# Seven experiments to measure the speed of sound

Updated: Feb 20

Calculating the speed of sound is one of the easiest experiments to perform with a smartphone. For students, this calculation is very satisfying, because although the sound wave is an often abstract concept, its many physical properties can be easily studied with a device that everyone has in their pocket. In this article we propose seven different experiments to compute the speed of sound with a smartphone.

**1. Sound waves and Acoustic velocity**
A sound wave is a mechanical vibration that propagates through a medium, such as air or a liquid. The speed of sound is the speed at which this wave propagates in this medium, it depends on the temperature, the pressure and the density of the medium through which it propagates.
In air, if we assimilate it to a perfect diatomic gas, we can calculate the speed of sound by the equation: c = sqrt(γ*RT/Ma),
c, the speed of sound,
γ, the ratio of heat capacities at constant pressure and volume. γ= 7/5 for air,
R, the ideal gas constant,
T, the absolute temperature of the medium,
Ma, the molar mass of air: Ma = 29g/mol.
Using previous formula we calculate the theoretical speed of sound at the usual conditions of temperature and pressure: c = 343 m/s for a temperature of 20 degrees, or approximately 767 miles per hour. In water, sound travels more than 4 times faster than in air, i.e. at about 1,482 meters per second, and in some metals like soft iron, it travels 15 times faster fast at around 5,960m/s (13,333 miles per hour).

**2. The different ways of measuring the speed of sound with a smartphone**

There are many different ways of measuring the speed of sound using a smartphone or a tablet. These methods fall into three broad categories, which, interestingly, use different characteristics of sound waves :

1. __Measuring the time it takes for sound to travel a certain distance__

2. __Measuring wavelength at a given frequency using interference experiments__

3. __Measuring the frequency in an Helmholtz resonator__

We offer for each category a different activities that can be carried out in class or at home using one or several smartphones.

**3. Measuring sound waves time of travel**

Like any speed calculation, the objective of this method is to determine the time it takes for the sound wave to travel a certain distance. The speed of sound being high, the measurement of time requires specific equipment: the sound stopwatch, or acoustic stopwatch. A sound stopwatch measures the time difference between two sounds whose sound level exceeds a certain threshold. This device is not on a lab bench but many smartphone applications exist that offer this functionality. In FizziQ, we preferred to let the students build it with triggers. This very simple construction also seems more didactic to us and triggers have many other applications as well.

The traditional protocol for measuring the speed of sound with a sound stopwatch is as follows: two cell phones are separated by a certain distance (at least 5 meters), and an operator is placed near each telephone. The operators take turns clapping their hands, starting and stopping the two audible stopwatches. We check that the time difference dt between the two stopwatches is dt = 2*d/c, where d is the distance between the smartphones, c the speed of sound. This experiment allows an accuracy between 5 and 10%, and can be improved by performing several measurements. An opportunity to revise the course on statistics !

This protocol works well, but is often difficult for students to understand because calculating the offset is not intuitive. We prefer a variation of this protocol developed by Aline Chaillou of the La main à la pâte Foundation and which you can consult on this video. In this second protocol, we start by synchronizing the chronometers by putting them side by side and we trigger the sound chronometers by clapping our hands. Then one of the two portables is moved by a distance d. The operator located near this second laptop then stops the two stopwatches by clapping his hands. The calculation of the shift is then very intuitive for the students because they have immediately put in relation the difference in distance which creates the phase shift with the displacement of one of the two portables.

The time difference dt is equal to: dt = d/c.

This second protocol also makes it possible to introduce the notion of clock synchronization. It is the same concept of synchronization that was used in the famous Hafele-Keating experiment in 1971 to prove special relativity. Be careful to calibrate the trigger level of the sound stopwatch so that it does not trigger when you move one of the two smartphones.

Watch our vidéo :

## 4. Measuring sound wavelength using interferences

This second type of protocol is based on measuring the wavelength of a pure sound of known frequency. We deduce the speed by the relation: c = l.f, with l the wavelength and f the frequency.

The method often used in the physics laboratory uses a sound source and two microphones placed at a certain distance from this source and connected to a double input oscilloscope. By moving the two microphones relative to each other, we find the distance for which the two waves are in phase, which is the wavelength. With a smartphone, this manipulation is not possible because it does not have a double entry... but with a little imagination you can find other ways!

