Question 461: Waves

Question

A stretched string fixed at both ends has ' $m$ ' nodes, then the length of the string will be

Accepted Answer

For a stretched string fixed at both ends and having ' $m$ ' nodes, the calculation for the length of the string can be broken down as follows:
The length of the string with $p$ loops (also known as harmonics or anti-nodes) is expressed by:
$ l = \frac{p \lambda}{2} \quad 	ext{.... (i)} $
In terms of harmonics:
Number of loops (or anti-nodes) $= p \quad 	ext{.... (ii)}$
The relationship between nodes and anti-nodes is given as:
Number of nodes $= 	ext{Number of anti-nodes} + 1$
Given that there are $m$ nodes, it follows:
Number of anti-nodes $= m - 1$
Thus, from Equation (ii), we have:
$ p = m - 1 $
Inserting this value of $p$ into Equation (i), we deduce:
$ l = \frac{(m-1) \lambda}{2} $
Therefore, the length of the string is $\frac{(m-1) \lambda}{2}$.

Answer

$(m-1) \frac{\lambda}{2}$

Answer

$\frac{(m+1) \lambda}{2}$

Answer

$\frac{m \lambda}{2}$

Answer

$(m-2) \frac{\lambda}{2}$

Waves

A stretched string fixed at both ends has ' m ' nodes, then the length of the string will be

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