# Where does the energy go?

## Summary:

Standing Waves and Resonance
In the case of young's double slit experiment. We can see the energy redistributing in double amplitude and zero amplitude. But in the case of stationary waves on a string when I am applying a frequency exactly between two harmonics, I am giving it kinetic energy and seeing nothing. Where does that energy go? The most Kinetic energy seen and the sound energy heard is at a harmonic. and none of it at middle of a harmonic. The answer I know is anticlimactic but I need a full description anyway.

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anorlunda
Staff Emeritus
I don't understand your question. Are you saying the string does not move, that it has no kinetic energy?

Perhaps a drawing or a video may explain better your scenario.

Mister T
Gold Member
But in the case of stationary waves on a string when I am applying a frequency exactly between two harmonics, I am giving it kinetic energy and seeing nothing.
When you say you are "applying a frequency" do you mean you have a frequency generator and amplifier connected to some device that in turn makes the string vibrate? If so, when I do this I see the string vibrate.

The most Kinetic energy seen and the sound energy heard is at a harmonic.
When the driving frequency matches one of the natural frequencies of vibration of the string (what you are calling the harmonics) you get large amplitude oscillations at that frequency. This is called resonance. If your driving frequency doesn't match one of the natural frequencies you still get vibrations and energy is still conserved.

When you say you are "applying a frequency" do you mean you have a frequency generator and amplifier connected to some device that in turn makes the string vibrate? If so, when I do this I see the string vibrate.

When the driving frequency matches one of the natural frequencies of vibration of the string (what you are calling the harmonics) you get large amplitude oscillations at that frequency. This is called resonance. If your driving frequency doesn't match one of the natural frequencies you still get vibrations and energy is still conserved.
Yes, I have a generator and I understand resonance. What I fail to comprehend is when it is not a harmonic instead exactly between harmonics, I see and feel almost no vibration. I know amplitude cancels out and stuff. But still that energy has to be in some form somewhere, what is that form? If not kinetic and sound.

ZapperZ
Staff Emeritus
Yes, I have a generator and I understand resonance. What I fail to comprehend is when it is not a harmonic instead exactly between harmonics, I see and feel almost no vibration. I know amplitude cancels out and stuff. But still that energy has to be in some form somewhere, what is that form? If not kinetic and sound.
But the destructive interference is also accompanied by constructive interference elsewhere!

The energy is in the ENTIRE wave, not just at one specific location, and over a period of time, not just at a specific instant in time.

This question has been addressed in many previous threads, so you're welcome to do a search. Here are a few of them:

etc..

Zz.

But the destructive interference is also accompanied by constructive interference elsewhere!

The energy is in the ENTIRE wave, not just at one specific location, and over a period of time, not just at a specific instant in time.

This question has been addressed in many previous threads, so you're welcome to do a search. Here are a few of them:

etc..

Zz.
JazakAllah Khairan (Thank You). I should have searched better before asking. Will get back to you if I have any query.

Henryk
Gold Member
Yes, I have a generator and I understand resonance. What I fail to comprehend is when it is not a harmonic instead exactly between harmonics, I see and feel almost no vibration. I know amplitude cancels out and stuff. But still that energy has to be in some form somewhere, what is that form? If not kinetic and sound.
The question should be rephrased a bit, I propose 'what is happening to energy'
You have observed that at resonance, there is a large amplitude of vibrations while away from resonance, the amplitude of vibration is much smaller, almost not detectable.
Now, let's talk about energy. You have a system (a vibrating thing), the amount of energy given to the system is equal to the work done on it. The work is given by the formula ##W = F \cdot ds## where F is the force applied and ds is the displacement under the force. This applies to all physical systems (ok, the force can be time dependent, so can the displacement, therefore, the work done is really equal to ##W = \int{ F(t) \cdot v(t) dt} ##)
Anyway, now you see, your generator provides a certain amount of force. At resonance, the string vibrates a lot, that is the displacement is large and the amount of energy transmitted from the actuator to the spring is large. Away from resonance, the amplitude of the vibration (displacement) is small and at the same actuator force, the amount of energy given to the string is much smaller.
And then, what happens to the energy?
some of it is send out as sound waves, the rest dissipates into heat.

Delta2
therefore, the work done is really equal to W=∫F(t)⋅v(t)dtW=∫F(t)⋅v(t)dtW = \int{ F(t) \cdot v(t) dt} )
Anyway, now you see, your generator provides a certain amount of force. At resonance, the string vibrates a lot, that is the displacement is large and the amount of energy transmitted from the actuator to the spring is large. Away from resonance, the amplitude of the vibration (displacement) is small and at the same actuator force, the amount of energy given to the string is much smaller.
Note that the work can also be negative. If the string moves in the opposite direction as the actuator. With an ideal string, the actuator can actually do no work at all on average, if the reflected wave comes back 180 degrees out of phase with the actuator. This might cause the actuator to use less energy, or produce more heat, depending on how it's built.

sophiecentaur
Gold Member
Yes, I have a generator and I understand resonance.
If you understand resonance then you will understand that, once the resonance is established and the amplitude is as high as it's going to get, the power in from your source will equal the power dissipated (equilibrium). The energy will be Q (quality factor) times the input power.
Well off-resonance, you will still have movement but no stored energy in the resonator so the amplitude will be very small - but the string will still be oscillating just at the level of the input actuator.

Note that the work can also be negative. If the string moves in the opposite direction as the actuator. With an ideal string, the actuator can actually do no work at all on average, if the reflected wave comes back 180 degrees out of phase with the actuator. This might cause the actuator to use less energy, or produce more heat, depending on how it's built.
Are you saying that an ideal actuator would just not do any negative work to not dissipate any energy because damping is not allowed and that it being real would just dissipate all that energy(please expand on it for me)?
I do not hear any sound and do not feel it heating up (It's a copper wire). Do you have any idea how long it will take to heat up or would it at all? Is the dissipated energy measurable in any form?

sophiecentaur
Gold Member
Are you saying that an ideal actuator would just not do any negative work to not dissipate any energy because damping is not allowed
If you don't allow damping then the amplitude will build up to a level corresponding to how long the actuating signal is applied (Actuator power times time). There will always be a finite Q so the final energy of the oscillation will be Q times the applied Power.
The resonance has a complex Impedance with a Resistive equivalent as well as Reactive equivalent. You seem to be looking for something wrong with the theory but there isn't.

PS I just read the final comment in the above post from @ZapperZ . He's got it just right imo!!! Try to be a bit more humble when you clearly have got something wrong in your understanding and you will make yourself happy and everyone else too.
Alternatively, pay for a private tutor to help you on your terms (if you can find one). PF is free and it needs to be treated nicely.