(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.
(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?
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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.
(a) Number
Units
(b) Number
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Units
(c) Number
i
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[B] Earthquakes generate two sorts of waves in rock, shear or 5 waves which travel at about 4.5
km/s and pressure or P waves traveling at about 8 km/s.
(1) After one such quake, a seismograph station picks up the S wave 3.0 minutes after the P wave.
How far away was the source, or epicenter, of the earthquake? |
(2) When a quake is detected, a steady 250 Hz alarm sounds in the hallway of a facility to alert the
person on watch. The alarm is also sent to the cell phones of the staff on duty which emit the same
250 Hz tone. One of the staff responding to the alert runs down the hall, approaching the hallway
speaker. She notices a 2 Hz variation in the loudness of the 250 Hz tone as she runs with her phone.
What frequency is she hearing from the speaker? From your answer, how fast was she running, if the
speed of sound is 344 m/s?
A wave traveling through granite comes to a boundary with sandstone. The effective bulk
modulus and density for each of these rocks is given in the table below.…
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