Velocity of transverse waves on a stretched string

VELOCITY OF TRANSVERSE WAVES ON A STRETCHED STRING

When a strecthed string between two points in plucked in a direction perpendicular to its length, then transverse vibrations are set up in the string.

This is due to the fact that tension in the string tends to bring back to its mean position but, due to its inertia the string overshoots and goes to the other extreme.

To determine the velocity of transverse wave in a string, let us consider a string PQ, stretched under a certain tension along the x axis as shown in Fig. 4.11.

Consider a small portion (AB) of the string of length dl, which is practically an arc of a circle of radius R. [of course, R would be different for different portions.]

As the string moves along the circular arc, a centripetal force acts on it along the radius towards the centre 'O'.

∴ The centripetal force is given by = mdlv² / R

Where m → Mass per unit length of the string

v → Velocity of the string (or) wave

This force is supplied by the tension T, which act tangentially at points A&B. These tensions are set up in the string when it is plucked aside and tends to restore the static equilibrium.

The resultant of two tensions along OC (i.e) one along OA and the other along OB shall be written as

 (i) Tension along OA (Fig. 4.12) is sin dθ = x/T

∴ x = T sin dθ                  ……………(2)

(ii) Tension along OB (Fig. 4.13) is sin dθ = x/T

∴ x = T sin dθ                  ……………(3)

∴ Resultant tension along OC = Tension along OA + Tension along OB              ……….(4)

Substituting equation (2) and equation (3) in equation (4) we get

Resultant tension along OC = Tsindθ + Tsinθ

Resultant tension along OC = 2T sin dθ

Since dθ is very small, we can write sin dθ ≈ dθ

Resultant tension along OC = 2T dθ               …………(5)

From Fig. 4.14 we can write Arc length dl = 2 dθR

 (or) 2dθ = dl / R               …………(6)

Substituting equation (6) in equation (5) we get

Resultant tension along OC = Tdl / R                …………(7)

Equation (7) represents the resultant force

∴ We can write equation (1) = equation (7)

[ mdl / R ] v² = Tdl /R

mv₂ = T

Velocity of the transverse wave along a stretched string v = √(T/m)         ……..(7)

This expression shows that velocity of a transverse wave along a stretched string is directly proportional to the square root of the Tension T and inversely proportional to the mass per unit length (m).