Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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The Absolute Temperature of Air in a Region Linearly Increases from T1 to T2 in a Space of Width D. Find the Time Taken by a Sound Wave to Go Through the Region in Terms - Physics


The absolute temperature of air in a region linearly increases from T1 to T2 in a space of width d. Find the time taken by a sound wave to go through the region in terms of T1T2d and the speed v of sound at 273 K. Evaluate this time for T1 = 280 K, T2 = 310 K, d = 33 m and v = 330 m s−1.

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The absolute temperature of air in a region increases linearly from T1 to T2  in a space of width d.
The speed of sound  at 273 K is v.
vT is the velocity of the sound at temperature T.
Let us find the temperature variation at a distance x in the region.
Temperature variation is given by:

\[T =  T_1  + \frac{\left( T_2 - T_1 \right)}{d}x        \] 

\[  v \propto \sqrt{T}\] 

\[ \Rightarrow \frac{v_T}{v} = \sqrt{\left( \frac{T}{273} \right)}\] 

\[ \Rightarrow    v_T  = v\sqrt{\left( \frac{T}{273} \right)}\] 

\[ \Rightarrow dt = \frac{dx}{v_T} = \frac{du}{v} \times \sqrt{\left( \frac{273}{T} \right)}\] 

\[ \Rightarrow   t = \frac{\sqrt{273}}{v} \int\limits_0^d \frac{dx}{\left[ T_1 + \frac{\left( T_2 - T_1 \right)}{d}x \right]^\frac{1}{2}}\] 

\[ \Rightarrow t = \frac{\sqrt{273}}{v} \times \frac{2d}{T_2 - T_1} \left[ T_1 + \frac{\left( T_2 - T_1 \right)}{d} \right]_0^d \] 

\[ \Rightarrow t = \frac{\sqrt{273}}{v} \times \frac{2d}{T_2 - T_1}\left( \sqrt{T_2} - \sqrt{T_1} \right)\] 

\[ \Rightarrow t = \left( \frac{2d}{v} \right)\left( \frac{\sqrt{273}}{T_2 - T_1} \right) \times \sqrt{T_2} - \sqrt{T_1}      \left( \because A^2 - B^2 = \left( A - B \right)\left( A + B \right) \right)\] 

\[ \Rightarrow T = \frac{2d}{v}\frac{\sqrt{273}}{\sqrt{T_2} + \sqrt{T_1}}         .  .  . (i)\]

Evaluating this time:
Initial temperature T1 = 280 K
Final temperature T2 = 310 K
Space width d = 33 m
v = 330 m s−1

On substituting the respective values in the above equation, we get:

\[T = \frac{2 \times 33}{330}\frac{\sqrt{273}}{\sqrt{280} + \sqrt{310}} = 96  \text { ms }\]

Concept: Wave Motion
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HC Verma Class 11, 12 Concepts of Physics 1
Chapter 16 Sound Waves
Q 14 | Page 353
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