Conditions for node and antinode

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Solution

Dear Student,
Nodes:

Certain points in a standing wave, which are permanently at rest.

Nodes are the positions of zero displacement.

Antinodes:

Certain points in a standing wave with the maximum amplitude of vibration.

Antinodes are the positions of maximum displacement.

Now, Let 2 waves traveling in opposite direction be

$Y_{1} = a \sin (k x - 2 π f t) Y_{2} = a \sin (2 π f t + k x) w h e n 2 w a v e s a r e s u p e r p o s e d t h e n r e s u l \tan t w a v e Y_{r} = Y_{1} + Y_{2} = a \sin (k x - 2 π f t) + a \sin (2 π f t + k x) = 2 a \cos (2 π f t) \sin (k x) (u \sin g \sin (a - b) + \sin (a + b) = 2 \sin (a) \sin (b)) N o w a m p l i t u d e o f t h i s w a v e i s 2 a \sin (k x) = 2 a \sin (\frac{2 π}{λ} x) I t w i l l b e 0 i . e . 2 a \sin (\frac{2 π}{λ} x) = 0 w h e n x = n \frac{λ}{2} h e n c e n o d e w i l l b e a t x = n \frac{λ}{2} a n d m a x i . e . 2 a \sin (\frac{2 π}{λ} x) = 2 a a n i t i n o d e a t x = n \frac{λ}{4}$

If a standing wave is formed on stretched string of length L

then fundamental frequency will be corresponding to node at L = $λ / 2$
L = 2 $λ$
this implies Fundamental frequency = $f_{o} = \frac{v}{λ} = \frac{v}{2 L}$

Regards

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