Moving with the Ripples
The drops from a leaky faucet form a steady wave in the water, a snapshot of which is shown in the diagram above.
Rob carefully lowers his paper boat into the water such that it starts at the bottom of the wave.
True or False?
Rob's boat will move away from the faucet along with the ripples.
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7 solutions
So when you start at the bottom, the wave will propagate away from the faucet, while the boat is on the wave, it will be pulled down towards the bottom of the wave (which has moved away) with the force of Gravity. This means the boat will indeed move away from the faucet. Granted not much, and not far, but it will.
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The boat makes small elliptical motions, but its position remains fixed on average. When it is in the bottom half of the ellipse, there is a force on it against the waves, and when it is in the top half, the force on it is along the waves.
You are right, the boat does move very slowly to the right due to the flow of the water, but not because of the ripples.
Agreed. Even if you took the idealised model of a floating object on a wave which accelerates when on a tilted surface sloping one way and equally decelerates when the the surface tilts backwards then there is still and overall motion. The boats starts IN A TROUGH and so will accelerate in the propagation direction on the tilted surface until the boat is floating on the crest. The boat will then be travelling with a small velocity and then decelerates as the wave tilts backwards. The net effect is that there is a propagation direction displacement over each cycle.
Have you seen a surfer with a long board riding the wave? There is no need for huge breaking waves for that kind of surfing. It is true that the waves do not carry water, but they can easily carry an object on the top of the water.
If the water does not move away from the faucet, a pile of water would form; this can't be right: the water must flow outward to keep the surface even.
However, this outward transport of water may happen near the surface as well as deeper down, and it is difficult to predict where exactly. It may well be that the water deep below the boat moves outward while the water around the boat moves inward. Surface waves tend to show circular flow, so this scenario is quite possible.
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This is a valid point, water must flow away from the faucet to avoid piling up.
The water is filling up very slowly, so the rate at which water flows away from the faucet would be really small. This would cause the boat to flow away from the faucet, but its speed would be much less compared to the speed of the ripples.
We know in a real experiment the boat won't stay at its initial location. What will move the boat away?
But won't the extra water droplets cause the surface of the water to rise under the faucet, and thus there will be a net movement or flow of water away from the faucet to distribute the water evenly, slowly carrying the boat with it? Therefore, even though the volume of the water droplets is minimal, won't the boat still move very slowly?
Although, I do understand why my answer is wrong because in this case, the boat doesn't move along with the ripples, and thus the statement is wrong.
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You are right. Water must flow away from the faucet to avoid piling up.
The water is filling up very slowly, so the rate at which water flows away from the faucet would be really small. This would cause the boat to flow away from the faucet, but its speed would be much less compared to the speed of the ripples.
Practically the boat will move slightly in the direction of wave propogation.
I don't see how the given answer can be correct. The solutions given ignore some factors. If you put an object in the water, that has the same density as the water, it will go up and down and not tend to progress in the direction of the wave anymore than the water will progress in the direction of the wave. But a boat that floats will tend to slide down the wave like a surfboard, due to gravity, in the direction of the wave. And then, it will slide back toward the faucet on the back side of the wave. The question is, will it be propelled backward with the same force as it was propelled forward. The answer is no. The principle involved here might best be demonstrated by a related example, which I hope to simplify with some conditions.
Suppose we have a road in hilly terrain that goes up and down in a sinusoidal fashion. Suppose we somehow reduce the friction to zero. We propel a ball forward, fast enough to get it over the first hill. Will the acceleration down each hill be as great as the deceleration up each hill? To be more specific, will the horizontal component of the acceleration down the hill have the same magnitude as the horizontal component of the deceleration up the hill? If the answer is not clear, speed the ball up enough to catapult the ball off the road at the top of each hill, so that it does't have as much time in contact with the road on the downward side of the hill -- meaning that it will not spend as much time in contact with the incline, which is necessary to increase the horizon component of the velocity. Even if the ball is not airborn at the top of the hill, the force of the ball against the road will not be as great initially on the decline, until the tended parabolic trajectory becomes steeper that the road.
But we have ignored friction. Let's complicate it with the simulation of the ball and road. Suppose we build a small model with a ball and a sinusoidal surface that we can move. Further more, we put a cloth over the sinusoidal surface that weights enough to maintain contact with the sinusoidal surface as it moves. We also attach the cloth to something stationary. Which means, as we move the sinusoidal surface the cloth moves up and down, but does not progress with the underlying surface. Now put the ball on the oscillating surface. What does the ball do, even if the cloth does not more in the direction of the waves?
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I agree the question does not seem to mimic reality. Drips will make surface ripples rather than deep waves but the effect will be much the same. Ripples propagate due to surface-tension <-> kinetic energy whereas deep waves propagate due to hydrostatic <-> kinetic energy. In both cases a floating object will experience surfing effects.
^--SCIENCE ANSWER--^ False: Waves move through the water, boat goes up and down but remains in place relative to the faucet.
^--ENGINEERING ANSWER--^ True: Based on the ratio of the boat length to the wave height in the problem pictured I would expect that the boat would slide down the front of the wave at least a few centimeters each time and thus "surf" in fits and spurts away from the faucet.
Correct, BUT the boat is shown to have an angled front and back. When the wave pushes the boat up (on the angled front) it will be propelled back away. Not because the water is moving away, but because the force of gravity on the boat is being exploited.
Try it in real life. The boat moves away. Poorly worded/illustrated question. A cork or even raft would have exhibited the intended behavior, but not a paper boat of the type illustrated.
