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The centripetal force is equal to the resultant of tensions on either side of the element and is given by :

F = 2 T sin θ

For small angle involved, we can approximate as :

sin θ = Δ l 2 R = Δ l 2 R

Combining above two equations, we have :

F = 2 T sin θ = 2 T Δ l 2 R = T Δ l R

On the other hand, the centripetal force for uniform circular motion is given by :

F = Δ m a R = μ Δ l v 2 R

Equating two expressions of centripetal force, we have :

μ Δ l v 2 R = T Δ l R

v = T μ

It is evident from the expression that speed on the stretched string depends on inertial factor (μ) and tension in the string (T)

Dependencies

We need to create a simple harmonic vibration at the free end of the string to generate a harmonic transverse wave in the string. The frequency of wave, therefore, is determined by the initial condition that sets up the vibration. This is a very general deduction applicable to waves of all types. For example, when light wave (electromagnetic non-mechanical wave) passes through a glass of certain optical property, then speed of the light changes along with change in wavelength – not change in frequency. The frequency is set up by the orbiting electrons in the atoms, which radiates light. Thus, we see that frequency and time period (inverse of frequency) are medium independent attributes.

On the other hand, speed of the wave is dependent on medium or the material along which it travels. In the case of string wave, it is dependent on mass per unit length (inertial attribute) and tension in the string. The speed, in turn, determines wavelength in conjunction with time period such that :

λ = v T

One interesting aspect of the dependencies is that we can change wave speed and hence wavelength by changing the tension in the string. We should note here that we have considered uniform tension, which may not be realized in practice as strings have considerable mass. Cleary such situation will lead to change in speed and wavelength of the wave as it moves along the string. On the other hand, the amplitude of wave is a complex function of initial condition and the property of wave medium.

Problem : A string wave of 20 Hz has an amplitude of 0.0002 m along a string of diameter 2 mm. The tension in the string is 100 N and mass density of the string is 2 X 10 3 k g / m 3 . Write equation describing wave moving in x-direction.

Solution : : The equation of a wave in x-direction is given as :

y x , t = A sin k x ω t

The amplitude of wave is given. Hence,

y x , t = 0.0002 sin k x ω t

Now, we need to know “k” and ”ω” to complete the equation. The frequency of the wave is given. The angular frequency, therefore, is :

ω = 2 π ν = 2 X 3.14 X 20 = 125.6 radian / s

In order to determine wave number “k”, we need to calculate speed. An inspection of the unit of mass density reveals that it is volume density of the string – not linear mass density as required. However, we can convert the same to linear mass density as string is an uniform cylinder.

linear mass density = volume mass density X cross section area

μ = 2 X 10 3 X π d 2 4 = 2 X 10 3 X 3.14 X 0.002 2 4

μ = 6.28 X 10 - 3 k g / m

Putting in the expression of speed, we have :

v = T / μ = 100 6.28 X 10 - 3 = 10 5 6.28 = 39 m / s

The wave number, k, is :

k = ω v = 125.6 39 = 3.22 rad / m

Substituting in the wave equation,

y x , t = 0.0002 sin 3.22 x 125.6 t

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Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
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