The dispersion relation for water waves is 1 - exp(-2kh) w = (gk+Tk/p). 1+ exp(-2kh)' where...

  

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The dispersion relation for water waves is 1 - exp(-2kh) w = (gk+Tk/p). 1+ exp(-2kh)' where g is the acceleration due to gravity, 7 ~ 0.072 kg/s is the coefficient of surface tension, p = 103 kg/m is the density, and h is the depth of the water. (a) "Deep-water" waves propagate along the surface of the ocean with properties that are independent of the depth of the water. After simplifying the dispersion relation, find the phase and group velocities in this limit. [Answer for phase ve- locity: (g/k)+(Tk/p).] Show that the phase and group velocities are equal when the wavelength is 1.7 cm. What is the frequency (in Hz) and velocity of deep-water waves with 1.7 cm wavelength? (b) Show that for deep-water waves with wavelengths much greater than 1.7 cm the phase velocity is twice the group velocity. All mechanical waves involve a competition between a "restoring force" and "inertia." Physically, what is the restoring force for long-wavelength, deep-water waves? (c) Show that for deep-water waves with wavelengths much less than 1.7 cm the phase velocity is 2/3 of the group velocity. Physically, what is the restoring force for short-wavelength, deep-water waves? (d) Show that, in shallow water, long-wavelength waves are nondispersive. What are the phase and group velocities? [Answer: gh.] (e) For big summer swells at Newport Beach, the breakers arrive every fifteen seconds. What was the wavelength of these breakers in deep water? About how long did it take them to arrive from the South Pacific? The dispersion relation for water waves is 1 - exp(-2kh) w = (gk+Tk/p). 1+ exp(-2kh)' where g is the acceleration due to gravity, 7 ~ 0.072 kg/s is the coefficient of surface tension, p = 103 kg/m is the density, and h is the depth of the water. (a) "Deep-water" waves propagate along the surface of the ocean with properties that are independent of the depth of the water. After simplifying the dispersion relation, find the phase and group velocities in this limit. [Answer for phase ve- locity: (g/k)+(Tk/p).] Show that the phase and group velocities are equal when the wavelength is 1.7 cm. What is the frequency (in Hz) and velocity of deep-water waves with 1.7 cm wavelength? (b) Show that for deep-water waves with wavelengths much greater than 1.7 cm the phase velocity is twice the group velocity. All mechanical waves involve a competition between a "restoring force" and "inertia." Physically, what is the restoring force for long-wavelength, deep-water waves? (c) Show that for deep-water waves with wavelengths much less than 1.7 cm the phase velocity is 2/3 of the group velocity. Physically, what is the restoring force for short-wavelength, deep-water waves? (d) Show that, in shallow water, long-wavelength waves are nondispersive. What are the phase and group velocities? [Answer: gh.] (e) For big summer swells at Newport Beach, the breakers arrive every fifteen seconds. What was the wavelength of these breakers in deep water? About how long did it take them to arrive from the South Pacific?