2.1 Transverse harmonic waves  (Page 6/6)

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

$$F=2T\sin \theta$$

For small angle involved, we can approximate as :

$$\sin \theta =\frac{\frac{\text{Δ}l}{2}}{R}=\frac{\text{Δ}l}{2R}$$

Combining above two equations, we have :

$$\Rightarrow F=2T\sin \theta =\frac{2T\text{Δ}l}{2R}=\frac{T\text{Δ}l}{R}$$

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

$$F=\text{Δ}m{a}_R=\frac{\mu \text{Δ}l{v}^2}{R}$$

Equating two expressions of centripetal force, we have :

$$\Rightarrow \frac{\mu \text{Δ}l{v}^2}{R}=\frac{T\text{Δ}l}{R}$$

$$\Rightarrow v=\sqrt{\left(\frac{T}{\mu }\right)}$$

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 :

$$\Rightarrow \lambda =vT$$

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.