Understanding resonance

This section is dedicated to helping you to understand resonance.  In separate sections I will discuss how your life relates to resonance, but for now the focus is on understanding resonance.

Do I really need to understand resonance?

You may already know something about resonance.  Just from the introductory sections you probably have a basic understanding.  But there are some very important points about resonance that you may not know, and we will discuss them in this section. You will gain the greatest benefit from this site if you do understand resonance, but most of the advice will make sense even if you don’t understand resonance.

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The actual definition of resonance

You will find a number of definitions on the Internet.  Some refer to the resonant quality of a person’s voice, and others refer to resonance as an amplification at certain frequencies – that’s what we are focused on:

“The increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic force whose frequency is equal or very close to the natural undamped frequency of the system.”

But as you can see, this definition is very complicated and you need a science degree to properly understand it.  So this section will describe it in a far more understandable manner

Let’s get started with the basics

Resonance is all around us.  Thanks to resonance the planets stay in orbit, clocks and watches keep time, microwave ovens heat our food, we can listen to beautiful music, doctors can see inside our body (MRI: Magnetic Resonance Imaging) and so much more.

Your child loves to resonate!

The simplest example of resonance is a child on a swing.  If you give the swing a push it will swing back and forward.   If you just give it one push it will swing back and forth a few times and then come to rest.   (That’s because of friction and damping.)  To keep the swing moving you have to push again each time the swing reaches the closest point to you.  You have to match the frequency of the swing to make it swing high (and to avoid having the swing slam into your hands, or to push just after the swing swings away.

Child pushed on swing by her Mother

But have you ever wondered why on one swing the child will move back and forth more slowly than on another swing?  If you go to a park where the rope on the swing is longer, your child will swing back and forth more slowly.  The natural frequency is lower when the rope on a swing (or pendulum) is longer.

You can experiment with our animation of a pendulum.  You can grab the ball and swing it, or you can click the curved arrow.  As expected, you will see swing back and forth, slow down, and stop.

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Music to your ears

Every musical instrument generates notes based on resonance.

The notes of each guitar string

When you pluck a string on a guitar, the string vibrates back and forth in resonance.  The natural frequency of the string is tuned to the desired note: E, A, D, G, B, and E.  By placing your finger on a string (holding the string against a fret) the string is shortened and thus makes the note higher in frequency.  By making this change (shortening the string) you are changing the resonant characteristics.
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You can change the frequency of the string in other ways too.  If you tighten the tuning machine (also known as tuning pegs, pegheads, and by other names) on the headstock, the note of the string will go higher.  Once again, you are changing the strings resonant characteristics.

Pianos also use strings.  When you hit the keys on the piano, a hammer hits the strings.  Each key on the piano corresponds to a note (A, B, C, etc.), which corresponds with a frequency.

Middle C corresponds to (approximately) 261 Hz.  The note A above Middle C is 440 Hz.

Piano keys and guitar frets

Each string in the piano is either a different length, or it has a different tension to set the frequency of the note correctly.

Wind instruments also use resonance.  We could go into LOTS of depth on a topic like this, but suffice to say that the resonance is established in the tube or pipe.  It is called a standing wave.

The tuning fork

A tuning fork provides another classic example of a resonance.

Three tuning forks that create three sounds

When you give the tuning fork a whack it will make a sound.

Different tuning forks will generate different sounds – you will notice that their “tines” have different lengths.

When you strike the tuning fork, the tines are vibrating side to side – you can’t see them vibrating, but if you touched the tine you would feel them vibrating.

In this animation you can click on the tuning fork and see the tines vibrate in resonance.

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How are you going so far?

I hope you understand what we have discussed thus far.

The key points so far are that resonance is all around us, and we can change the resonance by changing the length of the swing, changing the length of the guitar string, changing the tension of the piano string – and in many other ways.

Consider a spring

A spring can resonate too.  If we attach a ball to the end of a spring, pull it and let it go, the ball will bounce up and down for some time before stopping.  When we do this the spring is resonating.  The rate at which the ball bounces up and down is the resonant frequency, or the natural frequency.

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Of course, if you don’t pull it very hard and let go, it won’t bounce as much, and it will stop bouncing more quickly.

Three special properties of a resonance

Now we are going to dig just a little deeper.  What we learn later about how your life relates to resonance, and how to improve your life, will be easier to understand if you understand everything we cover in this section.

The spring has three properties or characteristics that dictate the rate at which the ball attached to the spring will bounce up and down (i.e. the natural frequency) and the time it takes for the ball to stop bouncing (the damping).  We can change these properties and see what happens to the spring when we give it a tug and make it start bouncing.

Change the mass of the ball

The first property we will look at is mass.  You might normally use the word weight when describing how heavy something is, but technically the correct term is mass.

