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Let speed of sound in the medium be $v$ when the source is at rest w.r.t the medium.

In the case of the observer moving towards the stationary source with a velocity of $u_1$, the speed of sound relative to the observer is $v+u_1$. And thus $T=T_o(1-\frac{u_1}{u_1+v})$

But in the case of the source moving towards the stationary observer with a velocity of $u_2$, the speed of sound $w.r.t$ the observer is taken as $v$ only. Thus, Time period, $T=T_o(1-\frac{u_2}{v})$ Why, though? As the source is moving, shouldn't the velocity of sound increase and be taken as $v+u_2$?

Taking the sound in air as $330ms^{-1}$, the speed of the sound emitted will remain the same even if the source is moving at relativistic speed in the medium ?

There is one similar question, but it hasn't been answered properly and I want to know if the source is moving with relativistic speed, either towards or away from the observer makes any difference to the speed of sound in the medium.

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First, I wrote an answer to a somewhat related question at: Can we apply simple mechanics laws on sound waves

In the case of the observer moving towards the stationary source with a velocity of $u_{1}$, the speed of sound relative to the observer is $v + u_{1}$.

No, this is the wrong way to think about it and it's likely the source of your confusion. The observe approaches a stationary source at $u_{1}$ and the sound wave is emitted by the source with a phase speed of $v$. That means, the observer is approaching the sound waves at $v + u_{1}$. The speed of sound does not change simply because the observer moves relative to the source.

But in the case of the source moving towards the stationary observer with a velocity of $u_{2}$, the speed of sound w.r.t. the observer is taken as $v$ only.

Yes, the speed of sound is still the same and does not change simply because the source moves. There is something else assumed here but not explicitly stated. The speed $u_{2}$ must be less than $v$ if the time of arrival equation you show is correct. This is why the approaching speed is simply $v$. The sound wave beats the source to the observer's location.

...the speed of the sound emitted will remain the same even if the source is moving at relativistic speed in the medium ?

If an object moved at relativistic speeds, it would create a shock wave and the emitting object would pass by the observer before the observer heard both the sonic boom and the emitted sound. Actually, if you managed to move an object at relativistic speeds in a neutral medium like Earth's atmosphere, the shock would generate enough heat to ionize the air creating a plasma. The obligatory reference to xkcd is necessary here too: https://what-if.xkcd.com/1/

There is one similar question, but it hasn't been answered properly and I want to know if the source is moving with relativistic speed, either towards or away from the observer makes any difference to the speed of sound in the medium.

No, the speed of the emitting object does not change the speed of sound in the ambient medium. A relativistic object would, however, change the medium immediately surrounding it so much that its local speed of sound would be drastically different than the ambient medium.

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