How Does Temperature Gradient Affect Sound Speed and Travel Time?

[aim1732]

Q:The absolute temperature of air in a region linearly increases from T₁ to T₂ in a space of width d. Find time taken by sound wave to go through the region in terms of T₁ T₂,d and speed v of sound at 273K.

Equations :

v_t = v( 1 + t/546)
dt = dx/v


Attempt:
I was basically looking to express speed of sound as a function of distance and integrating.

Assuming linear increase in temperature with distance
T(general) = T₁ + [T₂-T₁]*x/d

Therefore
v(general) = v[ 1 + (T₁ + [T₂-T₁]*x/d)/546 ]

dt = dx/v
= 546d (dx)/ v( 546d +T₁d + {T₂ -T₁}x )

Am I right with this? I was hoping for an easier way as I am not sure if this is correct.

 

[Coto]

I'm a little confused on what you are trying to do.

What is your $$v_t$$ formula? Does $$t$$ represent temperature for this formula? What is $$v$$?

 

[aim1732]

Actually v is velocity and t is temperature so v_t is velocity at that temperature.

v_t/v₂₇₃ = sqrt(T/273)
t=T+273 and using binomial expansion we have the result.