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(C) A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density p and depth so that the equation of the surface is z = n(x, t) where z is measured vertically upwards from the undisturbed surface (z = 0). You are given that for n(x, t) = E e sin(kx wt) with < 1, the pressure in the fluid can be written P= Papgz + p¶, where Pa is the constant atmospheric pressure and is given by = eg sin(kx - wt) cosh[k(z + h)]/cosh(kh). Here k and w are related through the dispersion relation c = gk tanh kh. Consider the form of this flow in the allow water limit, kh→0. 1. What is the dispersion relation in this limit? 2. What is the wave speed? 3. Show that to leading order in kh the horizontal component of velocity is the same at all depths. 4. Show that to leading order in kh the vertical component of velocity varies linearly with depth, satisfying the kinematic boundary conditions on y = 0 and y = -h.

(C) A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density p and depth so that the equation of the surface is z = n(x, t) where z is measured vertically upwards from the undisturbed surface (z = 0). You are given that for n(x, t) = E e sin(kx wt) with < 1, the pressure in the fluid can be written P= Papgz + p¶, where Pa is the constant atmospheric pressure and is given by = eg sin(kx - wt) cosh[k(z + h)]/cosh(kh). Here k and w are related through the dispersion relation c = gk tanh kh. Consider the form of this flow in the allow water limit, kh→0. 1. What is the dispersion relation in this limit? 2. What is the wave speed? 3. Show that to leading order in kh the horizontal component of velocity is the same at all depths. 4. Show that to leading order in kh the vertical component of velocity varies linearly with depth, satisfying the kinematic boundary conditions on y = 0 and y = -h.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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(C) A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density p and depth h, so that the equation of the surface is z = n(x, t) where z is measured vertically upwards from the undisturbed surface (z = 0). = e sin(kx - wt) with € < 1, the pressure in the fluid can be You are given that for n(x, t) written P= Papgz + po, where Pa is the constant atmospheric pressure and is given by = eg sin(kx - wt) cosh[k(z + h)]/cosh(kh). Here k and w are related through the dispersion relation c = gk tanh kh. Consider the form of this flow in the shallow water limit, kh → 0. 1. What is the dispersion relation in this limit? 2. What is the wave speed? 3. Show that to leading order in kh the horizontal component of velocity is the same at all depths. 4. Show that to leading order in kh the vertical component of velocity varies linearly with depth, satisfying the kinematic boundary conditions on y = 0 and y = :-h.
Transcribed Image Text:(C) A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density p and depth h, so that the equation of the surface is z = n(x, t) where z is measured vertically upwards from the undisturbed surface (z = 0). = e sin(kx - wt) with € < 1, the pressure in the fluid can be You are given that for n(x, t) written P= Papgz + po, where Pa is the constant atmospheric pressure and is given by = eg sin(kx - wt) cosh[k(z + h)]/cosh(kh). Here k and w are related through the dispersion relation c = gk tanh kh. Consider the form of this flow in the shallow water limit, kh → 0. 1. What is the dispersion relation in this limit? 2. What is the wave speed? 3. Show that to leading order in kh the horizontal component of velocity is the same at all depths. 4. Show that to leading order in kh the vertical component of velocity varies linearly with depth, satisfying the kinematic boundary conditions on y = 0 and y = :-h.
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