How does the linear density (µ) of the string affect the propagation speed (v) of a wave on a taut string?

Answer 1

Consider a pulse moving to the right on a taut string with uniform velocity v and measured with respect to a stationary frame of reference.

To facilitate understanding, take a frame of reference that moves along with the pulse at equal speed so that the pulse is "at rest" in this second frame.

Newton's Laws are valid in either a stationary frame or in a frame moving with constant velocity.

Picture a small segment of the string of length Δs, which forms the arc of a circle of radius R. In the pulse's frame of reference, this small segment of length Δs is moving toward the left with velocity v. This small segment will have a centripetal acceleration of v²/R, which is supplied by the force of tension (F) in the string.

The force F acts on each side of the segment and tangent to the arc. The horizontal components of F cancel, and each vertical component (of Fsin θ) acts radially inward toward the center of the arc.

The total radial force is then 2Fsin θ. Since the segment is small, θ is small and sin θ can be approximated by θ. The total radial force is therefore expressed as F_r = 2Fsin θ ≈ 2Fθ.

The small segment has a mass given by m = µΔs, where µ is the mass per unit length (or density) of the string. Since the segment forms part of a circle and subtends an angle 2θ at the center, Δs = R(2θ), and from this m = µΔs = 2µRθ.

Apply Newton's Second Law to this segment to obtain (from the radial component of motion) F_r = mv²/R or 2Fθ = 2µRθv²/R, where F_r is the force which supplies the centripetal acceleration of the segment and keeps up the curvature at this point. Solving for v yields v = √(F/µ).

Note that this derivation is based on the assumption that the pulse height is small relative to the length of the string. Assuming this allows use of sin θ ≈ θ. The model further assumes that the tension F is not affected by the presence of the pulse, so that F is the same at all points on the string.

Finally, this proof assumes no particular shape for the pulse. It is then concluded that a pulse of any shape will travel on the string with speed v = √(F/µ) without changing the shape of the pulse.