Mastering Physics Solutions: Surface Waves

Mastering Physics Solutions: Surface Waves

On December 26, 2011, in Chapter 13: Vibrations and Waves, by Mastering Physics Solutions

Surface Waves

Part A = sqrt((λg)/(2π))
Part B = 3.5 m/s Click to use the calculator/solver for this part of the problem
Part C = λ/sqrt((λg)/(2π))
Part D = 25,6.25 m, m/s Click to use the calculator/solver for this part of the problem
Part E = 350,23 m, m/s Click to use the calculator/solver for this part of the problem

Solutions Below:

The waves on the ocean are surface waves: They occur at the interface of water and air, extending down into the water and up into the air at the expense of becoming exponentially reduced in amplitude. They are neither transverse nor longitudinal. The water both at and below the surface travels in vertical circles, with exponentially smaller radius as a function of depth.

Both empirical measurements and calculations beyond the scope of introductory physics give the propagation speed of water waves as:

v = sqrt(g/k)

where g = 9.8m/s2 is the magnitude of the acceleration due to gravity and k is the wavenumber.

This relationship applies only when the following three conditions hold:

1. The water is several times deeper than the wavelength.
2. The wavelength is large enough that the surface tension of the waves can be neglected.
3. The ratio of wave height to wavelength is small.

The restoring force (analogous to the tension in a string) that restores the water surface to flatness is due to gravity, which explains why these waves are often called “gravity waves.”

Part A

Find the speed v of water waves in terms of the wavelength lambda.
Express the speed in terms of g, λ, and π.

Give your answer in meters per second to a precision of two significant figures.

k is given by 2π/λ. Since v = sqrt(g/k) and the formula is dividing by k, the formula for v in terms of g, λ, and π is:

v = sqrt((λg)/(2π))

Part B

Find the speed v of a wave of wavelength λ = 8.0 m.

Just use the formula from Part A:

v = sqrt((λg)/(2π))
v = sqrt((8*9.8)/(2*3.14159))
v = sqrt(78.4 / 6.28)
v = sqrt(12.48)

v = 3.5m/s

Part C

Find the period T for a wave of wavelength λ.

T is just λ/v, and using the formula from Part A (so that everything is in the terms this problem asks for) gives us:

T = λ/v

T = λ/sqrt((λg)/(2π))

Part D

On the East Coast of the United States, the National Weather Service frequently reports waves with a period of 4.0 s. Find the wavelength lambda and speed v of these waves.

Use the formula from Part C, above, to find λ:

T = λ/sqrt((λg)/(2π))
4 = λ/sqrt((9.8λ)/6.28)
4 = λ/sqrt(1.56λ)
42 = λ2/1.56λ
16 = λ/1.56
λ = 24.96
λ = 25 (2 significant figures)

Next, solve for v:

v = sqrt((λg)/(2π))
v = sqrt((25 * 9.8) / (2 * 3.14))
v = sqrt(39.013)
v = 6.25 (yes, it’s 3 sig figs, but MP gives this answer)

So the answer is:

λ, v = 25, 6.25

Part E

On the West Coast of the United States, the National Weather Service frequently reports waves (really swells) with a period of 15 s. Find the wavelength λ and speed v of these waves.

Express your answers numerically as an ordered pair separated by a comma. Give an accuracy of two significant figures.

Same method as Part D:

Find λ first:

T = λ/sqrt((λg)/(2π))
15 = λ/sqrt((9.8 * λ)/(6.28))
15 = λ/sqrt(1.56λ)
152 = λ2/(1.56λ)
225 = λ2/(1.56λ)
λ = 351
λ = 350 (2 significant figures)

Now find v:

v = sqrt((λg)/(2π))
v = sqrt((350 * 9.8) / (2 * 3.14))
v = sqrt(546.18)
v = 23.37
v = 23 (2 significant figures)

So the answer is:

λ, v = 350, 23

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