Question

8. How many independent standing waves with wavelengths between 9.5 and 10.5 mm
can occur in a cubical cavity Im on a side? How many with wavelengths between 99.5
and 100.5 mm?
​

Answer 1

Given:

A cubical cavity 1 m on a side?  

To find:

How many independent standing waves with wavelengths between 9.5 and 10.5 mm

can occur in a cubical cavity Im on a side? How many with wavelengths between 99.5

and 100.5 mm?

Solution:

From given, we have,

The number of standing waves is given by,

N (λ) d (λ) = 8πV/λ⁴ dλ

where, V = l³

The number of standing waves with wavelengths between 9.5 and 10.5 mm in a cubical cavity 1 m on a side is,

N (λ) d (λ) = 8πV/λ⁴ dλ

= [8π × (1 m)³ × (10^{-3} m)] / [ 10 × 10^{-3} m]⁴

= (8π × 10^{-3} m⁴) / (10^{-8} m⁴)

= 8π × 10⁵

= 2.5 × 10⁶

The number of standing waves with wavelengths between 99.5 and 100.5 mm in a cubical cavity 1 m on a side is,

N (λ) d (λ) = 8πV/λ⁴ dλ

= [8π × (1 m)³ × (10^{-3} m)] / [ 10 × 10^{-3} m]⁴

= (8π × 10^{-3} m⁴) / (10^{-8} m⁴)

= 8π × 10⁵

= 2.5 × 10⁶