The spring we have looked at thus far had a ball of a certain weight attached to the end of the spring.  The weight of the ball affects the spring.  If we change the weight of the ball, the natural frequency will change – the rate at which it bounces up and down.  Try this demonstration.  Use the slider to change the weight of the ball.   (Just click on different points along the slider to change the mass of the ball.)

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You will see that when you increase the weight, the natural frequency is reduced.  It bounces up and down more slowly.  But if you reduce the weight, the natural frequency is increased – the ball will bounce up and down more quickly.

Does that mean that my weight affects my natural resonance?

The mass of the spring-mass system does affect the natural frequency of the system.  And while a lot of people are personally affected by their weight, positively or negatively, that’s not the reason why we are discussing mass.  The mass affects the natural frequency and that is the main point to understand for now.

Change the stiffness of the spring

The spring we have used thus far has another property called stiffness.  A spring with more stiffness would be harder to bend or stretch.  Under your car, for example, are springs that are very stiff.  If you held one in your hands you would find it very hard to stretch it.  The springs in your bed a quite stiff – they are hard to bend, but nothing like a car suspension spring.  Whereas the old Slinky toys that you may have owned as a child are not very stiff – they are easy to stretch.

Car suspension spring

Bed spring

Slinky (R) spring













In the following animation you can control the stiffness of the spring.  The spring will appear to change visibly when you change the stiffness.  You will see that when the spring is more stiff, the ball bounces up and down more quickly – the natural frequency is higher.  And when the spring is made to be less stiff, the natural frequency is reduced.

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Change the mass and stiffness

In this animation you can change the mass and the stiffness.

<Spring – control mass and stiffness>

As you may have observed already, increasing mass lowers the natural frequency and increasing stiffness increases the natural frequency – and vice verse.  So you can have a very stiff spring and lots of mass (like a car spring carrying a car), or less stiffness and less mass (like a bed spring carrying a person) and have the same natural frequency.

Child bouncing on bed


If you jump up and down on a bed, or get in a car and jump up and down, the natural frequencies will be similar – even though the spring stiffness and the mass are quite different.

Why is any of this relevant to me?

That’s a good question!

There are really two points to make at this point.  Just as different springs have different natural frequencies, so do different people.  We are all different, therefore the way that one person manages to live in resonance could be different to the way another person lives in resonance.  The challenge you face is to find your resonance and to live in your resonance.  The mass and stiffness affect the natural frequency of objects like springs, but a huge number of factors affect your natural frequency.  But if you can find your natural frequency – look out; your life will change in so many ways.

The second point is that, just as we changed the mass and stiffness of the spring in order to change the spring’s natural frequency, you can make changes to your life in order to change your natural frequency.

The bottom line is that if you are not living in resonance now, you will need to make some changes so that you are living in resonance.  We will discuss those changes in other sections.  But before we do, there are two more topics we need to discuss – they are very important and they make your life even better!

The third property – damping

Damping is very important – possibly the most important quality of the resonance.

If you pull down on the mass and let go, it will bounce for a certain period of time and then stop.  If something is “lightly damped”, it will bounce for a long time.  If there is no damping it will bounce for ever.  But if it is “heavily damped” it will not bounce for a very long time.  In fact, if you have enough damping, the spring will not bounce at all…

In the following example you can change the value of damping and see how the ball bounces as a result.  Just click at different positions along the slider to set a different value, then pull down on the ball and see what happens.

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Later we will learn that damping is very important to your life.  Damping represents all of the factors that hold you back.  You see, you could make changes so that you are living in resonance – in fact, you could be living in resonance right now – but damping will make you feel as if you are not living in resonance.  You won’t achieve great results unless you reduce the damping in your life.

Shortly we will learn about the “life curve” (technically this is called the “frequency response curve”, or the “transmissibility curve”).  You will visually see how damping changes the amount of amplification achieved when you are in resonance.

Resonance, frequency and the “life curves”

OK, you are almost there.  We have one more characteristic to tell you about.

When you have a stationary spring and you pull on the mass and let it bounce, the spring will naturally bounce at its natural frequency.  When you cause a pendulum to swing (i.e. a child on a swing), the pendulum will swing back and forth at its natural frequency.  We see the same when you pluck a guitar string.  But now we are going to try a completely different experiment!

A new experiment

Rather than pulling on the ball at the end of the spring, we will now hold the other end of the spring in our hand and cause it to bounce – we will raise and lower the end of the spring in a smooth periodic motion.

Excite the spring below resonance

To begin with, let’s move the spring slowly.


As the spring is raised and lowered, the mass on the end also rises up and falls down.  Your hands and the mass rise and fall in time with each other – they are in-synch with each other.  But you will notice that if you raise and lower your hand by 6 inches (15 cm), the mass on the end of the spring also moves by 6 inches.  What you can see if that the work you put in is equal to the results you get out.

