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||| A water wave is called a deep-water wave if the water’s depth

is more than one-quarter of the wavelength. Unlike the waves

we’ve considered in this chapter, the speed of a deep-water wave

depends on its wavelength:

v = B

gl

2p

Longer wavelengths travel faster. Let’s apply this to standing waves.

Consider a diving pool that is 5.0 m deep and 10.0 m wide. Standing

water waves can set up across the width of the pool. Because

water sloshes up and down at the sides of the pool, the boundary

conditions require antinodes at x = 0 and x = L. Thus a standing

water wave resembles a standing sound wave in an open-open tube.

a. What are the wavelengths of the first three standing-wave

modes for water in the pool? Do they satisfy the condition for

being deep-water waves?

b. What are the wave speeds for each of these waves?

c. Derive a general expression for the frequencies fm of the possible

standing waves. Your expression should be in terms of m, g, and L.

d. What are the oscillation periods of the first three standing wave

Step by step solution

01

Description on wave lenghts.

It deals the distance between two lengths

02

Description on solution

$$\begin{gathered} f=\frac{1}{\mathrm{λ}}.\sqrt{\frac{g\mathrm{λ}}{2\mathrm{π}}}=\sqrt[]{\frac{g}{2\mathrm{π}\mathrm{λ}}} \\ {f}_m=\sqrt[]{\frac{g}{2\mathrm{π}}}\sqrt{\frac{1}{\mathrm{λ}}}=\sqrt[]{\frac{g}{2\mathrm{π}}}.\sqrt{\frac{m}{2L}} \end{gathered}$$

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