What do wave crests indicate about a boat's speed?

[Mohammed Sayanvala]

Homework Statement

*I cannot place the original image due to copyright reasons, but the image above is a good alternative.

"Wave crests spread out behind a boat as shown above. What do the wave crests indicate about the boat's speed?"

1. It is increasing.
2. It is less than the speed of the water waves.
3. It is equal to the speed of the water waves.
4. It is greater than the speed of the water waves.
5. The wave crests do not indicate anything about the boat's speed.

Homework Equations

v=fλ
v=Δx/Δt

The Attempt at a Solution

Perhaps, since the distance between the crests remains constant, the speed of the boat is equal to the speed of the water waves. But, then if the boat is moving at a constant speed (slower than the speed of water waves), won't the distance between water waves also be constant...?

 

[FactChecker]

Looking at the general shape of the water disturbance, can you say anything about the velocity of the boat versus the velocity of the spread of the waves? Can you detect any change in the disturbance shape near the boat (just made) versus the shape farther back (made earlier)? And what does that mean?

 

[berkeman]

Welcome to the PF.

Another way to start to think about this -- if the waves were spreading out from the boat at a 45 degree angle, what would that tell you about the speed of the boat versus the speed of the waves?

 

[haruspex]

since the distance between the crests remains constant, the speed of the boat is equal to the speed of the water waves.

Is there a good reason to suppose that the distance between the crests is affected by the speed of the boat?
Look at the direction of a wavefront and backtrack to see where it started. How far have the wave and boat traveled from that point?

if the waves were spreading out from the boat at a 45 degree angle

Ok, but the answer might not be quite what you think.

 

[gneill]

Research: "Kelvin wake".

 

[berkeman]

Ok, but the answer might not be quite what you think.

Interesting. I'll have to think about that more...

 

[FactChecker]

I briefly researched "Kelvin wake", as @gneill recommended and I think that the real answer is MUCH more complicated than I originally imagined. IMHO, the person who made up this question was probably not intending such a complicated answer and expected an answer based on $tan(\theta)$, even though that is wrong.

 

[haruspex]

I briefly researched "Kelvin wake", as @gneill recommended and I think that the real answer is MUCH more complicated than I originally imagined. IMHO, the person who made up this question was probably not intending such a complicated answer and expected an answer based on $tan(\theta)$, even though that is wrong.

The image posted (though I note this is not the one on which the thread is based) does show something which leads to one of the five answers. But that something I have in mind is not the pattern of wake behind the boat.
Hint: compare with the duck wake at https://en.wikipedia.org/wiki/Wake#Kelvin_wake_pattern.
The image in post #1 is also at that link.