Fluent Essays

Only problems 2-5 are needed. No need for problem1. Chapter 6 or 7(im not sure which one it is) in the textbook Im providing might be helpful and contain most of the contents you need to know. The problems should be easy if you read the chapter, but I dont have time to read or study it😢 Really need it to be done in 12 hrs and please guarantee the correctness as much as you can. Thanks a lot:) .homework.5.pdf .textbook.pdf Unformatted Attachment Preview SIO 111/Physics 111 Spring 2020 Homework #5. 1. (a) Suppose that the temperature is given by T ( x,t ) = β x where β = 8 deg / meter . At what speed and in what direction (toward positive or negative x) would an observer need to move in order to observe a temperature change of 20 deg / sec? (b) Next suppose that the temperature is given by T ( x,t ) = β x + γ t where β is as given above and γ = 2 deg / sec . At what speed and in what direction would an observer need to move in order to observe no temperature change at all? 2. In chapter 1 we showed that (assuming the shallow-water limit), the wavenumber of a shoaling wave can be predicted from ω 0 = gH ( x ) k ( x ) . Show that the same result is obtained by solving the equation (cf. 6.21) cg d ∂Ω d k ( x) = − H ( x) dx ∂H dx with Ω ( k, H ) = gH k . 3. Waves with wavelength 300m approach the coast at an angle of 60 degrees in deep water, as shown. What is the corresponding angle in water 5m deep? The full depth profile H(x) is unknown, but it does not depend on the longshore coordinate y. 4. Measurements of the bathymetry near a straight beach show that H ( x ) = −α x where α , the bottom slope, is a positive constant. A wave with frequency ω 0 and longshore wavenumber l0 approaches the shore at an angle, as shown. (In the figure, x1 ≡ x and x 2 ≡ y .) Show that, in sufficiently shallow water, the rays obey the equation − gα x dy = l0 dx ω 02 Using the fact that wave crests (or troughs) are everywhere perpendicular to the rays, write the corresponding equation for wave crests. Solve the equation for the wave crests, and show that in very shallow water the wave crests have the shape of parabolas, as seen in the figure. (All of this assumes that waves reach the shoreline without breaking.) Hint: If two curves are locally perpendicular, then the slope of one equals the negative reciprocal of the other. 5. An ocean on x<0 has the depth field H ( x, y ) = −α x (1+ ε cos(βy )) Write down the four coupled equations you would need to solve to determine the path of a ray, and the values of k and l along it, assuming shallow-water dynamics. Introduction to Ocean Waves Rick Salmon Scripps Institution of Oceanography University of California, San Diego Preface Wind waves, with periods of a few seconds, and the tides, with periods of twelve hours or more, are really two examples of the same physical phenomenon. They differ only in the source of their energy. For the shortestperiod waves—periods of, say, one to four seconds—the connection between the wind and the waves is obvious. On a windless day, the surface of Mission Bay is dead flat. But if the wind begins to blow, waves appear within a few minutes and grow steadily in amplitude to a saturation level that depends on the strength of the wind. When the wind stops blowing, the waves gradually decay. For the breakers on Scripps Beach, the connection with the wind is not so obvious, but these longer-period waves—periods of, typically, ten seconds— are wind-generated too. Their wind sources are powerful storms that may have occurred days ago and thousands of miles away. Only the very strong winds associated with these storms can generate these long, fast-moving waves. Because their wavelengths are so long, these waves experience very little dissipation; they lose little of their energy on their long cross-ocean trip to San Diego. The energy in these long waves travels at a speed that increases with the wavelength. Because of this, these far-traveling waves sort themselves by wavelength during the long trip, with the longest waves reaching San Diego first. This sorting explains why the breakers on Scripps Beach often seem to have a single, well-defined period. In contrast to the wind waves, the tides receive their energy from the gravitational pull of the Sun and Moon. This energy source imposes a scale—a wavelength—that is comparable to the Earth’s radius and therefore not directly observable by eye. However, ancient peoples recognized the connection between astronomy and the tides merely by observing the regular periods of the tides. Because tidal periods are comparable to, or longer than, a day, the tides are strongly affected by the Earth’s rotation. Because the spatial 1 Salmon: Introduction to Ocean Waves 2 scale of the tides is so large, the tidal response to astronomical forcing is very sensitively dependent on the irregular shape of the ocean basins. Until very recently, this fact, and a misunderstanding of tidal dissipation mechanisms, defeated attempts at a quantitative