Excite the spring above resonance

Now we will repeat the experiment, but this time move your hands up and down much more quickly.


You will see that the hand might again be moving by 6 inches, and the mass might also be moving approximately 6 inches – so again we are not getting much value for the work we are doing – what we put in is what we get out.

Notice that the timing of the motion is not completely different.  As your hand moves down, the mass moves up.  The two are always moving in the opposite direction.

In fact, we can try one more experiment; let’s go even faster.


Notice that the hand and mass are moving in opposite directions, but now the mass is not moving very much – we are now putting in a certain amount of work, but our results are pitiful.

How does this related to people?

Later I will discuss this in greater detail; you see, I believe there are basically two types of people (who do not achieve great success) – those who are easily manipulated, who go with the flow, and those who are negative, who resist change, who constantly complain.

Easy going: effort made equals results achieved

When we moved the spring up and down slowly, the hand (doing the work) and the mass (the result of the work) moved up and down together – the mass always did what it was told; it followed the hand blindly.  People can be like that; too optimistic (believing without questioning or thinking), too easy going, too agreeable.

Negative: effort made equals results achieved

When we moved the spring more quickly, the hand (doing the work) and the mass (the result of the work) moved the same amount, but now the mass always did the opposite of the hand.  Just like some people you know; if you suggest they go left they will go right.  If there is something good to be seen they will point out the bad.  They are very pessimistic, objectionable, and negative.

Negative: effort made is greater than the results achieved

And we also saw that if you are very negative, you will achieve nothing much at all – even if you work hard you will always struggle to achieve very much.

Let’s try something different

In this experiment we are going to do two things at once.  We will move the hand up and down more and more quickly and we will see what happens to the mass on the spring.  Click the start button and watch what happens.  You should watch how much the mass moves compared to the amount the hand moves.  In this experiment, the hand is always moving the same amount (e.g. 6 inches), but the rate of up-and-down movement is slowly increased.


If you watched carefully, something very interesting happened.  To start with the mass moved the same amount as the hand.  But as the hand began to move more quickly, the mass started moving more and more, and you may have noticed that the timing between the hand and the mass slowly changed as well.

At one magic point, the mass was bouncing quite a lot.  Even though the hand was moving by as much as it always did, now mass was bouncing far more.  When the mass was bouncing the greatest amount, the system was in resonance!  And this is the key to everything!

But you will have also seen that moving hand more quickly did not make the mass bounce with greater movement, instead it started bouncing with less movement!  And the faster we moved the hand up and down, the amount of bounce continued to drop.  At one point the amount of bounce was equal to the amount the hand moved.  Beyond that point, as we saw before, the amount of bounce was less than the amount the hand moved.

Can we create a graph to see how it changes?

We sure can!  This graph, although a little technical, is the most important graph on this website.  We will call it the life curve.  It shows us how much the mass moves in comparison to the amount the hand moves as we change the rate of bounce (the frequency).

We will repeat the experiment we just performed.  We will move the hand by a certain amount, let’s call it one inch, and we will graph the number of inches the mass moves as we increase the frequency of the bounce.


Initially the curve is flat, but it slowly ramps up to a maximum value.  It is at its maximum when the system is in resonance.

The curve then decreases – that’s where we saw the mass moving less and less.  At a certain point, we start to see less than one inch of movement.

Does this curve look familiar?  It is on our logo!

We are going to try this experiment again, however this time we will reduce the damping.


Did you see the difference in the movement of the mass?  This time it moved much more.  The hand still moved the same amount, but the mass moved more because we had less damping that would normally restrict the movement.

Can this really be correct?

Yes, resonance is natural and it is all around us.  Microwave ovens work by exciting water molecules in food to make them resonate and therefore increase in heat.  Trumpet players excite the natural frequencies in the tube of the trumpet, and violinists excite the natural frequency on the strings by vibrating them with a bow.  The examples go on and on.

Again, how is this relavent to my life?

There are three things we can do to achieve great results in life:

One: Live life at your natural frequency – live in resonance.  You have to make the necessary changes so that you are operating at your natural frequency – so that you get more out of life.  This may mean doing different work, or changing your situation so that the work you love to is better appreciated.

Two: Reduce your damping.  You need to make changes to remove everything from your life that is holding you back.  Certain people will be holding you back.  Your own mind will be holding you back.

Three: Work harder.  Living in resonance will amplify your efforts, and reducing damping will further amplify your efforts, but if you make greater effort, you will reap greater rewards.

So, there you have it – three steps to success.

Hopefully you understand enough about resonance and damping to understand the tremendous.