I was unfamiliar with the term "steady wave", but in https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.math.u-psud.fr/~fabre/Articlesm2/Groves.pdf&ved=0ahUKEwjT-_6Yn-TWAhUhjVQKHQG6D70QFgg5MAQ&usg=AOvVaw0MJhPINxL8oTDXM11VF3d0 I found "Steady waves are water waves ... uniformly translating [moving] in the horizontal direction", which is equivalent to what I found in other sources.
We are told "Rob carefully lowers his paper boat into the water such that it starts at the bottom of the wave."
Clearly, as the wave continues away from the faucet, its leading slope begins to lift the boat, but gravity pulls it down the slope, "away from the faucet along with the ripples."
How can this be False? What could prevent the boat from doing this?
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Truly "steady waves" or "solitons" are found in shallow-water channels. They are solutions of the 1x1-dimensional Korteweg-deVries partial differential equation.
It is unlikely that the waves generated here are truly solitons. Typically, ripples are concentric circles that grow bigger, which necessitates a decrease in wave amplitude. However, most traveling waves at least approximately fit the picture of uniform translation over time.
The assumption typically made with paper boats is that they move along with the surrounding water. The effect of gravity on the boat is not accounted for separately. This is reasonable because water resistance will be relatively strong.
Since water waves are longitudinal waves, the boat will travel along the direction of propagation of the wave and hence cannot move away from the faucet.
That reasoning does not work here. The waves are not waves inside water, but surface waves or gravitational waves on the water-air boundary; they behave differently. Typically, such surface waves combine longitudinal and transverse motion.
Initially, when the boat is placed in a wave trough, it is in equilibrium, but as the waves propagate away from the dripping faucet, the water surface under the boat will incline such that the side closer to the faucet is higher than the farther side. Then the bouyancy force, the resultant force of the water pressure, will have a horizontal component, pushing the boat away from the faucet. Effectively, the boat will slide down the slope of the wave like a surfer. Since the ripples from a dripping faucet are small, the horizontal force will be small, and as a paper boat is not very streamlined, its resistance will be relatively high, so the movement of the boat will be slow - much slower than the speed of propagation of the wave, but it will still move away from the faucet, so the statement is true, unless the phrase "along with the ripples" is taken to mean at the same speed as the ripples, in which case it is false.
But the waves are moving outward from their source, and the boat should be pushed forward in the direction of the wave, which is away from the faucet. I disagree
The top of the wave would move away from the faucet but the bottom of the wave would be moving towards it (circular motion) so the boat will not move down with the ripples but rather oscillate back and forth about its equilibrium position.
The waves do not affect the boat's motion. However, must flow from the faucet end of the tub to the far end. This flow must move the boat.
Please explain
I think there is some clarification needed here: Water waves are longitudinal waves for the most part, except for the ripples, which are transverse waves -- this phenomena only happens on the surface.
The waves on the surface on the water are not longitudinal waves.
As the droplets strike the surface of the water then their P.E converted into K.E. Here waves are generated upon the surface since they are free to oscillate harmonically.Then this energy is carried out by the particles induction method i.e., by transverse and longitudinal waves produced on water surface.
I have often observed that objects placed on the surface of the water near a dripping or running faucet are more frequently drawn into the dripping/running stream of water than floating away. While I cannot provide an explanation established in natural law, I can hazard a guess that the flow of water impacting on the surface is disrupting the surface tension creating some kind of backwards flow towards the point of impact, kind of like any surface being depressed by a force/weight, except in this case the surface is a flowing liquid.
A mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium,not transfers materials while waves can move over long distances, the movement of the medium of transmission — the material — is limited.
Therefore, the oscillating material does not move far from its initial equilibrium position.
... Mechanical waves can be produced only in media which possess elasticity ...
Once the horizontal oscillation of the water results in a series of waves. Vector analysis shows that when the boat is lifted by the first wave, gravity will send the boat away from the tap, then up and down on the next, smaller wave. As the boat encounters waves too small to keep it trapped in the resulting trough, momentum should be able carry it over the ensuing, smaller waves. Ultimately, momentum will carry it away from the tap, a wee bit.
Water wave is a mechanical wave so it only transfers energy not matter.
Waves in their 'sin wave' form don't actually move, they just go up and down. It's when they roll over you got a problem...
Rob's boat won't move away because in mechanical wave motion particles of medium do not displayed permanently.
I disagree with this answer. While it is true that a mechanical wave is just the oscillation of matter, it is also true that the faucet is adding water to the system. If there weren't a net of water molecules moving away from the faucet, that would mean that the water is slowly piling up at that point. Since we know that isn't happening, we know that there is SOME movement away. It's not fast, but there is some, and the boat would track it.
Note that the boat would be oscillating back and forth, towards and away from the faucet, as the waves move under it, because it always wants to move downhill. However, because of the trickle of water I mention above, the movements towards the faucet would always be a tiny bit less than the movement away.
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The front surface of the advancing ripple is inclined and the buoyant force of the displaced water has a component force of starting friction. If the vector sum of the boat's weight plus the resisting buoyant force exceeds the starting friction, the boat will be propelled in the direction of the ripple's . motion. If the surface inclination is steep enough, the boat's motion can equal the ripples motion
The mechanical waves carry only the energy not the matter. It means that the water doesn't move away from the faucet, but the water only moves up and down doing a harmonic movement and the paper boat will go up and down too, without move away from the faucet.