Confusion about Wave Motion

[ktmsud]

TL;DR Summary

Does one vibration produce infinitely long wave for infinite long time in perfect conditions?

Suppose a particle in a medium is displaced from its mean position by giving some energy and it is released. Due to property of elasticity and inertia it starts to vibrate about its mean position and due to interactions with other particles of medium they also vibrate in some later time. The energy initially given is converted back and forth between kinetic and potential energy and If there is no damping then the initial particle vibrates infinitely and so do particles near it and other particles of medium. In such scenario, wave travels to infinite distance and infinite time.
While deriving equation of progressive wave, amplitude of vibration is taken constant. Does it mean there is no damping? If Given energy is converted between K.E and P.E of initial particle then how is energy travelling? If energy is not travelling then how are other particles vibrating? If energy is travelling and making particles to vibrate why is not it decreasing?

 

[Baluncore]

While deriving equation of progressive wave, amplitude of vibration is taken constant. Does it mean there is no damping?

You are studying the simpler mathematics of steady-state transmission. It means the vibration is being driven by a source of energy that maintains the amplitude, while radiation and damping remove that energy from the system.

If Given energy is converted between K.E and P.E of initial particle then how is energy travelling?

That energy is not travelling, it is circulating in place. Since conversion between KE and PE is a quadrature phenomenon, real power is not involved. The product of sine and cosine changes sign as energy circulates. It is the damping of the oscillation that extracts real energy from the system.

If energy is not travelling then how are other particles vibrating?

The small amount of energy is being radiated by the damping of the resonator. That energy is coupling to, and circulating in similar resonators nearby. Some energy is travelling.

If energy is travelling and making particles to vibrate why is not it decreasing?

Because there is assumed, in the mathematics, to be a power source, a flow of energy that maintains the vibration.

 

[ktmsud]

Because there is assumed, in the mathematics, to be a power source, a flow of energy that maintains the vibration.

Is there a need for source in such case? Energy is not going anywhere so one push is sufficient to vibrate particles forever.

 

[Baluncore]

You asked ...

If energy is travelling and making particles to vibrate why is not it decreasing?

I answered ...

Because there is assumed, in the mathematics, to be a power source, a flow of energy that maintains the vibration.

You then asked ...

Energy is not going anywhere so one push is sufficient to vibrate particles forever.

Which contradicts the travelling energy assumption in your original question.

Assuming no radiation and no damping, an excited set of particles will vibrate forever.

If a steady state is to be maintained, conservation of energy requires energy to be provided as energy is radiated.

 

[jbriggs444]

If you have an infinite string that is plucked once then energy travels outward forever. The particles at the point of the plucking will return quickly to [near] rest. The resulting wave form will propagate in both directions.

Particles further out will vibrate briefly when the wave form passes and then go quiescent again.

An easy way to see something similar is to throw a stone into the middle of a still pond.

 

[ktmsud]

I asked:

Is there a need for source in such case? Energy is not going anywhere so one push is sufficient to vibrate particles forever.

Only because you said

That energy is not travelling, it is circulating in place. Since conversion between KE and PE is a quadrature phenomenon, real power is not involved.

 

[sophiecentaur]

TL;DR Summary: Does one vibration produce infinitely long wave for infinite long time in perfect conditions?

Given energy is converted between K.E and P.E of initial particle then how is energy travelling? If energy is not travelling then how are other particles vibrating?

In a wave (ignore EM, for the time being) each vibrating particle / element) interacts with adjacent particles. This is unlike a mass on a spring, rigidly attached at the other end, which cannot pass its energy onto another mass. The coupling between adjacent string particles will cause particles on either side to move - and so on, along the whole wave carrying medium. The maths will tell you that, as each particle along the line will launch its own little wave. Because of the delay in the interactions, you find that, in one direction, all these little waves will cancel each other out but, in the other direction, they augment each other. This causes a progressive wave, carrying all the original energy along. A rigid 'stop' on the string will cause a reflection because no energy can transfer to a 'solid wall'.
If the wave is constrained so that, for example on a string, the energy stays just on the string, the wave will go on for ever, except for the small losses of internal friction and viscosity of the air. On a pond's surface (2D wave) the energy will spread out over the ever increasing circumference so the flux will reduce as 1/r. For a spherical wave, the area goes up as r²so the flux drops as 1/r² (inverse square law).

For EM waves, there are no particles so that mechanical explanation is sadly lacking and you have to use (for example) Maxwells laws to predict the same result.

 

[Tom.G]

Lets try the simple interpretation:

TL;DR Summary: Does one vibration produce infinitely long wave for infinite long time in perfect conditions?

it starts to vibrate about its mean position and due to interactions with other particles of medium they also vibrate in some later time.

then the initial particle vibrates infinitely and so do particles near it and other particles of medium.

Here you have the energy of one particle transfering its energy to other particles, which then transfer to more particles.

It seems, from your description, that transferring energy to multiple particles would spread some fraction that original energy to an infinite number of other particles.

In the limit, the energy density would be so low that it could not be detected (either Absolute Zero or the temperature of the particles local environment).

 

[tech99]

TL;DR Summary: Does one vibration produce infinitely long wave for infinite long time in perfect conditions?

Suppose a particle in a medium is displaced from its mean position by giving some energy and it is released. Due to property of elasticity and inertia it starts to vibrate about its mean position and due to interactions with other particles of medium they also vibrate in some later time. The energy initially given is converted back and forth between kinetic and potential energy and If there is no damping then the initial particle vibrates infinitely and so do particles near it and other particles of medium. In such scenario, wave travels to infinite distance and infinite time.
While deriving equation of progressive wave, amplitude of vibration is taken constant. Does it mean there is no damping? If Given energy is converted between K.E and P.E of initial particle then how is energy travelling? If energy is not travelling then how are other particles vibrating? If energy is travelling and making particles to vibrate why is not it decreasing?

