derive the equations for harmonic frequencies of standing waves formed in a string due to the superposition of waves.Draw the first two harmonies

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Solution

$D e a r s t u d e n t L e n g t h o f s t r i n g = L V e l o c i t y o f w a v e o n s t r i n g = V a) w h e n w i r e v i b r a t e f o r m i n g o n e l o o p o r f i r s t h a r m o n i c L = \frac{λ}{2} s o λ = 2 L V = f_{1} \times λ = f_{1} \times 2 L f_{1} = \frac{V}{2 L} . . . 1 b) w h e n w i r e v i b r a t e i n t w o l o o p s o r s e c o n d h a r m o n i c L = \frac{2 λ}{2} s o λ = L V = f_{2} \times λ = f_{2} \times L f_{2} = \frac{V}{L} . . . 2 c) w h e n w i r e v i b r a t e i n t h r e e l o o p s o r t h i r d h a r m o n i c L = \frac{3 λ}{2} s o λ = \frac{2 L}{3} V = f_{3} \times λ = f_{3} \times \frac{2 L}{3} f_{3} = \frac{3 V}{2 L} = 3 f_{1} . . . 3 w h e n w i r e v i b r a t e f o r m i n g n l o o p s o r n^{t h} h a r m o n i c t h a n L = \frac{n λ}{2} s o λ = \frac{2 L}{n} V = f_{n} \times λ = f_{n} \times \frac{2 L}{n} f_{n} = \frac{V}{\frac{2 L}{n}} = \frac{n V}{2 L} = n f_{1} . . . (n) R e g a r d s$

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