Spread the love

Stationary Waves In A Stretched String

Stationary Waves In A Stretched String :- Stationary waves are commonly produced in stretched strings of musical instruments such as guitars, violins, and sitars, where the reflected wave combines with the incident wave to form fixed vibration patterns.

Consider a string of length stretched under tension between two fixed points. When the string is plucked, transverse waves travel along the string and are reflected back from the fixed ends. The reflected waves combine with the incident waves, and under suitable conditions, stationary waves are formed on the string.

Since both ends of the string are fixed, the displacement of particles at the ends is always zero. Therefore, nodes are formed at the two ends of the string. Depending upon the manner in which the string vibrates, different modes of vibration are possible, as shown below :

(1). First Mode Of Vibration

The stretched string should be plucked at its middle point

)

to produce the fundamental mode.

Stationary Waves In A Stretched String - Curio Physics

The stretched string should be plucked at

)

to produce the third mode (second overtone).

In general if the string is plucked at length $\displaystyle \frac{l}{2p}$ , then it vibrates in “p” loops and we have the p^thmode of vibration and the p^th harmonic [or (p-1)^th overtone] is given by :

$\displaystyle {{\nu }_{p}}=\frac{p}{2l}\sqrt{\frac{T}{\mu }}$

🔬 Curio Capsules

(1). Overtones are the frequencies obtained after leaving the first harmonic (fundamental frequency). The frequency appearing immediately after the first harmonic is called the first overtone, irrespective of its harmonic number. Hence overtone numbering is always consecutive as 1st, 2nd, 3rd, etc., and therefore contains both odd and even numbers.

For example, in a closed organ pipe the harmonics are :

f, 3f, 5f,…

Since the harmonic appears immediately after the harmonic, it is called the 1st overtone.

(2). Harmonics are frequencies that are integral multiples of the fundamental frequency. Depending on the system, harmonics may be only odd, only even, or both odd and even. Example :

String fixed at both ends → both odd and even harmonics)
Closed organ pipe → (only odd harmonics)

Stationary Waves In A Stretched String

Like this: