High School

When waves generated by tsunamis approach shore, the height of the waves generally increases. Understanding the factors that contribute to this increase can aid in controlling potential damage to areas at risk. Green's law explains how water depth affects the height of a tsunami wave.

If a tsunami wave has height \(H\) at an ocean depth \(D\), and the wave travels to a location of water depth \(d\), then the new height \(h\) of the wave is given by:

\[ h = H R^{0.25} \]

where \(R\) is the water depth ratio given by \(R = \frac{D}{d}\).

(Round your answers to two decimal places.)

(a) Calculate the height of a tsunami wave in water 20 feet deep if its height is 7 feet at its point of origin in water 20,000 feet deep.

(b) If water depth decreases by a third, the depth ratio \(R\) is increased by 1.5. How is the height of the tsunami wave affected?

The new height of a tsunami wave is \(x\) times the height before \(R\) is increased by 1.5.

Answer :

a). The height of the tsunami wave in water 20 feet deep is approximately 40.68 feet.

b). The new height of the tsunami wave, h₂, is x times the height before R is increased by 1.5, where [tex]x = (R + 1.5)^{0.25}[/tex].

(a) To calculate the height of a tsunami wave in water 20 feet deep if its height is 7 feet at its point of origin in water 20,000 feet deep, we need to find the water depth ratio R and then use it in the formula

[tex]h=H*R^{0.25}[/tex]

Given:

H = 7 feet (height at the point of origin)

D = 20,000 feet (ocean depth)

d = 20 feet (water depth)

We can calculate the water depth ratio R using R = D/d:

R = 20,000 feet / 20 feet

R = 1000

Now, substitute the values of H and R into the formula to find the new height h:

h = 7 feet * 1000^0.25

Using a calculator or mathematical software to evaluate the expression:

h ≈ 40.68 feet

Therefore, the height of the tsunami wave in water 20 feet deep is approximately 40.68 feet.

(b) If the water depth decreases by a third, the depth ratio R is increased by 1.5.

We need to determine how this change in R affects the height of the tsunami wave.

Let's say the height of the tsunami wave before the change in R is denoted as H₁, and the new height after the change is denoted as H₂.

We have the relationship: H₂ = x * H₁,

where x is the factor by which the height is affected.

Given that the depth ratio R increases by 1.5, we can write the new depth ratio R₂ as:

R₂ = R + 1.5

We can express R₂ in terms of the original depth ratio R as:

R₂ = R + 1.5

= (D/d) + 1.5

From Green's law, we know that [tex]h_2 = H_2 * R_2^{0.25}[/tex].

Substituting H₂ = x * H₁ and

R₂ = R + 1.5, we get:

[tex]h_2 = (x * H_1) * (R + 1.5)^{0.25[/tex]

To find the relationship between the new height h₂ and the original height H₁, we can divide both sides of the equation by H₁:

[tex]h_2 / H_1 = x * (R + 1.5)^{0.25[/tex]

Therefore, the new height of the tsunami wave, h₂, is x times the height before R is increased by 1.5, where [tex]x = (R + 1.5)^{0.25}[/tex].

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The new height of the tsunami wave is 0.93 times the height before R is increased by 1.5.

(a) Calculation of height of tsunami wave in 20 feet deep water, given that its height is 7 feet at the origin (in water 20,000 feet deep) is as follows:

First, we need to calculate the ratio of the depth of water at origin to the depth of water at the given location.

The ratio is R = D/dR

= 20000 / 20R

= 1000

The new height of the tsunami wave h is given by

h = HR0.25h

= 7 x (1000)0.25h

= 7 x 5.62h

= 39.34 feet

Therefore, the height of a tsunami wave in water 20 feet deep is 39.34 feet. (rounded to two decimal places)

(b) Given that the depth ratio R is increased by 1.5 when water depth is decreased by a third. The new height of a tsunami wave is x times the height before R is increased by 1.5 is to be determined.The formula to find the new height is:

h = HR0.25

The depth ratio R is increased by 1.5, which means that the new value of R is R + 1.5h = H(R+1.5)0.25

Hence, the new height of the tsunami wave is x times the height before R is increased by 1.5 is given by

x = h / h'

where h is the original height and h' is the new height.

From the above formula, h' = H(R+1.5)0.25

Therefore, x = h / [H(R+1.5)0.25]

Substitute the given values to calculate x.

We know that H = 7, R = 1000 and the new value of R is

R + 1.5 = 1001.5x

= 7 / [7(1001.5)0.25]x

= 0.93

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