A tsunamiis a fast-moving ocean wave caused by an underwater earthquake. The water first goes down from its normal level, then rises an equal distance above its normal level, and finally returns to its normal level. The period is about 15 minutes. Suppose a tsunami with an amplitude of 10 meters approaches a pier where the normal depth of the water is 9 meters.
a)If the water depth varies sinusoidally with time as the tsunami passes predict the water depth at the following times after the tsunami hits the pier. 2 minutes. 4 minutes. 12 minutes.
b)what will the minimum depth of the water be
c)between what two times is there no water at the pier
d)if a tsunami travels at 1200 kilometers per hour what is its wavelength
To answer these questions, we can use the equation for a sinusoidal wave:
y = A * sin(2π/T * t + ϕ)
Where:
- y is the height of the water (depth)
- A is the amplitude of the wave
- T is the period of the wave in seconds
- t is the time in seconds
- ϕ is the phase constant
Given that the amplitude A is 10 meters and the period T is 15 minutes (or 900 seconds) for the tsunami, we can now solve each part of the question:
a) To predict the water depth at specific times after the tsunami hits the pier, we can substitute the given time values (in seconds) into the equation and calculate the depth.
- At t = 2 minutes = 2 * 60 = 120 seconds:
y = 10 * sin(2π/900 * 120 + ϕ)
- At t = 4 minutes = 4 * 60 = 240 seconds:
y = 10 * sin(2π/900 * 240 + ϕ)
- At t = 12 minutes = 12 * 60 = 720 seconds:
y = 10 * sin(2π/900 * 720 + ϕ)
To find the value of ϕ, we need initial conditions or information about the phase constant at the time the tsunami hits the pier.
b) The minimum depth of the water would occur when the sinusoidal wave reaches its lowest point (trough). In this case, the minimum depth would be -10 meters because the normal depth of the water is 9 meters below sea level.
c) The water at the pier will be absent when the depth is below the normal depth (9 meters). So, we need to find the times when the depth is less than 9 meters.
d) The equation given does not provide enough information to directly determine the wavelength of the tsunami. To calculate wavelength, we would need either the frequency or the wave speed. The given information only provides the speed of the tsunami, which is 1200 kilometers per hour. Without the frequency, we cannot determine the wavelength.
To solve the remaining parts of the question and accurately predict the water depth, we would need additional information, such as the initial conditions or the value of the phase constant at the time the tsunami hits the pier.
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Thanks.