For \(y(x,t)=0.020\cos(4x-12t)\), find the amplitude.

For \(y(x,t)=A\cos(6x-18t)\), is the wave moving left or right?

For \(y(x,t)=0.030\cos(5x-20t)\), find the wavelength.

For \(y(x,t)=0.030\cos(5x-20t)\), find the wave speed.

A wave is described by \(y(x,t)=0.040\cos(8x-32t+\pi/3)\). Find the phase difference between two points separated by \(0.25\,\mathrm{m}\) at the same instant.

A pulse shape at \(t=0\) is \(y(x,0)=g(x)\). Write expressions for the same pulse moving right and left with speed \(v_w\).

For \(y(x,t)=A\cos(kx-\omega t+\phi)\), derive the equation relating \(x\) and \(t\) for a fixed phase point and hence show that the phase speed is constant.

A sensor at \(x=0\) records \(y(0,t)=A\cos(\omega t)\). A second sensor at \(x=d\) records the same oscillation delayed by a time \(\tau\). Build a right-moving wave equation consistent with these observations and express \(k\) and \(v_w\) in terms of \(\omega\), \(d\), and \(\tau\).