Subject: Physics
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.
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.
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
$$ f: f_1 : f_2 : f_3:\dots = 1: 2: 3\dots $$
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;
$$ f\propto \frac {1}{\sqrt {\mu }} $$
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.
Reference
Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.
S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.
A progressive sound wave is produced in the surrounding air having a frequency equal to that of the stationary wave in the string.
This is fundamental frequency or frequency of the first harmonic is the lowest frequency produced by the vibrating string.
Sonometerconsists of a wire under tension which is arranged in a hollow wooden board.
According to the law of transverse vibration of stretched string, the fundamental frequency is inversely proportional to the resonating length, L of the string.
According to the law of transverse vibration of stretched string, the fundamental frequency is directly proportional to the square root of the stretching force or tension.
According to the law of transverse vibration of stretched string, the fundamental frequency is inversely proportional to the square root of the mass per unit length.
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