A problem related to Doppler Effect Will Doppler Effect of sound waves be observed if the waves are observed from a frame that is moving with a constant velocity with respect to the frame in which the waves are being produced ( where the source is located )?
I came across a problem that asked me to find the wave equation of a sinusoidal wave with respect to the moving frame. Now, since the source of the wave is moving backwards with respect to the moving frame, so Doppler effect should be observed and the frequency should change. But the solution didn't make any changes to the frequency or the wavelength. How is this possible?
 A: If I am understanding your question correctly, the answer is yes.
The Doppler effect is a change in the observed frequency (of sound here) due to motion of either the source or the observer.
The following equation for Doppler shifted frequency is very well known $$\tag 1 f_{o} = f_{s} \left(\dfrac{v}{v \mp v_{s}}\right)$$ where  $f_o$, $f_s$ are the observer and source frequencies respectively, $v$ is the velocity of sound, and $v_s$ is the velocity of the source (the minus sign in the denominator is used for an approaching source, and the plus sign is used for a receding source).

Will Doppler Effect of sound waves be observed if the waves are observed from a frame that is moving with a constant velocity with respect to the frame in which the waves are being produced ( where the source is located )?

This is probably a less familiar concept, but indeed the Doppler effect is also noticed for a stationary source and moving observer, such that the observer frequency $$\tag 2 f_{o} = f_{s} \left(\dfrac{v + v_{o}}{v}\right)$$
if the observer is moving toward the source and $$\tag 3 f_{o} = f_{s} \left(\dfrac{v - v_{o}}{v}\right)$$  if the observer is moving away from the source.
Note how in the "well known" equation (1) for the Doppler effect, we have velocity of the source $v_s$ on the RHS, where for cases where the observer is moving toward/away from the sound source, equations (2) and (3) respectively, we have the observer velocity $v_o$ on the RHS.

But the solution didn't make any changes to the frequency or the wavelength. How is this possible?

This is not possible. If the source and observer are in relative motion, there will be Doppler shifting.
