A seismic sensor buried in the ground records the vertical displacement of Earth’s surface at a single location. The table lists the times, measured from t=0, at which the sensor output first reaches successive maximum positive displacements.
Time of nth positive crest at the sensor
n 0 1 2 3 4
t(s) 0 1.80 3.60 5.40 7.20
A geological survey indicates that the surface wave responsible for these readings is traveling horizontally toward the sensor with speed v=3.2kms−1 and has a peak-to-equilibrium vertical amplitude of 2.5mm.
(a) Determine the period, frequency, and angular frequency of the wave. Provide units.
(b) Calculate the wavelength.
(c) Construct a mathematical expression y(x,t) for the vertical displacement of the ground surface that satisfies the sensor data and is consistent with the stated propagation direction. Define coordinates so that the sensor is located at x=0.
(d) Using your expression, find the horizontal distance from the sensor to the nearest point that has a displacement of +1.0mm at the instant t=0.
(e) Describe how engineers could use the phase constant in your expression to synchronize data from multiple sensors placed at different positions along the wave’s line of travel.