SMS 204: Physics for marine sciences

Instructor: E. Boss

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Review: Waves

• What characterizes waves:
• Periodic behavior in time or space.
• Examples: ???
• We will focus on gravity waves (where gravity is the restoring force).

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Anatomy of a wave:

Wave packet:

Approximated as:

Asin{2π(x/λ-t/T)}+Bcos{2π(x/λ-t/T)}=C sin{2π(x/λ-t/T)+φ}

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Wave propagation:

• Plane wave propagating in positive x direction:

h=Asin{2π(x/λ-t/T)+φ}

Phase: 2π(x/λ-t/T)+φ

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• Speed of propagation:
• Phase speed (celerity): c=λ/Τ
• Group speed: speed of energy propagation.
• Not necessarily the same (or even in the same direction).

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Deep and shallow-water waves (H-water depth):

• Different behavior

Deep water waves: λ<2H, C=[gλ/2π]^1/2

Different waves (λ) have different speed-dispersion.

c_g=0.5C

Implications for waves leaving a storm

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Shallow water waves: λ>20H, C=[gΗ]^1/2

Different waves have the same speed-

non-dispersive.

c_g=C

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What is a Tsunami [movie]?

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How is the water moving when a wave passes?

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What determines how deep a surface wave is felt?

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Why do wave break?

How about in the middle of the ocean?

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What forces create gravity waves in the ocean?

Capillary waves - the smallest and steepest waves.

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Restoring force: surface tension + gravity

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η=Asin{2π(x/λ-t/T)}

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p={ρg+γ(2π/λ)²}η

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c=[(2π/λ) (g+γ(2π/λ)²/ρ]^1/2

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When λ>>2π(γ/ρg)^1/2 (about 2cm for pure water) the wave is a gravity wave.

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Some questions about waves:

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1. What do wave transport?

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• How and why do they evolve (attenuate) as they propagate?

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Example: the wake of a boat.

From wiki-media

Movie

• Hull speed
• Speed of wave formed at the surface by moving an object (boat, duck)=(gL/(2π))^1/2

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• Going faster than hull speed results in excessive drag.

http://students.washington.edu/~ukc/library/052902-1notes.pdf

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Surface: superposition of many waves:

Demo: beating.

Resonance

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• Physical construct have natural frequencies based on their dimensions (think about musical instruments).

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• Forcing at these frequencies (among others) result in large response at the resonant (s) frequency (ies).

Breaking news:

Example: Seiches – the natural modes of a basin

http://www.islandnet.com/~see/weather/almanac/arc2004/alm04jun.htm

In an enclosed basin: the 1^st (usually dominant) mode is half wavelength long.

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In a basin open at one side: the 1^st mode is a quarter wavelength long.

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The seiche is the ‘natural’ wave of the basin.

How do we determine the period of a seich for a basin of depth H and length L?

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Internal waves:

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Largest gravity waves in the ocean.

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Very un-intuitive [link].

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Wave energy

• E=(ρga²)/2, per L² both kinetic and potential
• Propagates with c_g
• energy flux (per L of beach): c_gE

Class exercise:

How much energy flux is generated by 1m high 8s deep wind waves per 1m of wave front?

Doppler shift

• Change in frequency due to the motion of the source and/or the receiver
• Allows for determination of movement of target.

Stationary source:

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f = c/λ

f’=(c±u_r)/λ

🡪 Δf =±u_r/λ=fu_r/c

Stationary receiver:

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Δf=±fu_s/(c±u_s)~ ±fu_s/c

Both moving:

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Δf~f(±u_s±u_r)/c

c-speed

λ-wavelength

f=frequency

u_r,s-speed of receiver or source

Most common method to measure currents

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Summary:

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Wave carry information and energy.

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Waves do not carry matter effectively.

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