Dimensional analysis. Waves on the surface of the ocean do not...

Question

Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary 'return force' for water piled up in the wave crests is due to the gravitational attraction of the Earth. Thus the speed v (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave's wavelength λ. Assume the wave speed is given by the functional form v = Cgᵅ hᵝ λᵞ, where α , β , c and C are numbers without dimension. In deep water, the water deep below the surface does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7 and Appendix D), determine the formula for the speed of surface ocean waves in deep water.

Accepted Answer

Hey, everyone in this problem, we're asked what the dimensions of the quantity R should be. And where R is formed through the following combination, R is equal to the square root of H multiplied by C divided by G where G is the gravitational constant, H is planks constant and C is the speed of light. OK. Three of those universal constants, we're given four answer choices. Option A is that the dimensions are in mass. Option B, the dimensions are length multiplied by time. Option C dimensions are length and option D time. OK? All right. So let's start by writing out our equation. We have that R is going to be the square root of H multiplied by C divided by capital G. Let's take a look at each of these constants on their own and figure out the dimension of them before we move forward. OK. So if we look at the dimension of age, recall that H is going to be measured N and let me, let's do this first. OK. H is in jewel seconds. OK. Recall that we can write a jewel as a kilogram meter square per second. Quit. And so our unit is gonna be kilogram meter squared per second squared multiplied by seconds or we can write it as kilogram meters squared per second. OK. So when we're looking at the dimensions, the dimensions of H are going to be, well, we have kilogram which is a mass multiplied by meter squared length squared divided by seconds, which is time. OK. So we have meter or sorry mass multiplied by length squared divided by time. That's the dimension of age. We're gonna do the same for the other two. So C is measured in meters per second. It's a speed in those standard units. OK? And so the dimension of C it is going to be the meters which is a link divided by seconds, which is time length divided by time. OK? And finally the gravitational constant G OK. This is measured in Newton meters squared per kilogram squared. Hm. Recall that a Newton can be written as a kilogram meter per second squared multiplying by meters squared per kilogram squared still. And so gravitational constant G in terms of units can be written as meters cute. And one of these kilograms is going to divide it. So we have meters cubed divided by second squared multiplied by kilogram. All right. So if writing the dimensions of G, what do we have? Well, we have a length cubed in the numerator. OK. Those meters cute, we have second squared in the denominator. So that's time squared. And we have kilogram in the denominator as well. That's a mass. So we have L cube divided by T squared M. OK? So we have all of the dimensions of all of these constants. Now, we can put them all into this equation. OK. It's almost like substituting in values. OK. We're gonna substitute these into this equation. Simplify as much as we can and see what the units of R are going to be. OK? Let's do it. I'm gonna do it in blue just so it's a little bit different from each of these individually. We can just see the difference. OK. So the dimensions of R are going to be equal to the dimensions of everything on the right hand side. OK. So we are going to get the square root and I'm gonna draw that square root at the end when we know how much space we need of H which has dimension meters multiplied by length squared, divided by time multiplied by the units of the planks constant, not planks constant. Sorry, the speed of light, which has dimensions L divided by T and all of that is gonna be divided by the dimensions of G which is led divided by T squared. Yeah. OK. And again, all of this is in under a square root. All right. So let's simplify where we can, we have a fraction. Hm Actually, we're dividing two fractions. So we can use our rules, we can flip the denominator and multiply And so we end up with M multiplied by elsewhere, divided by T, multiplied by L, divided by T multiplied by T squared, multiplied by M divided by LC. OK. And again, when I say MLT, I'm talking about the dimensions here. OK. So now we've had have this all written out and we can go ahead and simplify. No, let's collect like terms in the numerator and the denominator and see what we have. So we have that the dimensions of R are gonna be equal to the square root in the numerator. We have an M term and then another M term. So we're gonna have M squared, we have an L square term and then an L term that's gonna give us L cute. And then we have a T square. So we've collected all light terms in the numerator that's gonna be divided by and let's collect light terms in the denominator. We have T multiplied by T that's gonna give us T squared. And then we have our L cube term and now that everything is collected, it's easier to look at. And what we can see is that we have a T squared term in the numerator and denominator that's gonna divide out. We have an L cube term numerator and denominator that's gonna divide out and we're gonna be left with the dimension of R being equal to the square root of the dimension M squared. OK? Taking the square root of something that's squared, we're just going to get that something back. And so we get that the dimensions of R are just going to be equal to mass. OK? And so if we were to get a value for R, we're gonna have a mass. OK? And if we use the standard unit for all of those constants, that's going to be in kilograms. All right. So let's compare this to our answer choices. Now, what we can see is that the correct answer is going to be option A. We have the dimensions of mass or R. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Dimensional Analysis

Wave Speed in Deep Water

Non-dimensional Parameters

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