Question

Sound exits a diffraction horn loudspeaker through a rectangular opening like a small doorway. Such a loudspeaker is mounted outside on a pole. In winter, when the temperature is 273 K, the diffraction angle θ has a value of 17o. What is the diffraction angle for the same sound on a summer day when the temperature is 311 K?

Answer 1

We know that diffraction angle is given by:

sin $\theta$ = lambda/D

lambda = wavelength of sound = v/f

D = width of the rectangular opening

v = speed of sound

f = frequency, So

sin $\theta$ = v/(f*D)

Now Assuming air is an ideal gas, speed of sound will be given by:

v = sqrt ($\gamma$kT/m)

$\gamma$ = Cp/Cv = ratio of specific heat capacities

k = Boltzmann constant,

T = temperature

m = average mass of the molecules of air

So,

sin $\theta$ = v/(f*D) = [1/(f*D)]*sqrt ($\gamma$kT/m)

sin $\theta$ = $\sqrt {\frac{\gamma\: kT}{mf^2D^2}}$

Now Since $\gamma$, k, m, f, and D are constant, So

$sin \theta \propto \sqrt T$

$\theta$_s = diffraction angle in summer = ?

$\theta$_w = diffraction angle in winter = 17 deg

T_s = Temperature in summer = 311 K

T_w = Temperature in winter = 273 K

So,

$\frac{sin \theta_{s}}{sin \theta_{w}} = \sqrt {\frac{T_{s}}{T_{w}}}$

${sin \theta_{s}} = {sin \theta_{w}}\sqrt {\frac{T_{s}}{T_{w}}}$

${sin \theta_{s}} = {sin 17^{\circ}}\sqrt {\frac{311}{273}}$

${sin \theta_{s}} = 0.312$

$\theta_{s}} = arcsin (0.312)$

$\theta_{s} = 18.2^{\circ}$

Diffraction angle in summer = $\theta_{s} = 18.2^{\circ}$

Please Upvote.

Comment below if you have any query.