Slope of string equals speed ratio

Let a transverse wave travel along a string in the +x direction with speed v. Its displacement can be written as a function of (x − vt):

y = f(x − vt)

(for example y = A sin(kx − ωt) = A sin k(x − vt), with v = ω/k).

Proof

Let u = x − vt.

1. Slope of the string at a point:

∂y/∂x = f′(u) · ∂u/∂x = f′(u)

2. Transverse speed of the particle at that point:

∂y/∂t = f′(u) · ∂u/∂t = −v f′(u)

3. Compare:

∂y/∂t = −v (∂y/∂x)

4. Taking magnitudes:

| ∂y/∂x | = | ∂y/∂t | / v

So the slope at any point of the string is numerically equal to the ratio of the particle speed to the wave speed at that point. For a wave travelling in the −x direction, y = f(x + vt) and ∂y/∂t = +v ∂y/∂x, which gives the same magnitude relation.

Check with y = A sin(kx − ωt): slope = Ak cos(kx − ωt); particle speed = −Aω cos(kx − ωt); ratio = ω/k = v.