Waves in a Raingutter

The more you practice blowing a "typical" regatta boat in a raingutter filled with water, the more you learn to respect the waves that are created. Without using a stop watch, they can help you determine your speed and if you don't watch, they can stop your speed!

First we'll look at the waves, then we'll see how your boat makes them and how they affect your boat.

Surface waves in a trough

Water waves in a trough are surface waves. There are other kinds of waves that form under water where there are currents, but those won't be a factor in a regatta race. If you float a small peice of paper, styrofoam or cork in a trough with waves, you can see an up-down, forward-backward motion. It's circular!

In general for a forward travelling wave, there are up-down (or side-to-side) waves called transverse waves as in light; forward-backward waves called longitudinal waves; "slinky" waves called compression waves as in sound; and twisting waves called tortional waves - you can do this with a slinky by quickly twisting one end one way and then back again then stopping to watch the "twist" travel to the other end.

Water waves are a combination of transverse and longitudinal waves. Water is nearly incompressible, so it's compression waves don't carry sound very well. What this means is that as a crest forms, water is pulled in from both sides to make it higher. The water pulled into the crest from the sides makes the forward-backward motion of the wave. If water compressed - that is if its molecules bunched up closer together - the water around a crest would still get pulled into it, but the crest would not get very high.

A water wave passing a point on the surface can be modeled mathematically using the trigonometric functions sine and cosine.

The sine function is written sinO for "sine of theta" where theta is a greek letter representing the phase of the wave. Using radians to measure the angle of the phase, O, theta, takes on values between 0 (zero) and 2pi.

For z in the vertical direction and y in the horizontal, the model looks like:

z = HsinO; x = ScosO for H the wave height and S half the forward-backward "slosh" distance.

Our particle on the water surface starts out at x = S, z = 0 with its phase at O = 0. As O, theta, goes from a phase of 0 to pi/4 (45 degrees) our particle on the surface lifts up and starts moving to the left a bit. At pi/2 (90 degrees), the particle is at height z = H (above where the surface would have been if there was no wave) and a distance S to the left of where it started, so x = 0.

Between pi/2 (90 deg) and pi (180 deg) the particle continues to move to the left and down to the surface. At O = pi, z = 0 and x = -S.

Between pi and 3pi/2 (270 deg) the particle continues to move down and starts returning to the right. At O = 3pi/2, z = -H and x = 0.

Between 3pi/2 and 2pi (360 deg) the particle continues to move to the right and starts up to the surface. At O = 2pi, z = 0 and x = S; the same as when the phase was O = 0.

This is why sine and cosine are called circular functions, they come back to where they started. So instead of increasing O past 2pi, we almost always just start over with O = 0.

Because water has mass, it has momentum when it moves. For a water wave, it is the resistance to motion and how much water can "stretch" that gives us a way to determine how the phase, O, changes with time in real waves. It turns out that once water waves are started in a shallow trough, the height and forward speed of waves depends only on gravity; they are really pretty simple waves!

Since 10 feet is a fairly small distance for a wave to travel and because the race is usually over in a few seconds, we won't worry about the change in the height of waves because they won't change much. We are interested in waves that bounce off the end of the trough because they can burry a boat, but we need to know how fast they are going to tell when they will bounce back.

The speed of a shallow wave in a trough is given by:

v = \[gh] for g = 386.088 in/s2

This is so simple that a table would be useful to use if we measure the depth of water in our trough. We can get the time t it takes to travel the ten foot length of the raingutter from t = L/v with L being the length of the raingutter (L = 120 inches = 10 feet). We'll put that in the table too.

The water in a trough should never get below 1.5 inches or some boats might srape the bottom. At that depth, the water is moving 24.1 inches per second which is two feet a second so it goes the 10 foot distance in 5 seconds. When the water is at the top of the trough, 3 inches, the waves are traveling a foot a second faster and bounce back in 3.5 seconds.

Making waves

How does the wave get started? The wave is created when your boat pushes water out of its way at the start of the race. Generally, the deeper your boat starts in the water, the more water it will push. Energy from your breath moves the boat and is transfered to disturbing the water. The water tries to pile up since it doesn't compress much. But water stacked on water makes the water under and surrounding it push down and spread out - it has to go somewhere!

If the boat stopped, the reaction it started would still continue as a wave down the trough. It would be a single wave. Down the trough and back it would go, then bounce back again and again until it dies out. Note that as the wave becomes lower in height, the time of travel doesn't change!

Of course, as you continue to blow, more waves will be generated until the water is quite chopy from the waves bouncing around in the raingutter. Ultimately, the moral of the story is to find a way to beat the wave to the other end of the trough. If you do this, you'll be well on your way to beating your opponent - unless he's read these manuals!

Breakers!

Suppose now, you can not blow your boat faster than the initial wave. If you manage to blow your boat at a constant speed v and the first wave is moving at a speed w that is faster, then when the wave hits the end 10 feet away, your boat will be a distance y from the start given by,

y = vt = v(L/w) = Lv/w

Now the wave will travel at a speed -w with respect to your boat since it just bounced off the end of the raingutter. We can solve the distance equations for your boat and the wave to determine the time T when and the place Y where they will meet.

Y = y + v(T-t) = L-w(T-t)

Solve for collision time T to get

y + vT-vt = L-wT+wt

Move the terms in collision time T to the right and the others to the left being sure to change their signs,

vT+wT = L+vt+wt-y

Simplify by collecting terms and dividing through by v+w

T = ((v+w)t + L-y)/(v+w)

Simplify by distributing the denominator on the right,

T = t + (L-y)/(v+w)

The term (L-y)/(v+w) is the time it takes the boat and wave to colide after the wave is reflected off the end. We can substitute Lv/w for the distance y to get

T = t + (L-Lv/w)/(v+w)

Which simplifies to

T = t + L(1-v/w)/(v+w) = t + L(w-v)/w(v+w) = t + (L/w)(w-v)/(w+v)

Since t = L/w, we can still further reduce this to,

T = t(1+(w-v)/(w+v))

This shows that the closer the boat speed is to the speed of the wave, the closer to the end of the trough will be the collision with the wave.

Swamped!

The force of the wave on your boat is determined by the amount of energy transfered to your boat from the wave and how quickly it is done. Since the wave doesn't lose much energy bouncing off the end of the raingutter, it still has the energy you gave it from your initial breath! The boat is probably still in the same depth of water and at the same angle. If you have tired and are not blowing STRONGER, that wave will STOP your boat or move it BACKWARD - unless ...

The trick is to pass over the wave by blowing down on the back of your boat or the bottom of your sail to lift the bow (nose) over the wave. Your boat goes up and over instead of stopping!