins, with the Boltzmann constant $$k$$

Gas:  $$v_{sound} = \sqrt{ \gamma p / \rho } = \sqrt{ \gamma k T / m }$$

In dry air at low altitude where the Ideal Gas Law holds, there is no dependence on pressure!  There is a large variation with temperature however, and this is approximately

Air:   $$v_{sound} = 331.3 \sqrt{ T / 273.15 }$$ m/s with T in kelvin

#### Measure the speed of sound

A simple direct way to measure the speed of sound in air is to send a pulse of sound out to a surface that is a known distance away and measure how long it takes for the sound to be echoed back. In this experiment we have recorded the sharp impulse of a two blocks of wood on one another and the echoes from objects around it, including a concrete block wall 32 meters away. The sound propagates out to the wall and comes back to the sensor and we identify the return pulse and the time taken. The distance traveled is 32 meters taken twice.  This is an aerial view from Google Maps of the site where the measurement was made.

The sound source and the microphone recording the sound were at the  weather station which is at the top of the photograph.  Sound propagated to the back wall of the observatory where it reflected again to the same microphone.  This the recording has the initial pulse and the return pulse, as well as sound reflections from all the other structures and trees at the site.    We have a precise ground measurement of the distance  with a steel measuring tape, as well as a distance taken off of Google Earth that agree.  It is 32.0 meters from the microphone and source to the wall, or 64.0 meters round trip.

You may be able to listen to the sound at this link depending on your computer's resources and the browser you are using:

The first sounds you hear are rustling of turning on the recorder and preparing to strike two wood blocks together to make the impulses.  After that is a clear strike, and it is followed by several others.  The echos are too fast to discern by ear.

The digital data measuring the variation in air pressure from the sound wave was captured and is in the interactive plot you will see at this link in a separate window, or use the embedded version below it.

Move the mouse cursor over this plot and use the tools at the upper right to zoom, pan, and inspect the data.

This video shows the same plot and plays the sounds of this impulse and several others.

We will analyze this data to find the speed of sound in air at the time this measurement was made on the afternoon of November 2, 2020.  The environmental conditions have a small effect on sound speed and these were

• Air temperature 10.6 °C
• Humidity 25%
• Pressure at site 1.012 kPa
• Wind  SW at 8 km/h but the site is sheltered by forest with no wind at ground level

Sound speed in air depends primarily on temperature and is, at low levels of the atmosphere, largely independent of air pressure.  However the measurement of air pressure raises the interesting question of how it depends on altitude and varies with time.  These web pages that may be helpful to understand this

Conversion of Earth's atmospheric pressure from and to sea level to pressure at an altitude
National Weather Service station records  for Louisville, Kentucky
Atmospheric pressures on other planets
Wikipedia on the "bar" unit and atmospheric pressure

There are two issues to consider in this regard, the conventional temperature for quoting air pressure and its dependence altitude.  Chemists and physicists use "standard temperature and pressure" in Système International (SI) units  for air to be  at  $$0 \, ^\circ C, \; 273.15 \, K, \; 32 ^\circ F$$) and $$10^5 \; pascals, \; 100 \, kPa \; or 1 \, bar$$. However in Imperial units, used by some engineers and largely only in the United States, this may be  $$60 \, ^\circ F, \; 15.6 \, ^\circ C$$ and $$14.696 \, psi, \; 1 \, atm, \; or \; 1.01325 \, bar$$.  Some caution or thoughtful explicitness is advised when the situation is critical. An example of that is the second issue, which is the dependence on altitude and the use of atmospheric pressure to measure altitude in small aircraft.  In that case the pressure is referenced to the pressure at ground level and it is directly converted to an "altimeter" reading indicating height above the ground.  That's obviously not where you would want to make an error or forget the sign!

Now to our case, the National Weather Service reported  pressure at this time for the their station closest to our site as a "station pressure" of 29.88 inches of mercury,  a "sea level pressure"  of 1030.8 millibar,  and an "altimeter reading pressure" of 30.43 inches of mercury. The inches unit refers to the height of a column of mercury in the barometers of the last century which remains the unit used by NWS and aircraft. The millibar is thousands of a bar, so in SI units we would have a station pressure 758.952 mm of Hg and  sea level pressure of 1.0308 bar for the atmosphere at our site at the time the data were taken.  The other pressures are reduced to sea level and a reference  used by aircraft. These both use the observed conditions  to remove the effects of elevation from the pressure readings and estimate the pressure it would be at sea level at that moment.  The dependence of pressure on temperature in the atmosphere obviously affects this conversion, and the accuracy  of pressure as an altitude indicator.  In our case the air pressure is 758.852 mm Hg or 1.0117196 kPa, taking the SI unit of bar as 1 kPa.  The sea level pressure, which we do not need,  is higher because we are at a higher altitude. We will not correct for the difference between the elevation used by the National Weather Service for its station is 500 feet or 152 meters and the elevation of the observatory where our data were taken that is  nominally 215 meters.

