Question

assuming small amplitude ( n p lane waves Consider a quiescent body of water of depth has shown in Fig. 4. A small amplitude plane wave I travelling along the surface of the water with velocity c. The form of waves is taken to be sodal in which the amplitude of the wave is and its wavelength is i. The equation of the free surface will be y (x ) = sin (x-e!) If the value of s , and h are known, find the expression of e as a function of c, 2, and h. Please discuss the cases when 2> or h i. (12%) Fig. 4 Hint: solve the governing equation together with velocity conditions at the free surface and bottom wall and the dynamic condition at the tree surma

Answer 1

As we know that the governing equation for wave propagation of small amplitude on surface of fluid are:

ᎣᎣ , ᎧᎧ . * Ꭷy2 2ᎢC 2Ꭲ , , ( Ꮖ , 0, t} = -£ , cos ,( Ꮖ - ct) ( , , ) + 9 , , ) = 0 Ꭷ , -h, ) = 0 y .

where $\phi$ is velocity potential and rest of the variables have same terminological meaning as question.

as the equations are stated over free surface, hence the solution will be either trigonometric or hyperbolic function. So, assuming the following solution,

6(x, y, t) = cos 2+ (x – ct)C, sinh 2014 + Cocosh Lay

for bottom boundary, i.e at y=-h, .

д/ду = 0

this gives,

2πη C = Ctanh -

for free surface, i.e y=$\eta$,

δώ Ρ 1 Οι ++5 φ· φ + gη = 0

we obtain the following expression:

Cy cos 24" (x – ct) – 27€ +924 tanh 24 h = 0

this equation is satisfied for all x and t. Also imposing small amplitude approximation, we get

gh = , taal. 27/

for :

h>>

2 gh 2 h

and for $h<<\lambda$ :

tanh 23h – 23h

which gives:

$\frac{c^{2}}{g h}=1$

where c = speed of wave