physical explanation of the tides. This course was originally planned to cover both wind waves and tides. However, it was soon realized that ten weeks is barely to sufficient to cover either of these topics in any detail. Wind waves were selected as being of greater general interest. You can find lots of books about ocean waves. Nearly all of them fall into one of two categories: popular books full of pictures and sea lore, and textbooks written for people with a bachelor’s degree in a physical science. All of the latter assume a prior knowledge of fluid mechanics, or contain a general introduction to fluid mechanics as part of the book. In a ten-week course, we cannot afford to learn fluid mechanics before embarking on waves. Therefore, this textbook, which has been written especially for the course, avoids the need for a background in fluid mechanics by basing our study of waves on two fundamental postulates. These two postulates are: 1. The dispersion relation for ocean waves, which is introduced and explained in chapter 1. 2. The principle of wave superposition, which is explained and illustrated in chapters 2 and 3. Strictly speaking, these two postulates apply only to ocean waves of very small amplitude. Nevertheless, a great many useful facts may be deduced from them, as we shall see in chapters 4, 5, 6 and 7. Not until chapter 8 do we justify our two postulates on the basis of first physical principles—the conservation of mass and momentum by the fluid. Chapter 8 is a whirlwind introduction to fluid mechanics, but its primary goal is a very limited one: to justify the two fundamental postulates that will have already proved so useful. Chapters 9 and 10 apply the newly derived equations of fluid mechanics to tsunamis and to the physics of the surf zone. However, no great expertise in fluid mechanics is required for this course. In fact, the course is designed to give you just a taste of that subject, enough to decide if you want to learn more about it. However, you do need to know some math: basic differential and integral calculus, a bit of vector calculus, a good bit about ordinary differential Salmon: Introduction to Ocean Waves 3 equations, and at least a wee bit about partial differential equations. To determine if your math background is sufficient, have a look at the first few chapters. Contents 1 Basic waves 5 2 Two waves 20 3 Many waves 31 4 Waves generated by a distant storm 39 5 Wave measurement and prediction 50 6 Shoaling waves 62 7 Rogue waves and ship waves 76 8 Hydrodynamics and linear theory 90 9 The shallow-water equations. Tsunamis 103 10 Breakers, bores and longshore currents 117 4 Chapter 1 Basic waves To describe ocean waves, we use a right-handed, Cartesian coordinate system in which the z-axis points upward. The x- and y-axes point in horizontal directions at right angles. In the state of rest, the ocean surface coincides with z = 0. When waves are present, the surface is located at z = η(x, y, t), where t is time. The ocean bottom is flat, and it is located at z = −H, where H is a constant equal to the depth of the ocean. Refer to figure 1.1. Our first basic postulate is this: Postulate #1. If A|k|  1, then the equation η = A cos(kx − ωt) (1.1) describes a single, basic wave moving in the x-direction, where A, k, and ω are constants; and ω and k are related by p ω = gk tanh(kH). (1.2) Here, g = 9.8 m sec−2 is the gravity constant, and tanh(s) = es − e−s es + e−s (1.3) is the hyperbolic tangent function. This first postulate, summarized by (1.1) and (1.2), needs some elaboration and a lot of explaining. It will take us a while to do this. But before saying anything more about Postulate #1, we go on to state: 5 Salmon: Introduction to Ocean Waves 6 Figure 1.1: Ocean surface elevation in a basic wave. Postulate #2. Still assuming A|k|  1, we may add together as many waves satisfying Postulate #1 as we like; the result will be a physically valid motion. For example, η = A1 cos(k1 x − ω1 t) + A2 cos(k2 x − ω2 t) (1.4) is physically valid if the pairs (k1 , ω1 ) and (k2 , ω2 ) each satisfy the requirement (1.2), that is, if p p (1.5) ω1 = gk1 tanh(k1 H) and ω2 = gk2 tanh(k2 H) Using only these two postulates, we can explain quite a lot about ocean waves. Eventually we shall go deeper into the underlying physics, but for quite some time, these two postulates will suffice. Our immediate task is an elaboration of Postulate #1. Equation (1.1) describes a single basic wave with amplitude A, wavenumber k, and frequency ω. The frequency ω is always positive; the wavenumber k can be positive or negative. If k is positive, then the wave moves to the right, toward positive x. If k is negative, the wave moves to the left. The wave height—the vertical distance between the crest and the trough—is equal to 2A. See figure 1.2. The wavelength λ is related to the wavenumber k by λ= 2π |k| (1.6) and the wave period T is related to the frequency ω by T = 2π ω (1.7) Salmon: Introduction to Ocean Waves 7 Figure 1.2: The wave height