Waves and Laws of Transverse Vibration in Stretched String

Waves in Stretched Strings

A string is a tight wire. When it is plucked or bowed, progressive transverse waves travel along the wire and is reflect at the fixed ends. These waves superpose with the incident waves and produce a stationary wave in the wire. A progressive sound wave is produced in the surrounding air having a frequency equal to that of the stationary wave in the string.

Modes of Vibration

A stretched string can produce different frequencies. Since the ends are fixed, these are the position of nodes in the wire. When the string is plucked at the middle, an antinode is formed at the middle. This is the simplest mode of vibration and the distance between the consecutive nodes is λ/2 where λ is the wavelength of the transverse wave in the string.

$$\begin{align*} L &= \lambda /2 \\ \text {or,} \: \lambda &= 2L \\ \text {The frequency of vibration is given by} \\ f &= \frac {v}{\lambda } = \frac {v}{2L} \\ \end{align*}$$

where v is the velocity of the transverse wave. This is fundamental frequency or frequency of first harmonic. It is the lowest frequency produced by the vibrating string.

Overtone in Stretched String

If the string is plucked at a point one-quarter of its length from one end, the string vibrates in two segments. This mode of vibration is called the first overtone. This vibration can be also be set when the vibrating antinodes are formed in the string as shown in the figure.

If λ₁ is wavelength and f₁ is the frequency of the resulting stationary wave, we have

$$\begin{align*} L &= \frac {\lambda }{2} + \frac {\lambda }{2} = \lambda \\ \text {The frequency of the wave,} \: f_1 &= \frac {v}{\lambda } = \frac v L = 2f \\\end{align*}$$

Thus the frequency of the first overtone is two times the fundamental frequency. This is also called second harmonics.If the string is made to vibrate in three segments by touching it at one-third of the length from one end, additional nodes are produced in it.

$$\begin{align*} \text {If } \: \lambda \: \text {is the wavelength and} \: f_2 \: \text {its frequency of the wave, then} \\ L &= \frac {\lambda }{2} + \frac {\lambda }{2} + \frac {\lambda }{2} = \frac {3\lambda }{2} \\ \text {or,} \: \lambda &= \frac {2L}{3} \\ \text {and the frequency is given by} \\ f_2 &= \frac {v}{\lambda } = \frac {3v}{2}L = 3f \\ \end{align*}$$

Hence, the frequency of second overtone is three times the fundamental frequency which is also called third harmonics. Similarly we can obtain other overtone in the same string with more segments. The ratio of the frequency of string is

$$ f: f_1 : f_2 : f_3:\dots = 1: 2: 3\dots $$

Laws of Transverse Vibration in Stretched String

The velocity of a transverse wave travelling in a stretched string is given by

$$ v = \sqrt {\frac {T}{\mu }} $$

where T is the tension in the stretched string and µ, the mass per unit length. Since the frequency, f = v/2L in fundamental mode, then

$$ f = \frac {1}{2L}\sqrt {\frac {T}{\mu }} $$

From this expression, it follows that there are three laws of transverse vibration of stretched string;

1. The length of length: The fundamental frequency is inversely proportional to the resonating length, L of the string.
   $$ f \propto \frac {1}{L} $$
2. The law of tension: The fundamental frequency is directly proportional to the square root of the stretching force or tension.
   $$ f \propto \sqrt {T} $$
3. The law of mass: The fundamental frequency is inversely proportional to the square root of the mass per unit length.

$$ f\propto \frac {1}{\sqrt {\mu }} $$

Verification of the Laws of Transverse Vibration

These laws can be verified experimentally using a sonometer. This device consists of a wire under tension which is arranged in a hollow wooden board as shown in the figure. The vibration of the wire are passed by the movable bridges to the box and then, to the air inside it.

1. To verify \( f \propto 1/L\)
   To verify this law, take a tuning fork of known frequency, such as 320 Hz. Taking a load of 1 kg on the string, find the resonating length of the wire between the bridges C and D. this is found by using a small paper on the wire. Let L₁ be the resonating length for this tuning fork. The same process is repeated for next tuning fork having a different frequency. Let L₂ resonating length for the second tuning fork. It will be found that the product \(f_1 \times L_1 = f_2 \times L_2 \) at constant tension on and mass per unit length of the string. This follows that
   $$F \propto \frac {1}{L} $$
   
   
2. To verify \(F \propto \sqrt {T} \)
   To verify this law, a length L of experimental wire AB is fixed between bridges C and D, and load W on it is varied to alter tension T. to measure the frequency of vibration in this wire, an auxiliary wire PQ is used which runs parallel to the experimental wire AB. The tension on this wire is kept constant which is not shown in a figure. The bridges M and N are moved until the note on length l of auxiliary wire between MN is same as that in experimental wire CD. Since the tension in the wire MN is constant, we have \(f \propto 1/l \). By varying W, the tension I in AB is varied. A graph between \(1/l\) and \(\sqrt {T}\) is a straight line passing through the origin, which shows that
   $$ F \propto \sqrt {T} $$
   at constant µ and resonating length.
   
   
3. To verify \( f \propto 1/\sqrt {\mu} \)
   To verify this law, the wires of different diameters and materials are used under same vibrating length and same tension. Parallel to the experimental wire AB, an auxiliary wire PQ with known tension is fixed on the sonometer as shown in a figure. The experimental wire is plucked at the middle between two bridges C and D and the length of the auxiliary wire PQ is set into resonance by varying the position of the bridges M and N on the auxiliary string, let the resonating length of the auxiliary wire be L₁ and the frequency of vibration of the experimental wire is proportional to the length. The second experimental wire is taken with the same load and same vibrating length and the auxiliary wire is again made in resonance with the experimental wire. The resonating length is again noted as L₂. When a graph between \(1/L \) and \(\sqrt {\mu } \) is plotted, a straight line is obtained which passes through the origin.
   $$ f \propto 1/\sqrt {\mu} $$