(II) Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary “return force” for water piled up in the wave crests is due to the gravitational attraction of the Earth. Titus the speed υ (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave’s wavelength λ . Assume the wave speed is given by the functional form υ = Cg α h β λ γ , where α , β , γ , and C are numbers without dimension. ( a ) In deep water, the water deep below does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7), determine the formula for the speed of surface waves in deep water. ( b ) In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e.. γ = 0). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.
(II) Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary “return force” for water piled up in the wave crests is due to the gravitational attraction of the Earth. Titus the speed υ (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave’s wavelength λ . Assume the wave speed is given by the functional form υ = Cg α h β λ γ , where α , β , γ , and C are numbers without dimension. ( a ) In deep water, the water deep below does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7), determine the formula for the speed of surface waves in deep water. ( b ) In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e.. γ = 0). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.
(II) Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary “return force” for water piled up in the wave crests is due to the gravitational attraction of the Earth. Titus the speed υ (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave’s wavelength λ. Assume the wave speed is given by the functional form υ = Cgαhβλ γ, where α, β, γ, and C are numbers without dimension. (a) In deep water, the water deep below does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7), determine the formula for the speed of surface waves in deep water. (b) In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e.. γ = 0). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.
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Tsunamis are fast-moving waves often generated by underwater earthquakes. In the deep ocean their amplitude is barely noticab
but upon reaching shore, they can rise up to the astonishing height of a six-story building. One tsunami, generated off the Aleutian
islands in Alaska, had a wavelength of 679 km and traveled a distance of 3410 km in 4.93 h. (a) What was the speed (in m/s) of the
wave? For reference, the speed of a 747 jetliner is about 250 m/s. Find the wave's (b) frequency and (c) period.
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(c) Number
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(I) AM radio signals have frequencies between 550 kHz and 1600 kHz (kilohertz) and travel with a speed of 3.0 x 108 m/s What are the wavelengths of these signals? On FM the frequencies range from 88 MHz to 108 MHz(megahertz) and travel at the same speed. What are their wavelengths?
A creature can detect very small objects, such as an insect whose length is approximately equal to one wavelength of the sound the bat makes. If a bat emits chirps at a frequency of 56 kHz, and if the speed of sound in air is 315 m/s,
what is the smallest insect (in mm) the bat can detect?
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