Transverse Vibration in a Stretched String:
A stretched string is a thin metal wire claimed between two rigid supports with tension. Such a string is the source of sound emitted by musical instruments like sitar, violin, piano, etc. To generate vibrations in the string, it is initially disturbed in a direction normal to its length. This transverse disturbance is initiated usually by one of the three following methods:
1. Plucking: This method is used for sitar, guitar, etc.
2. Striking: This method is used to play a piano.
3. Bowing: This method is used to play a violin.
Features of Transverse Vibration in a Stretched String:
The vibration in a stretched string has two salient features:
i. When a stretched string is displaced from its equilibrium position, two identical progressive waves are produced. These two waves travel toward the two ends of the wire. After getting reflected from the two ends, they again travel towards the opposite ends. Thus, the two waves are reflected again and again from both ends. Hence, a stationary wave is generated in the string. This is known as a stationary wave in a stretched string.
ii. When the stretched string is vibrated in one or more than one loop, every point on the wire vibrates in a direction normal to the length of the wire. So, it is a transverse vibration.
Laws of Transverse Vibration:
Let a thin, uniform, and flexible wire be claimed with a tension between two rigid supports. Here, l = length of the wire, M = mass of the wire, r = radius of the wire, ρ = density of the material of the wire, T = tension along the wire.
∴ So, the area of cross-section = πr2
Volume of unit length of the wire = πr2.1 = πr2
∴ Mass per unit length of the wire, m = πr2ρ = M/l
This mass per unit length is often called the linear density of the wire. It is a characteristic property of the wire used because, for a given wire, the linear density is fixed. It is independent of the tension applied and the total length of the wire.
As proposed by the French scientist Marlin Mersenne, the frequency (n) of transverse vibration in a stretched string varies with the relevant properties of the string according to the following laws:
Law of length:
If the tension and the mass per unit length remain constant, the frequency of transverse vibration is inversely proportional to the length of the string.
Let, n ∝ 1/l when T and m are constants.
Law of Tension:
If the length and the mass per unit length remain constant, the frequency of transverse vibration is proportional to the square root of the tension in the string.
Let, n ∝ √T when l and m are constants.
Law of Mass:
If the length and the tension in the wire remain constant, the frequency of transverse vibration is inversely proportional to the square root of the mass per unit length of the string.
Let, n ∝ 1/√m when l and T are constants.
Law of Radius:
If the material of the string remains the same, the density remains constant. Thus, the frequency of transverse vibration is inversely proportional to the radius of the string, when the length and the tension in the string are constant.
Let, n ∝ 1/r when l, T, and ρ are constants.
Law of Density:
For strings of the same radius, but made of different materials, the frequency of transverse vibration is inversely proportional to the square root of the density of the material if the length and the tension in the string remain constant.
Let, n ∝ 1/√ρ when l, T, and r are constants.