is twice the wave amplitude A. The wave described by (1.1) moves in the x-direction at the phase speed c= ω k (1.8) which has the same sign as k. The restriction A|k|  1 is an important one. It says that the wave height must be small compared to the wavelength. In other words, the wave must have a small slope. Only then are (1.1) and (1.2) an accurate description of a physical wave. The restriction to small amplitude A means that we are considering what oceanographers call linear waves. (The logic behind this terminology will be explained later on.) The theory of linear waves cannot explain such things as wave breaking or the transfer of energy between one wave and another. Nevertheless, and because nonlinear wave theory is so much more difficult, this course is largely limited to linear waves. There is a further restriction on (1.1) and (1.2) which must be explained. These equations assume that the wave is neither being forced nor dissipated. That is, (1.1) and especially (1.2) describe a free wave. The equations apply best to the long ocean swells between the point at which they are generated by storms and the point at which they dissipate by breaking on a beach. Equation (1.1) could be considered a general description of almost any type of wave, depending only on the interpretation of η. It is the dispersion relation (1.2) that asserts the physics, and tells us that we are considering a water wave. The dispersion relation is a relation between the frequency ω and the wavenumber k. Alternatively it can be considered a relation between the phase speed c and the wavelength λ. The physical description (1.1-2) is incomplete. To have a complete description, we must specify how the fluid velocity depends on location and time. The fluid velocity is a vector field that depends on (x, y, z, t). We Salmon: Introduction to Ocean Waves 8 write it as v(x, y, z, t) = (u(x, y, z, t), v(x, y, z, t), w(x, y, z, t)) (1.9) For the wave described by (1.1) and (1.2), the y-component of v vanishes. That is, v = 0; there is no velocity out of the page. The x- and z-components are given by cosh(k(H + z)) u = Aω cos(kx − ωt) (1.10) sinh(kH) and w = Aω sinh(k(H + z)) sin(kx − ωt) sinh(kH) (1.11) These are somewhat complicated expressions. However, two limiting cases will claim most of our attention. From now on, we let the wavenumber k be positive. This means that we are focusing on waves moving to the right, but the generalization to left-going waves is obvious. Our first limiting case is the case of deep water waves, in which kH  1; the water depth is much greater than a wavelength. In the limit kH  1, ekH ekH − e−kH → =1 ekH + e−kH ekH ek(H+z) + e−k(H+z) ek(H+z) cosh(k(H + z)) = → = ekz sinh(kH) ekH − e−kH ekH sinh(k(H + z)) ek(H+z) − e−k(H+z) ek(H+z) = → = ekz sinh(kH) ekH − e−kH ekH tanh(kH) = (1.12) (1.13) (1.14) Thus the deep-water wave is described by DW DW DW DW η = A cos(kx − ωt) p ω = gk (1.15a) (1.15b) kz (1.15c) kz (1.15d) u = Aωe cos(kx − ωt) w = Aωe sin(kx − ωt) The abbreviation DW is a reminder that the equation applies only to deep water waves. Salmon: Introduction to Ocean Waves 9 Our second limiting case represents the opposite extreme. It is the case kH  1 of shallow water waves, in which the depth H is much less than the wavelength. In the limit kH  1, ekH − e−kH (1 + kH + · · · ) − (1 − kH + · · · ) = kH −kH e +e (1 + kH + · · · ) + (1 − kH + · · · ) 2kH → = kH (1.16) 2 tanh(kH) = cosh(k(H + z)) ek(H+z) + e−k(H+z) = sinh(kH) ekH − e−kH (1 + k(H + z) + · · · ) + (1 − k(H + z) + · · · ) = (1 + kH + · · · ) − (1 − kH + · · · ) 1 2 = (1.17) → 2kH kH ek(H+z) − e−k(H+z) sinh(k(H + z)) = sinh(kH) ekH − e−kH (1 + k(H + z) + · · · ) − (1 − k(H + z) + · · · ) = (1 + kH + · · · ) − (1 − kH + · · · ) 2k(H + z) → = (1 + z/H) (1.18) 2kH Thus the shallow-water wave is described by SW SW SW SW η = A cos(kx − ωt) p ω = gHk Aω u= cos(kx − ωt) kH w = Aω(1 + z/H) sin(kx − ωt) (1.19a) (1.19b) (1.19c) (1.19d) Swell far out to sea certainly qualifies as DW. According to (1.15b), the DW phase speed—the speed of the wave crests and troughs—is given by r r ω g gλ DW c= = = (1.20) k k 2π Salmon: Introduction to Ocean Waves 10 Figure 1.3: Fluid velocity in a deep-water wave moving to the right. Hence, waves with longer wavelengths travel faster. The phase speed (1.20) is independent of the wave amplitude A, and it is much larger than the fluid velocity (1.15c-d). The latter is proportional to A and is therefore infinitesimally small in linear theory. Because of the factor ekz in (1.15c-d), the fluid velocity decays with distance below the surface (becoming smaller as z becomes more negative). According to (1.15c), the fluid velocity is in the direction of wave propagation under the crest, and in the opposite direction under the wave trough. According to (1.15d), the fluid is rising ahead of the crest, and descending behind it. See figure 1.3. In the shallow-water limit (1.19), the horizontal velocity u is independent of z (figure 1.4). The vertical velocity w varies