Holooly Plus Logo
Holooly Plus Logo

Question 4.8: Find the periods of free oscillations in the following cases......

Find the periods of free oscillations in the following cases:

1. A long narrow lake of uniform depth 15 m and length 5 km.

2. An open-ended channel of depth 20 m and length 10 km.

 

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.
Report Answer

In the first case we may use the 1-dimensional version of Equation 4.55

T=\frac{2}{\sqrt{g h} \sqrt{\left(m^2 / A^2\right)+\left(n^2 / B^2\right)}} \quad(m=0,1, \ldots ; \quad n=0,1, \ldots)    (4.55)

as the lake is narrow. The periods of free oscillations are given by

T=\frac{2}{\sqrt{gh} }\frac{A}{m}   \left(m=0,1,…\right)\\ =\frac{10,000}{m\times 12.1} \\=826.4,413.2,275.5,206.6,…\left(sec\right).

In the second case as the channel is open-ended we use Equation 4.56, giving the periods of free oscillation as

T\frac{4A}{\sqrt{gh}\left(2m+ 1\right) }  \left(m=0,1,…\right)  (4.56)\\=\frac{40,000}{\left(2m+ 1\right)\times 14.0 }\\ =2857,952.4,571.4,408.2,…\left(sec\right).

Related Answered Questions

Question: 4.1

Calculate the tide-generating force on the Earth due to the Earth-Moon and Earth-Sun systems, given that the mass of the Earth, (m), is = 5.98 × 10^24 kg, the mass of the moon, (M), = 7.35 × 10^22 kg, the major semi-axis of the lunar orbit around the Earth, (r), is 3.84 × 10^8 m, and the mean radius ...

Verified Answer:

We may estimate the magnitude of the effect of the...
Question: 4.2

The tidal constituents for four harbours are given in the following table. Classify the tidal regime at each harbour using the tidal ratio. Estimate the maximum tide level at each harbour. Calculate the length of the spring-neap cycle at harbours A and D. The mean water level relative to the local ...

Verified Answer:

We calculate F from the amplitudes of the constitu...
Question: 4.6

Derive a suitable approximation for the Coriolis parameter for regional scale motions centred on a latitude of θ0. ...

Verified Answer:

We expand the Coriolis parameter in a Taylor serie...
Question: 4.4

This example illustrates how information on spring and neap tidal information can be related to tidal constituents to derive further information to assist in a practical problem of safely docking a ship in harbour. A port is located in an estuary, upstream of a bridge whose deck base is at +20 m CD ...

Verified Answer:

a. From the tidal range information we can determi...
Question: 4.5

Given the following information for observed ocean tide levels (semidiurnal) at an established gauge and a new survey site, calculate the sounding datum for the new site. ...

Verified Answer:

Following the steps outlined in the section above:...
Question: 4.9

Nonlocal forcing results in an oscillation of frequency ω0 whose amplitude at the mouth of a channel is Q. If the channel is of constant depth H and of length A, determine a general expression for the amplitude of the oscillation at the head of the channel. Calculate specific values for the case ...

Verified Answer:

The governing equation is \frac{\partial²\e...
Question: 4.11

A wave in a tsunami has a period of 30 minutes and a height, Ho, of 0.5 m at a point where the ocean has a depth of 4 km deep. Calculate the phase speed, co, and wavelength, Lo, of this wave. Calculate its phase speed, ci, wavelength, Li, and height, Hi, in a coastal water depth of 15 m accounting ...

Verified Answer:

We assume the wave is a shallow water wave so that...
Question: 4.3

This example illustrates the differences between tidal range, amplitude and level with a nonstandard choice of chart datum. The amplitudes of the main tidal constituents (in metres) at a location (X) are shown in the table below, where the charted mean water depth (h) is also given. The chart datum ...

Verified Answer:

We calculate F from the amplitudes of the constitu...
Question: 4.7

This example demonstrates how a cotidal chart can be used to reveal information about the depth of the seabed. Use the phase information for M2 in the cotidal chart below to estimate the speed of the tidal wave along the coast of Finnmark and Kola. You may assume that 1° longitude at 70° latitude ...

Verified Answer:

Remember we have been asked to estimate, not calcu...