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Name: Arpan
Status: student 
Grade: 6-8
Country: USA
Date: Winter 2013-14

I noticed at various places, for example, when my brother and I would walk, we would see that his 1 step after which 1 of my steps would come but after a few seconds and that the difference in our step would decrease until our steps would be in sync; but then it would again gradually decrease to his 1 step and before his second step my first step. How does this work?

Pretty cool, isn't it? It happens because your brother's stride is not exactly the same length as yours. A similar thing happens when you are walking on a sidewalk made of separately poured blocks of concrete, if the length of a block is not exactly twice (or three times, etc.) the length of your stride. Also when two swings on a swing set have different length chains. This kind of thing happens all the time in nature, and it is usually most noticeable when two frequencies are close but not exactly the same. It is responsible for the beating sound you hear when two guitar strings are almost the same pitch, for example.

Tim Mooney

Form what you say in your question, your brother must have a shorter step than you do.

Your brother must take more steps to cover the same ground than you do. Let us say he takes 4 paces to cover the ground that you cover in three. Every time you have taken three full steps, he has taken an even 4 and you are back in sync. Look at the chart below. Start with the step where both of you are in sync

You have taken 1 step Your brother has taken 1 1/3 steps

" 2 " 2 2/3

" 3 " 4 and you are back in sync.

Now the difference in steps is really a lot less for you and your brother. Maybe he takes 21 steps to your 20. But every time you complete 20 steps he has taken a full 21 and you are in sync again.

One other place you see this is when the turn signal in you car is on and you see the light flashing from the car in front. Every once in a while, the click of your signal comes exactly at the same instant the light flashes. It then slips out of sync until a bit later and, bingo it's in sync again!

The technical term for what you are seeing is harmonics.

Hope this helps. Bob Avakian Oklahoma State University Institute of Technology

Hi Arpan,

Thanks for the question. What you are describing is the phenomena of beats. Beats, not to be confused with the tasty purple vegetable, occur when two different frequencies (or patterns) come into alignment and then go out of alignment. Because these two different frequencies are periodic, so the alignment happens regularly and in a predictable manner. If you look on the Internet, you should be able to hear some examples of beats with tuning forks or blowing over glass bottles filled to different heights with water.

I hope this helps. Thanks Jeff Grell

Hi, Arpan:

Thank-you for your excellent question. If you observe other things in the physical world as you get older, you will see that your observation of walking steps is similar to many other things. I will mention a couple of examples later.

Usually a taller person will take longer steps than a shorter person. A normal walking speed is about 3 miles per hour. This is about 80 meters in a minute.

If the taller person covers 80 centimeters per step, that means that they will take 100 steps per minute to cover the 80 meters.

If the shorter person covers 70 centimeters per step, that means that they will take

114 steps per minute to cover the 80 meters.

When the taller person takes 7 steps, he or she will walk 560 centimeters or 5.6 meters. The shorter person covers the same distance with 8 steps in the same amount of time. So every 8 steps of the shorter person happens at the same time and place as every 7 steps of the shorter person. This picture is a simple illustration: (see attached .pdf file or .png file)

In your situation, the two step lengths are probably closer than 70 cm and 80 cm. If the shorter person has a step size of 78 cm instead of 70 cm, it will take more time and more distance between the steps that happen together. This would be more difficult to show in a picture.

If you can get some assistance from friends or a music teacher, there are a couple of things which you might be able to try in school:

Music can demonstrate a similar principle. Do you know someone who plays a violin or cello for example? A violinist can play two notes at the same time. If those two notes are almost the same, you can hear a "beat" between the two notes. This "beat" is similar to the difference between your steps and your brother's steps.

If you can borrow two metronomes and place them next to each other, you can set them to operate at rates which are almost the same. Gradually the two will tick at the same moment, and then at different moments and back. This will be another demonstration of the same principle.

Many years ago, airplanes had two propellers. If the two propellers rotated at slightly different rates, you could hear a slow "beat" between them.

Best regards, Bob Zwicker


It is difficult to give a definitive answer to this question as two human beings may have a variety of reasons to synchronize their steps (or to not). There is a large component of psychology involved with human behavior, and so it may not be a straightforward physics-based answer. Why do you smile when someone smiles at you? Why do you raise your hand to meet that of someone holding theirs out to greet you? Trying to break this down into a formula may not be very explanatory or even useful.

Now, your question is reminiscent of a physics problem that can be explained. If you have two pendulums swinging, each at slightly different frequencies, they will tend to synchronize as long as they are connected in some way. We call that "coupling." If we think of a rubber band connecting these two pendulums, we can imagine that the pendulum swinging slightly ahead of the other will pull its partner forward. Conversely, the pendulum falling slightly behind will pull back on the other pendulum slowing it down slightly with every swing. Over time, the two pendulums will swing together. The rubber band represents "coupling," and it can come in different forms.

The idea of the pendulums swinging and synchronizing represents the problem of "coupled oscillators." The oscillation is simply the pendulums swinging back and forth. We can express the oscillation using mathematics (based on a physics model), and show how over time the two pendulums (two oscillators) synchronize when they are swinging almost together but not quite. This problem shows up in many systems from oscillating atoms to ticking clocks. In fact, there are many articles written on this subject (try "coupled oscillators"), and you will see that a simple case of synchronizing is not the only one possible.

If you consider your observation with your brother, there may be some connection along these lines, although again, making physics models for human behavior can be difficult. If you tell your brother, for example, that you believe that walking together will always cause you to walk in sync (explaining pendulums), he may simply walk the opposite. How will your model take this behavior into account, in this case?

Kyle J. Bunch, PhD, PE

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