91Ó°ÊÓ

First, we need to write the equation for the surface seismic wave. The general form of a harmonic wave equation is: \(y(x,t) = A \sin(kx - \omega t + \phi)\). In this exercise, we are given the amplitude, period, and wavelength, so we can find the wave number (\(k\)) and angular frequency (\(\omega\)) using the following formulas: \(k = \dfrac{2\pi}{\lambda}\) and \(\omega = \dfrac{2\pi}{T}\), where \(\lambda\) is the wavelength and \(T\) is the period. Using the given values, we have: \(A = 2.0\,\text{cm}=0.02\,\text{m}\); \(\lambda = 4.0\, \text{km} = 4000\,\text{m}\); \(T = 4.0\,\text{s}\) Now, let's calculate the wave number (\(k\)) and angular frequency (\(\omega\)): \(k=\dfrac{2\pi}{\lambda}=\dfrac{2\pi}{4000}\, \text{m}^{-1}\) and \(\omega=\dfrac{2\pi}{T}=\dfrac{2\pi}{4}\, \text{s}^{-1}\) Assuming the wave is harmonic and the phase constant (\(\phi\)) is \(0\), we can write the wave equation: \(y(x,t)= 0.02\,\text{m}\, \sin\left(\dfrac{2\pi}{4000} x - \dfrac{2\pi}{4} t \right)\) Since the wave is moving along the \(-x\) -axis, we can modify the equation to: \(y(x,t)= 0.02\,\text{m}\, \sin\left(\dfrac{2\pi}{4000} (-x) - \dfrac{2\pi}{4} t \right)\).

The maximum ground speed (i.e., particle speed) can be obtained by computing the derivative of the wave function \(y(x,t)\) with respect to time (\(t\)) and setting \(x=0\): \(V_{max} = \left|\dfrac{\partial y(x,t)}{\partial t}\right|_{x=0}\) We differentiate the wave function with respect to time: \(\dfrac{\partial y(x,t)}{\partial t} = -0.02(\dfrac{2\pi}{4})\,\text{m}\, \cos\left(\dfrac{2\pi}{4000} (-x) - \dfrac{2\pi}{4} t \right)\) Evaluate the expression for the maximum ground speed at \(x=0\): \(V_{max} = \left| -0.02(\dfrac{2\pi}{4})\,\text{m}\, \right| = 0.01\pi\, \text{m/s}\) The maximum ground speed as the wave moves by is \(0.01\pi\, \text{m/s}\).

Finally, to determine the wave speed (\(v\)), we can use the equation: \(v = \dfrac{\lambda}{T}\) Using the given values, we have: \(v = \dfrac{4000\,\text{m}}{4\,\text{s}}= 1000\,\text{m/s}\) The wave speed is \(1000\, \text{m/s}\).