Question

show that all harmonics are present on a stretched string between two rigid supports​

Answer 1

Answer:

the hormonic is type of numerical average it is calculate by dividing the number of observation by the reciprocal of each number

Answer 2

Answer:

• The vibration of string for different modes:-

a. Imagine pulling on a string that has been stretched between two solid supports. Due to plucking, the string vibrates and loops are formed in the string. The vibration of the string is as follows.

b. Let, p = number of loops

l = length of string

∴ Length of one loop = I/p                                          ......       (1)

c. Two successive nodes form a loop. Distance between two successive nodes are λ/2                                                               ......      (2)

From equation (1) and (2)

λ/2 = I/p

∴  λ = 2I/p                                                                         ....... (3)

d. Transverse wave velocity is determined by,

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

e. Frequency of string is given by,

n = v/ λ

Substituting  λ from equation (3),

$n = \frac{\sqrt{\frac{T}{m} } }{\frac{2I}{p} }$

∴ $n = \frac{P}{2l}\sqrt{\frac{T}{m} }$                                                                        ...... (4)

f. Fundamental mode or first harmonic:-

In this case, p = 1

∴ From equation (4)

$n = \frac{1}{2l} \sqrt{\frac{T}{m} }$

This frequency is called fundamental frequency.

g. First overtone oe second harmonic:-

In this case, p = 2

∴ From equation (4),

$n_{1} = \frac{2}{2l}\sqrt{\frac{T}{m} } = 2 * \frac{1}{2l}\sqrt{\frac{T}{m} } = 2n$

∴ $n = 2n$

h. Second overtone or third harmonic:-

In this case, p = 3

Using equation (4),

$n_{2} = \frac{3}{2l}\sqrt{\frac{T}{m} } = 3 * \frac{1}{2l}\sqrt{\frac{T}{m} } = 3n$

∴ $n_{2} = 3n$

i. $(p-1)^{th}$ overtone or $p^{th}$ harmonic:-

$n_{p-1}=p*\frac{1}{2l} \sqrt{\frac{T}{m} } = pn$

For $p^{th}$ overtone,

$n_{p}=\frac{p+1}{2l}\sqrt{\frac{T}{m} } = (p+1)n$

j. As a result, the vibrational frequencies of a stretched string are n, 2n, 3n, and so forth.

Thus, the stretched string's vibrations contain both even and odd harmonics.