# How much water is transported as a shallow water wave travels into a sea cave?

How much water is transported (volume) as a wave travels into a sea cave?

The wave has a height of 1m and a period of 12 seconds. The average water depth at the cave mouth it 5M and the width is 15m.

How do I calculate this? This is a real world problem and if there is any other information needed for this problem I will happily collect it.

This isn't a home work problem. This is a real cave and I assumed water is transported because it does "pile up" in the back of the cave until the wave is reflected and exits the cave in the opposite direction. Just as on beaches where waves push water up on to the beach and gravity pulls the water back to ocean creating the near shore current. I understand that deep water waves have an almost circular orbit but due to stoke's drift a particle will be slightly displaced in the direction the swell is moving. In this cave we can assume the wave is well with in the sallow water wave criteria and almost at the critical depth in which the wave deformation is to extreme and wave energy is about to be converted in to turbulent kinetic energy as the wave breaks. The orbit therefore would linear, as the wave moves in and out. The net change would be zero 0m^3, but how much water flows in and then flows back out of the cave?

• Is there any more water in the cave after the wave? Where is the transport exactly? Mar 6, 2017 at 15:14
• why don't you ask this at earthscience.se earthscience.stackexchange.com Mar 6, 2017 at 17:20
• Off hand it should be very little, as most of the the motion is transverse and it is mainly energy that goes forward.en.wikipedia.org/wiki/Wind_wave Mar 6, 2017 at 17:28

There's not enough detail here to really exact what's wanted, and if this is a HW problem, then it's maybe a 'trick' question.

Surface gravity waves do not transport ANY water. They rather transport energy. Any movement of water for shallow water waves is local to the water surface; the particles travel in elliptical orbits.

If waves moved water, then think of all the water that would 'pile up' at the shore!

Forward, backward motion of water is only realized when the depth becomes so shallow the wave height increases to a critical height and breaks on the shoreline, then runs back into the retreating void.

• I added some info to the question that hopefully clears up some confusion. Mar 7, 2017 at 17:37
• @surfer dude, if it's not a homework problem, then I suggest you edit your question and remove that tag Mar 7, 2017 at 17:50
• @surfer also willing to wager that the wave in the cave, no longer a shallow water wave in any sense of the canonical model for ocean waves. The model does not admit reflections and interference. So my guess is you need to start from first principles and develop a model. I doubt this one in any book; maybe in someone's experience. Good luck Mar 7, 2017 at 17:54
• Thanks a lot. I think I may be able to figure it out with a turbulent wave model but I need to do a bit more research to see whether it would be accurate under these parameters. Mar 7, 2017 at 18:34

If you knew the volume of water flowing through the mouth of the cave the job would be done.

You can assume the wave shape at the cave mouth to be sinusoidal (it's an estimate). You measured the peak of the sinusoid. The average height with the same volume is the peak height/sqrt(2).

The volume of water moving temporarily (rather than drifting) into the cave is the volume of this deviation for half the period - even though it all flows back again in the next half period.

You need to know the length of the sinusoid. You need to attempt to measure the distance between wave peaks.

period it gives you a length height of water. The width of the cave mouth. The volume of water deviation is therefore mean height*cave width (5m) * half your measured/estimated wave length.

Thus you could arrive at, I think at least some kind of rough volume estimate.

Does that seem about right to you?