**Laws of transverse vibration in a string:**

The velocity of a transverse wave traveling in a stretched string is given by \(v=\sqrt{\frac T\mu}\), where T is the tension in the stretched string and μ is the mass per unit length. Since the frequency, f = v/2l in fundamental mode, then

\(f=\frac1{2l}\sqrt{\frac T\mu}\)

From the expression, it follows that there are three laws of transverse vibration of stretched string:

**The law of the length:**The fundamental frequency is inversely proportional to the resonating length,*l*of the string*.*

\(f\propto\frac1l\)

**The law of the tension:**The fundamental frequency is directly proportional to the square root of the stretching force or the tension.

\(f\propto\sqrt T\)

**The law of the mass:**The fundamental frequency is inversely proportional to the square root of the mass per unit length.

\(f\propto\frac1{\sqrt\mu}\)

**Modes of Vibration of a Stretched String**

Let us consider a string that is stretched between two ends P and Q and plucked between these points produces a transverse wave. The stationary waves are thus set up in the string and the wire emits vibrations of different frequencies which are called modes of vibrations as shown in figures.

**The First mode of vibration**

Let a stretched string is plucked between two points P and Q in such a way that it forms only one segment. In this case, two nodes and one antinode are formed. If l be the length of string and λ_{1} be its wavelength, then the frequency is

\(f_1=\frac v{\lambda_1}=\frac v{2l}\)

Or, \(f_1=\frac1{2l}\sqrt{\frac T\mu}\)

This vibration is called fundamental mode or first harmonics and frequency is the fundamental frequency.

**The Second mode of Vibration**

Let a stretched string is plucked between the two points P and Q in such a way that it forms two segments. In this case, three nodes and two antinodes are formed. If *l* be the length of string and λ_{2} be its wavelength, then the frequency is

\(f_2=\frac v{\lambda_2}=\frac vl\)

Or, \(f_2=\frac12\sqrt{\frac T\mu}\) , \(\left[\because v=\sqrt{\frac T\mu}\right]\)

∴ f_{2} = 2f_{1}

This vibration is called the second mode of vibration or second harmonics or the first overtone. The frequency in the mode is twice the frequency of fundamental frequency.

**The Third mode of Vibration**

Let a stretched string is plucked between two points P and Q in such a way that it forms three segments. In this case, four nodes and three antinodes are formed. If *l* be the length of string and λ_{3} be its wavelength, then its frequency is

\(f_3=\frac v{\lambda_3}=\frac{3v}{2l}\) , \(\left[\because\lambda_3=\frac{2l}3\right]\)

Or, \(f_3=\frac3{2l}\sqrt{\frac T\mu}\), \(\left[\because v=\sqrt{\frac T\mu}\right]\)

∴ f_{3} = 3f_{1}

This vibration is called the third mode of vibration or the third harmonics or the second overtone. The frequency is three times the fundamental frequency. In general, for the n^{th} mode of vibration, the frequency of vibration is f_{n} = nf_{1} = f_{1} , 2f_{1} , 3f_{1} , ...., nf_{1}