The Nortonville River is known for its periodic tidal waves that occur every 12 hours. One day, a science team measures the wavelength and frequency of the tidal wave at a specific point of the river. They find that the wavelength of the wave is 40 meters and the frequency is 0.1 Hz.
a) Calculate the speed of the tidal wave.
b) If the amplitude of the wave is 2 meters, determine the wave's maximum displacement and minimum displacement at the given point.
c) Assuming the tidal wave propagates in a medium with a density of 1000 kg/m³, calculate the wave's energy density.
Answer:
a) The speed of a wave can be calculated by multiplying its wavelength (λ) with its frequency (f). Therefore, the speed (v) is given by the formula:
v = λ * f
Substituting the given values:
v = 40 m * 0.1 Hz
v = 4 m/s
Therefore, the speed of the tidal wave is 4 m/s.
b) The maximum displacement of a wave is equal to its amplitude (A), while the minimum displacement is equal to the negative of its amplitude (-A). Thus, the maximum and minimum displacements at the point are:
Maximum displacement = 2 m Minimum displacement = -2 m
c) The energy density of a wave can be calculated using the formula:
Energy density = (1/2) * ρ * (A²) * v²
where ρ is the density of the medium, A is the amplitude, and v is the speed.
Substituting the given values:
Energy density = (1/2) * 1000 kg/m³ * (2 m)² * (4 m/s)²
Energy density = 8,000 J/m³
Therefore, the wave's energy density at the given point is 8,000 J/m³.