[lal_2oo5]

the wavefunction of for a travelling wave on a string is given by y(x,t) = Asin(kx-wt). At any arbitrary point x and time t the maximum slope of the wave is s(max) and the maximum particle speed is vp(max). If the wave speed is v show that v = vp(max)/s(max).

[harum]

Every single particle of the rope moves in the Y direction only, with the wave moving along X. A snapshot of the string at any arbitrary time t will be a sinusoid. At any other time it will be the same sinusoid shifted along the axis X by a certain amount dependent on the difference between the two time points. This is exactly what the equation says. The crest of a traveling wave moves by definition with the speed of:
v = L/T, here L - wavelength (the distance between two adjacent crests), T - the period of the wave (or the sinusoid).

Look up in your textbook to see that by definition: k=(2*pi)/L and w=(2*pi)/T. This makes v = L/T = w/k.

Draw the graph of the equation as y vs x to see where the maximum slope of the sinusoid (a.k.a. wave) might be (it is the same as where dy/dx is maximum). Then draw y vs t and see where dy/dt, i.e. the speed of a particle, is maximum -- looks like, similarly to the maximum slope, it reaches the maximum where y=0 as well (either on the way up or down). You can express both maximum slope and maximum particle speed using the constants from the wave equation.

See if you get the same values:
s(max) = kA, vp(max) = wA.

From above: v = L / T = w / k = (w*A) / (A*k) = vp(max) / s(max).