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Chapter 16: Q85. (page 454)

A water wave is a shallow-water wave if the water depth d is less than 鈮� l/10. It is shown in hydrodynamics that the speed of a shallow-water wave is v = 1gd, so waves slow down as they move into shallower water. Ocean waves, with wavelengths of typically 100 m, are shallow-water waves when the water depth is less than 鈮� 10 m. Consider a beach where the depth increases linearly with distance from the shore until reaching a depth of 5.0 m at a distance of 100 m. How long does it take a wave to move the last 100 m to the shore? Assume that the waves are so small that they don鈥檛 break before reaching the shore

Short Answer

Expert verified

t = 28.571 sec

Step by step solution

01

Given Info:

Given Info:

v=gd

Depth increases from 0 to 5m from distance 0 to 100m.

Slope of this surface = 5100=0.05

If x is the distance from shore, depth of water can directly be found as d(x) = 0.05x

similarly velocity can be stated as a function of x as v=gd(x)

02

Finding time

We know that time taken can be found as integral of inverse of velocity.

鈬�t=鈭�01001vdx

Substituting values in above equation

t=鈭�01001gd(x)dx

鈬�t=鈭�01001g脳0.05xdx鈬�t=10.05g鈭�01001xdx鈬�t=10.05g2x0100鈬�t=10.05g脳20鈬�t=28.571sec

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