linearly with z, but√it is smaller than u by a factor kH. The shallow-water phase speed c = gH depends on H but not on the wavelength λ. In shallow water, waves of all wavelengths move at the same speed. Stand at the end of Scripps pier (if you can get past the gate) and watch the swells roll toward the beach. Because their vertical decay-scale is comparable to their wavelength, the waves extend downward a distance comparable to the spacing between wave crests. Where the ocean depth is greater than Figure 1.4: Fluid velocity in a shallow-water wave moving to the right. Salmon: Introduction to Ocean Waves 11 a wavelength, the depth might as well be infinite; the waves don’t feel the bottom. But where the ocean depth becomes smaller than a wavelength, the DW description becomes inaccurate, and we must pass over to the general description (1.1-2,1.10,1.11), which is valid for arbitrary H. When the waves reach shallow water—H much less than a wavelength—the simpler, SW description (1.19) applies. The foregoing paragraph ignores a subtlety in all the preceding equations: Strictly speaking, these equations apply only to the situation in which H is a constant. We may talk about deep water, and we may talk about shallow water, but we cannot—strictly speaking—use these equations to describe a situation in which H varies. However, common sense suggests that we may use our general description in the case where H changes very gradually. If the mean water depth H changes by only a small percentage in a wavelength, then the wave ought to behave as if the depth were constant at its local value. This turns out to be correct. Let x be the perpendicular distance toward shore. Let the mean water depth H(x) decrease gradually in the x-direction. Then x-directed, incoming waves ought to obey the slowly varying dispersion relation, p (1.21) ω = g k(x) tanh(k(x)H(x)) obtained by replacing k with k(x), and H with H(x), in (1.2). As H(x) decreases toward the shore, k(x) must increase; the wavelength shortens as the wave shoals. (To see this, take the derivative of (1.21) to show that dk/dx and dH/dx must have opposite signs.) In very shallow water (1.21) becomes p (1.22) SW ω = gH(x)k p and the phase speed c = gH(x). Why, you may ask, is it the wavenumber k and not the frequency ω that must change to compensate the change in H? The frequency is proportional to the rate at which wave crests pass a fixed point. Consider any two fixed points—any two values of x. If the frequency were different at the two points, then the number of wave crests between them would continually increase or decrease (depending on which point had the higher frequency). No steady state could exist. This is not the situation at the beach. Hence ω must be constant. (In chapter 6 we will prove that this is so, provided that the waves have small enough amplitudes, and that the depth changes gradually on the scale of a wavelength.) Salmon: Introduction to Ocean Waves 12 Back on the pier, look closely to see if this is true. As the waves approach shallow water, the wavelengths really do get shorter. The phase speed really does decrease. Later, when we develop this slowly varying theory more completely, we will show that other things are happening too. In particular, the wave amplitude A increases as the wave energy is squeezed into a smaller depth. Now lean over the pier and drop an object into the water. Observe where it goes. If the object is a surfboard, it may move at the phase speed all the way to the beach. If the object is a piece of tissue paper, it will move forward with each wave crest, and backward with each wave trough, with a very slow net movement toward the shore. The tissue paper is moving with the water—not with the wave. Its velocity is the same as the velocity of the surrounding fluid particles. What, then, is the trajectory of the fluid particles? First of all, what do we mean by a fluid particle? Fluid mechanicists usually ignore the fact that a fluid is composed of molecules and instead regard the fluid as a continuum—a continuous distribution of mass and velocity in space. This, it turns out, is a valid idealization if the fluid’s molecules collide with each other frequently enough. It is the continuum velocity to which (1.15c-d) refer. By fluid particle, we mean an arbitrarily small piece of this continuum. Let (xp (t), zp (t)) be the coordinates of a particular fluid particle, selected arbitrarily. We find the motion of the fluid particle by solving the coupled ordinary differential equations dxp = u(xp (t), zp (t), t) dt dzp = w(xp (t), zp (t), t) dt (1.23a) (1.23b) For DW, the velocity fields are given by (1.15c-d), so we must solve DW DW dxp = Aωekzp cos (kxp − ωt) dt dzp = Aωekzp sin (kxp − ωt) dt (1.24a) (1.24b) The exact solution of (1.24) is quite difficult (and also somewhat pointless because the right-hand sides of (1.24) are themselves approximations, valid only for small A). However, if A is small, the fluid part ... Purchase answer to see full attachment