#### What to do

1.  Given the temperature at this time, what is the expected speed of sound in m/s when the sound impulse measurement was made?

2.  Listen to the sound recording and pay attention to the first whack which is the the one accurately digitized above.  Given the speed of sound that you calculated, and assuming a distance of propagation of 32 times 2 meters, what requirements on the sound pulse would you place to make a measurement of the transit time possible?

Of the cases on the sound recording, the first one is shown here.  The other impacts were not as clean with flat surfaces of the wood meeting face on.  In this plot of the sound pressure amplitudes for this event, you can see in the unmagnified figure that there is an impulse that occurred near the beginning when two blocks of wood impacted.  Nearby structures produce multiple echoes after that impulse..  The scruff before the impusle is sound from the motion prior to impact.  We will measure from the leading (earliest) edge of the impulse the duration  from that start to when the pressure wave arrived back at the sound recorder after reflecting off of the main building's back wall. You can see that event as second pulse as a small blip at about 0.2 seconds in the data.  Use the plot tools to magnify and explore the data which has rich detail that is not apparent in the overview.  The tools are selected from the upper right side of the interactive plot.

3. At what time in seconds did the "start" pulse arrive at the sound recorder?

4. At what time in seconds, as precisely as you can determine, did the echo pulse from the observatory wall return to the recorder?

5.  The distance from the recorder to the wall is 32 meters.  Use the time you found and calculate the speed of sound in air.

6.  Examine again your determination of the start and return events.  Estimate for each one the minimum and and maximum time give the uncertainty in locating a precise moment.  With those in mind, what are the minimum and maximum transit times including this uncertainty?  You can express this as at time plus or minus a possible range.  If you were to repeat this measurement for many events and make unbiased determinations of the times this uncertainty should be the standard deviation of the measurement.  In this case we only have one example and the errors are in our determination of the points of interest from that.

7.  Given what you found in 6, what is the uncertainty in your measurement of the speed of sound such that you have a measured speed and a plus or minus estimate of error.

8.  What would happen to the speed of sound with a 10 kelvin change of temperature either way?  That is if the temperature were 0 Celsius or 20 Celsius, based on your measurement and the theory of dependence on temperature, which one would be faster and what is their ratio?  Do some online research and see what the effect of humidity is on sound speed in air too.  Comparing 50% humidity to 100% humidity at the same temperature, and the ratio you found here, which has a bigger effect?

9.  Inspect the beginning of the sound impulse just after impact and you'll see a sinusoidal oscillation that is the vibration of the block resonating at a dominant frequency.  Look at it carefully, find the the period of the oscillation (peak to peak) by timing several cycles and dividing the time interval by the number of cycles.  The frequency is 1/period.  What is the frequency of the oscillation.

10.  We can determine distance by using the difference between the arrival of a light pulse or another indicator of the occurrence of an event and the later arrival of the sound from it.  You have surely noticed that if you are a fireworks display the flash comes first, and then the boom from the explosion.  How long would it take for a large sound impulse to travel all the way around Earth and return to its starting point?  Low frequency sounds, so-called infrasound, will do this and are used to monitor for nuclear. The meteor impact at Chelyabinsk in 2013 was detected days later by sensors when the sound of the impulse had made trips around the globe.

Here's the shock wave from it recorded in real time by a local video camera.

#### Methods

This is for those who may be interested in experimenting with sound or in how this measurement was made.  The observatory site is in Oldham County, Kentucky, near Louisville.  You can visit it on Google Earth at this link.  Data were taken by standing at the weather station 32 meters from the back wall and creating a sudden sound impulse by knocking two wood blocks face to face.

The sound measurement was made with an Android cell phone using its internal microphone and sound digitizing electronics.  The capabilities of modern cell phones for physics measurements are there for us to exploit, and with the right software they can detect very subtle effects from motion, light and sound.  In this use we had a program "Visual Audio" installed. The code is from "Now Instruments and Software" and it:   digitizes data at 44,100 measurements per second, determines the relative change in pressure at the sound sensor in calibrated units over a 16-bit dynamic range, saves the data in its original form (which we use here) and also produces visual plots of frequency and amplitude, as well as logarithmic plots that allow rapid real-time audio analysis.  The company that produced the software makes  its code available for free for both Android and IOS in case you may want to try some experiments on your own. There are other similar programs available too.

VisualAudio for Android uses the microphone and audio system of the phone

VibSensor vibration analysis for Android uses the accelerometer of the phone

The interactive display of data is done with a code called "plotly" and might be of interest to those who are learning programming and may need tools for visualizing data in their own projects.