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Question

The pressure wave in a closed organ pipe of length 40 cm is given by p=5sin(2πx32)cos(50πt) where x,p and t are in cm,N/m2 and seconds respectively. The pressure nodes are located at which of the following distances along the pipe (from the open end) ?

A
8 cm , 24 cm
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B
16 cm , 32 cm
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C
16 cm , 24 cm
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D
18 cm , 32 cm
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Solution

The correct option is B 16 cm , 32 cm
Given,
The pressure wave is
p=5sin(2πx32)cos(50πt)

General equation for pressure wave in a closed organ pipe is

p=2pocos(2πλx)sin(2πTt)

Comparing both equations, we get

k=2πλ=2π32

λ=32 cm

Positions of zero pressure variation (or pressure nodes) are given by
x=nλ2 where n=0,1,2,3....
x=0 , λ2 , λ , 3λ2......
λ=32 cm
we can conclude that,
For, n=0 , x=0 cm
n=1 , x=16 cm
n=2 , x=32 cm
n=3 , x=48 cm

So, among the given options, only option (b) satisfies the position of pressure node.

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