Understanding the Energy Distribution of Expanding Water Ripples

[toothpaste666]

Homework Statement

A boat is on a quiet lake 200 m from a paddle wheel that creates waves which pass the boat every 1.8 s with an amplitude of 3 cm.

A) If the wave velocity for water waves is 1.5 m/s, what is the wavelength?
B) Write an equation for the displacement of the water as a function of position and time
C)What is the amplitude of the wave if the boat is moved 60 m closer to the paddle wheel?

Homework Equations

v = lambda/T
D = Asin(kx - wt)
k = 2pi/lambda
v = w/k

The Attempt at a Solution

A) I don't think this question is worded very clearly but I am assuming they mean that the period is 1.8 s
if that is true then v= lambda/T so lambda = Tv = 1.8s * 1.5 m/s = 2.7 m

B) D = Asin(kx-wt)
k = 2pi/lambda = 2pi/2.7 = 2.32
v = w/k so w = vk = 1.5 * 2.32 = 3.48
a is given as .03 m
so D = .03sin(2.32x-3.48t)

C) the wave is going 1.5 m/s so it goes 60 m in 60/1.5 = 40 s
D = .03sin(2.32*60 - 3.48 * 40) = .03sin(0) = 0
the water is flat at that point

I am not confident at all that I did this right. please help?

 

[toothpaste666]

One of the main reasons I am not confident is that my solution didnt use the info that the paddle wheel is 200 m away. I can't figure out where that would fit in though

 

[haruspex]

One of the main reasons I am not confident is that my solution didnt use the info that the paddle wheel is 200 m away. I can't figure out where that would fit in though

You're right, the question is not terribly clear, but I think you are supposed to consider the paddle wheel as a point source, like dropping a stone in a lake. As the ripples spread out, what happens to the amplitude?
Note that this will affect your answer to B.

 

[toothpaste666]

it lowers? my first thought is to add a damping term to the equation for the wave.
Ae^(-yt) sin(kx-wt)
where y = b/2m
but i don't have any info on the damping constant b or the mass so i don't think this is the right solution.
is there another way to describe damping for a wave?

 

[haruspex]

it lowers? my first thought is to add a damping term to the equation for the wave.
Ae^(-yt) sin(kx-wt)
where y = b/2m
but i don't have any info on the damping constant b or the mass so i don't think this is the right solution.
is there another way to describe damping for a wave?

It isn't a matter of damping (which would be non-conservation of work) so much as thinning out. If you stand twice as far from a sound source, what does it do to the intensity? Why is the relationship of that form? What does that imply for how the intensity of a surface wave attenuates?

 

[toothpaste666]

doesnt it decrease logarithmically?

 

[haruspex]

doesnt it decrease logarithmically?

I wouldn't think so. Not negative-exponentially either. Can you answer my question about sound intensity?

 

[toothpaste666]

well the reason i say that is because the formula i learned for sound intensity is
intensity (dB) = 10log(I/I0)

looking through my notes the only other formula i have for the intensity of a wave is
I = 2pi^2 d v f^2 A^2
where d is the density, v is the velocity, f is the frequency and A is the amplitude.

if neither of those fit the situation I would say the intensity of the wave would decrease the farther you get

 

[haruspex]

intensity (dB) = 10log(I/I0)

That's just a definition of decibels. It's to do with the way humans perceive sound, not how sound actually behaves.

I would say the intensity of the wave would decrease the farther you get

Sure, but we need to get to the algebraic relationship.
Think about this... a stone is dropped in a lake. It creates a ripple that starts off at a small radius and spreads out (ignore the fact that there will be multiple ripples). The energy only declines gradually, so pretend it's constant. The energy is related to the height of the ripple (we need to think how, exactly). As the ripple expands, that energy is spread over a longer distance.

 

[toothpaste666]

E is proportional to the amplitude right? E = 1/2kA^2

 

[haruspex]

E is proportional to the amplitude right? E = 1/2kA^2

Ok, so what about how the energy per unit length of perimeter will change?

 

[toothpaste666]

the perimeter being the 200 m?

 

[haruspex]

the perimeter being the 200 m?

No. What shape does a spreading ripple make? What is meant by a perimeter?

 

[toothpaste666]

the perimeter is the outside length of an object. I believe the shape of the ripple would be sinusoidal

 

[haruspex]

I believe the shape of the ripple would be sinusoidal

I meant, when looking from above. In that context, what do I mean by perimeter? How will the energy per unit length of perimeter change as the ripple expands (assuming total energy is constant)?