The lengths of the two organ pipes open at both ends are 'L' and (L+L₁ ) . If they are sounded together, the beat frequency will be ( v = velocity of sound in air)

Step-by-step solution

The fundamental frequency for an organ pipe open at both ends is f = v 2L where L₁' = L and L₂' = L + L₁ for the two pipes. The frequencies are f₁ = v 2L and f₂ = v 2(L + L₁) giving a beat frequency of f_ beat = |f₁ - f₂| Substituting the expressions yields f_ beat = | v 2L - v 2(L + L₁) | = v 2 | 1 L - 1 L + L₁ | Combining fractions inside the absolute value gives f_ beat = v 2 | (L + L₁) - L L(L + L₁) | = v 2 L₁ L(L + L₁) since L₁ > 0 The expression v L₁ 2L(L + L₁) corresponds to option D .