Calculating the speed of sound becomes a simple and engaging task with the use of a smartphone, transforming this commonly abstract concept into a tangible learning experience. For students, this exercise is particularly gratifying as it demystifies the complexities of sound waves, allowing them to explore its various physical properties through a device that's a familiar staple in their everyday lives. In this article, we introduce seven experiments, each utilizing a smartphone and the __FizziQ app__, to determine the speed of sound, making science interactive and accessible.

**Contents**

__Sound waves and their propagation__ - __Methods for measuring the speed of sound__ - __Measurement by time of flight__ - __Measurement by wavelength__ - __Measurement by resonance frequency__ - __Conclusion__

**Sound waves and their propagation**

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 the previous formula we can 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 significantly faster at close to 6,000 m/s (13,333 miles per hour).

**How to measure the speed of sound with a smartphone?**

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

These are the methods that have been used by generations of scientists to determine the speed of sound:

➡️ **Marin Mersenne,** the first, evaluated in 1635 the speed of sound in air at 448 m/s by the propagation time method. Value further refined by the scientists Viviani and Borelli in 1656 with a value of 344 m/s.

**➡️ Isaac Newton took a different approach through an analytical method** by determining it from the resonant frequencies of sound waves in a U-tube and details his method in the first edition ofIt begins (1687).

➡️ Over the centuries, as estimations were more accurate, one uncertainty remained: could humans go faster than the speed of sound? This question will be resolved in 1947 when **American aviator Chuck Yeager reached Mach 1 aboard the X-1 aircraft**. Once again, the human had crossed an impassable barrier.

Now, the iconic measurement of the speed of sound is readily accessible to anyone. You can delve into these experiments using one or more smartphones, with no need for specialized equipment. Embark on a journey of discovery right at your fingertips, grab your cellphones and let's dive into this exciting venture!

**Measuring the Time of Flight (ToF)**

Like any speed calculation, the objective here is **to determine the time it takes for a sound wave to travel a certain distance**. The speed of sound being high, the measurement of time requires a specific equipment: an acoustic stopwatch.

An acoustic stopwatch measures the time difference between two sounds which sound level exceeds a certain threshold. This device cannot be found on a lab bench but many smartphone applications exist that offer this functionality. In FizziQ, you will find the acoustic stopwatch in the Tools menu. You can also __build your own acoustic stopwatch using triggers__.

**The traditional protocol for measuring the speed of sound **with an acoustic stopwatch is as follows: two smartphones are separated by a certain distance (at least 5 meters), and an operator is placed near each telephone. The operators clap their hands one after the other. The first clap starts both stopwatch and the seconds stops them. Students then 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 do a bit of statistics as well !

The protocol works well, but younger students find it **often difficult to understand the offset formula calculation which is not very intuitive.** We prefer a variation of this protocol developed by Aline Chaillou of the La main à la pâte Foundation.

In this second protocol, **we start by synchronizing the chronometers **by putting them side by side and trigger the sound chronometers by clapping our hands. Then, we move one of the two smartphone by a certain distance d without making noise. An 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 smartphones.

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 relativity. Be mindful 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 :

**Measuring the speed of sound with Helmholtz resonators**

The second method of** calculating the speed of sound 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 cylinders or a bottles are easy to determine by calculus. This frequency depends on the speed of sound and the shape of the object. By measuring the resonance frequency we can infer the speed of sound.

**A very simple first protocol consists of blowing on the edge of a graduated test tube. This emits a sound for which we can measure the fundamental frequency using FizziQ.** For a closed tube, the fundamental resonance frequency is: f₀ = c(4*L+1.6*D), where L is the length of the tube, D is the diameter of the tube.

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 which always surprises 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 sounds 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 : f₀ = c(2*L+1.6*D)

**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. Examples of of pink noise can be found on internet.

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

**Measuring the speed of sound with waves interferences**

This third 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.

This method is the one usually used in the school labs. It uses **a sound source and two microphones placed at a certain distance from this source and connected to an oscilloscope with a dual input. **By moving the two microphones relative to each other, the operator finds the distance for which the two waves are in phase, which is the wavelength.

With smartphones, this protocol is not possible because they do not have dual sound inputs... However with a little imagination we 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 along the two smartphones axis where waves add up and places where they cance.

With FizziQ one can use the sound at 680 hertz from the sound library. Two smartphones are placed about 3 meters from each other. **A third smartphone is used to 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 an interesting 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%.

**To conclude**

We have identified a number of different ways to estimate the speed of sound. These experiments can be classified into three categories that relate to different physical 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!**

where is the seventh one u liarrrrrr