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Classical Mechanics Level 3

A sound wave starting from source $S$ follows two paths $SPTO$ and $SPQRTO$ to reach an observer at point $O$ . If $PQRT$ is a square of side length $\ell$ and the observer at point $O$ is unable to detect any sound, then what is the maximum possible wavelength $\lambda$ of the sound? (There is no loss of energy of sound wave due to any collisions.)

4\ell

8\ell

6\ell

2\ell

1 solution

Steven Chase
Jan 6, 2018

Path length difference:

$\Delta{L} = 5 l - 3l = 2 l$

For perfect destructive interference, the path length difference must be an integer number of half-wavelengths:

$2 l = N \frac{\lambda}{2} \\ \lambda = \frac{4 l}{N}$

The smallest integer value $N$ is one, yielding a maximum possible wavelength $\lambda_{max} = 4l$

Actually, the lengths of paths $SP$ and $TO$ were not mentioned to be $\ell$ , however, those two would cancel each other and still, the path difference would be same. Nice solution!

Tapas Mazumdar - 3 years, 5 months ago

Hey, I guess, for destructive inteference we have odd multiples of half wavelength that is

$\lambda \times \dfrac{2n-1}{2}=2l$

$\implies \lambda=\dfrac{4l}{2n-1}$

For $n=1$ , we have this

$\implies \lambda=4l$

Md Zuhair - 3 years, 4 months ago

Seems like I missed it. Yes it needs to be precisely 'odd'.

Tapas Mazumdar - 3 years, 3 months ago

@Tapas Mazumdar – Thanks for clarification :)

Md Zuhair - 3 years, 3 months ago

@Tapas Mazumdar – Have you deleted whatsapp?

Md Zuhair - 3 years, 3 months ago

I can't hear anything!

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