Surface Waves

Part A = sqrt((λg)/(2π))

Part B = 3.5 m/s

Part C = λ/sqrt((λg)/(2π))

Part D = 25,6.25 m, m/s

Part E = 350,23 m, m/s

Solutions Below:

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/s^{2} 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

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

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

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

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)

4^{2} = λ^{2}/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

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)

15^{2} = λ^{2}/(1.56λ)

225 = λ^{2}/(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