Velocity Of A Transverse Wave On A String

Velocity Of A Transverse Wave On A String

A transverse wave is produced in a taut string as shown in Fig.(14.8 a). Let us observe a small segment, such as AB, on the string from the frame that moves with the wave. In this frame, the wave is stationary while the string moves to the left at speed v.

The segment AB may be treated as a circular arc of some radius R, as shown in
the Fig.(14.8 b).

Length of AB = R(2θ) = 2Rθ

If μ is the linear mass density of the string, then

m = μ(2Rθ) = 2μRθ

The vertical component of tension in the string must provide the centripetal force, therefore,

2Tsinθ = mv²/ R

Since the angle θ is small, sinθ ≈ θ

∴2Tθ = (2μRθ)v²/ R

or v = √T/μ  (14.16)

Note that the velocity is measured with respect to the medium.

Example: 14.3

A harmonic wave with a wavelength of 20 cm, an amplitude of 3 cm and a velocity of 2 m/s travels on a string (μ = 0.25 g/m) to the left along the x – axis.

(a) What are the frequency and period of the wave motion?

(b) What is the tension in the string ?

(c) What is the position function for the wave ?

Solution

(a) f = v/λ =  (2 m/s)/0.2m = 10 Hz

and T = 1/f = 1/10 = 0.1s

(b) Using equation (14.16)

T = μv2 = (0.25 × 10^-3 kg/m)(2 × m/s)²

orT = 10^-3 N

(c) In order to write the wave equation we need to know ω and k.

∴ω = 2πf  = 2π(10) = 20π rad/s

and k = 2π/λ = 2π/0.2 = 10π rad/m

Thus,the wave equation is

y = (3 × 10^-2)sin[(10π)x + (20π)t]

where x and y are in metre and t is in second.