The first protocol that we propose consists in using two smartphones that emit the same pure sound, for example at a frequency of 680 hertz. By placing the smartphones at a certain distance, we will calculate the places of addition and cancellation of the two sound waves. With FizziQ you can use the sound at 680 hertz from the sound library. These two smartphones are placed about 3 meters from each other. With a third smartphone, we measure the sound intensity (oscillogram instrument on FizziQ) along the axis of the two smartphones. The interference of the two waves creates zones of very high intensities, the antinodes, and other very weak ones, the nodes. The distance between the nodes (about 50 cm) is equal to the wavelength of the sound wave for the frequency 680 hertz. By measuring the difference between the nodes (or the bellies), we calculate the speed of sound.

This experience also opens the discussion on how active noise reduction headphones work by carrying out a small activity: __https://www.fizziq.org/en/team/noise-cancellation__

The experiment can also be carried out with only two mobile phones. One of the two smartphones then serves as a transmitter, and also as a tool for measuring the sound volume. A second mobile that emits a pure sound of the same frequency is approached to the first, and the distance between the knot and the belly is noted by measuring the sound volume on the first smartphone, identified by the variations in intensity. To carry out this experiment with FizziQ, we prefer to use the sound intensity measured with the Oscilloscope instrument and which is more precise than the sound volume in decibels.

Finally, if you only have a smartphone, it is also possible to carry out this experiment by placing a reflective surface in place of the second smartphone from the previous experiment. The precision is further reduced but the calculation is nevertheless possible!

These different experiments make it possible to calculate the speed of sound with an accuracy of about 10%.

**5. Measuring sound waves frequency in Helmholtz resonators**

The third method of calculating the speed of sound that we study is based on the principle of acoustic resonance, which is a phenomenon in which an acoustic system amplifies sound waves whose frequency corresponds to one of its own frequencies of vibration. The resonance frequencies of certain cavities like a cylinder or a bottle are easy to determine and depend on the speed of sound and the shape of the object. By measuring the resonance frequency, for certain types of cavity, we can thus deduce the speed of sound.

A very simple first protocol consists of blowing on the edge of a graduated cylinder to emit a sound whose fundamental frequency is measured. For a closed tube, the fundamental resonance frequency is:

where L is the length of the tube, R the radius and a is an adjustment depending on the diameter of the tube: a = 0.62.R.

By measuring with the frequency meter of the application the fundamental frequency emitted by the tube, we can deduce the speed of sound. To make more precise measurements, we can measure the frequency for different heights of water in the test piece, and by doing a linear regression of the results, we can accurately determine the speed of sound to less than one percent.

If you are a Bordeaux lover and have an empty bottle, you can use a bottle from this region whose volumetric characteristics are immutable. Ulysse Delabre in this video details the calculations for measuring the resonance frequency when blowing into the bottle.

What if the bottle is unopened? It is still possible to carry out the experiment and, paradoxically, in an even simpler way: by uncorking it! When the cork is removed, a "pop" is heard which is due to the resonance of the air in the part between the liquid and the top of the bottle. If we measure the frequency of pop with the frequency meter, we can use the previous formula of the resonant frequency of a tube to deduce the speed of sound.

A last protocol always surprising for the students uses the fact that if several frequencies are emitted simultaneously in a cavity, the harmonics of the resonant frequency of the cavity will be amplified compared to the other emitted frequencies. If we measure the spectrum of a white noise emitted in this cavity, the harmonic frequencies of the resonant frequency are highlighted compared to the others. It is recalled that white noise is a random succession of sound emitted in all frequencies. White noise can be found in FizziQ's sound library.

So let's take a tube open at both ends, such as a paper towel roll or a vacuum cleaner hose. At one end of the tube, we will emit a white noise that can be generated with the FizziQ sound library or by using the sound of a video emitting white or pink noise. At the other end of the tube, we measure the frequency spectrum. Measuring the white noise spectrum through a tube will show peaks for the fundamental frequency and its harmonics. We deduce the resonance frequency then the speed of sound by the formula of the resonance frequency of an open tube.

Better results are often obtained with pink noise, which is similar to white noise, but with a reduced loudness for high-pitched sounds. The use of pink noise makes it possible to reinforce the intensity of the fundamental resonant frequency compared to its higher harmonics.

Finally, one can make different measurements with different sizes of the tube, and deduce c by measuring the slope on the graph.

**6. To conclude**

We have proposed different different experiments to calculate the speed of sound. These experiments can be classified into three categories that appeal to different properties of sound waves. All of these experiments can be done with FizziQ, or with other mobile or tablet apps, depending on your preference. The smartphone is one of the best tools available for measuring the speed of sound, offering multiple ways to approach the same problem, and easily accessible to students. Happy experimenting!