The infinite medium has inertia and springiness, but appears to the initial particle as a resistor (or purely dissipative load). Energy propagates away from the initial particle for ever. The particle delivers half of its initial energy to the medium, and oscillates with a damped motion, falling towards zero. We also see the damped wave train propagating through the medium for ever.

 

[sophiecentaur]

It seems, from your description, that transferring energy to multiple particles would spread some fraction that original energy to an infinite number of other particles.

I see where you're going with this. BUT remember that there is a delay in the transfer of the energy. You have appreciated that the wave consists of disturbances in both displacement and particle speed along the way. The speed of the wave is the speed of the point where all these disturbances ('fractions') add up. At other points (earlier of later) the net disturbance is zero so the energy from one place will arrive at just one place along the wave. Think in terms of the phases of all your fractions.
Huygen's principle can be used to predict what happens to a wave as it passes through an aperture. 'Secondary Wavelets' from all points across the wave front can be added add up to find the direction of the resulting wave front. This idea also works for a one dimensional wave on a string; enhancement only happens at a certain point along the string from the origin.

In a medium that's not uniform, a pulse will become distorted (dispersed) because the fractions "don't add up right",

 

[Tom.G]

It seems like we are talking/thinking about different circumstances.

I was describing that the finite initial energy gets distributed over a steadily increasing area or volume as it propagates. The same effect that causes distant sound, radio, and light to be weaker with distance; i.e. the increasing size of the wavefront with distance.

 

[sophiecentaur]

distributed over a steadily increasing area or volume

This happens for 2 and 3D waves but, for a linear wave, the energy stays within the region of a wave as it travels. This assumes no dispersion or dissipative loss.

The same applies to a plane wave of course. Most elementary treatments start with a linear wave and then a plane wave, before moving on to more real circumstances. As usual, the thread wanders from one to the other without a sheepdog to keep it on track.

 

[ktmsud]

I am extremely sorry for confusion that i have created. Let me clerify about how.

Suppose there is a one dimensional array of particles having property of elasticity (so that particles when disturbed tend to go to their initial position. i.e. restoring force on particle when it is disturbed from mean position) and inertia( in usual sense).

First assume only one particle, when this particle is disturbed from its mean position by giving some energy, restoring force pulls it back to its original position increasing its speed. Due to inertia this particle, doesn't just reach to original position and stop there but overshoots and goes on other side untill its total K.E at original position gets converted into potential energy. In this way, if there is no damping force, particle execute simple harmonic motion forever. Right?

Now consider whole array and also suppose there is no damping. Due to interactions between particles (let's say a bond) particle next to the vibrating particle also gets vibrated and vibration travels forward forming a wave. As energy is required for vibration, it seems that energy from first particle is travelling in the array.

Sometimes i think, as the particles are vibrating, their energy is with them just getting converted back and forth between K.E and P.E. So, energy of the system as a whole (with given energy included) remains constant and some kind of interaction energy between particles is disturbed and that disturbed energy is travelling along the array. When total energy of each particle is same - same as before start disturbance as i changed interaction energy only and that is travelling. So, all particle vibrate all the time till infinite time and if the array is infinite long wave should travel infinite distance as well.

And sometimes i think like when energy i have given to first particle is totally transfered to next particle/s it comes in its original position and only a pulse travels forward. Leaving particles behind at rest.

 

[sophiecentaur]

I am extremely sorry for confusion that i have created.

Don't worry; it happens all the time and threads tend to have a mind of their own.

So, all particle vibrate all the time till infinite time and if the array is infinite long wave should travel infinite distance as well.

This is not actually wrong but you could say it's a jump too fast.
We are always dealing with a 'real' situation. The medium is not infinitely rigid and it has a finite linear density; displacement is propagated at a finite speed. Start with one particle being displaced by the motion of the wall. As soon as it starts to move it will affect the next one - and the next and the next. Energy is spread out to all the particles in the pulse. It is mutually shared amongst them - backwards and forwards, according to the propagation speed of the disturbances. As I mentioned earlier, if the string is uniform, the effect of each particle produces a speeding up and or slowing down of the other particles in the pulse. Only particles in the region of the pulse end up with any energy so the pulse stays intact and travels at the 'wave speed'.
Particles way out ahead of the pulse cannot move because the influence hasn't arrived yet. The tail of the pulse dies away to nothing because net energy can't flow backwards from the particles in the pulse so the pulse shape will stay the same as it flows along. Only particles in the region of the pulse will be vibrating.

Your "infinite" concern: If there is no loss and if the wave is constrained along a line then the energy will be contained within the region of the pulse. It can only transfer energy forward at its wave speed and the sum of energy flowing 'backwards" from the pulse (left behind) will be zero. So it goes on for ever without reducing or changing shape (ideally).

The maths of Wave Theory describes it all very well and predicts what happens but it's a bit of a pain to leap into that without a lot of basics.

 

[sophiecentaur]

if there is no damping force, particle execute simple harmonic motion forever. Right?

I just saw this. In a wave, there is always a form of 'damping' force on one of the particles because the particles on either side are supplying and removing energy all the time. Only when you have a standing wave can you consider the particles in your way. Yes; they perform simple harmonic motion but it is not a simple resonance- it's forced by the neighbours to which it is coupled.
Once again, the maths describes it. (Arrrghhh)