Final answer:
The problem you are describing involves the analytical solution for sound wave propagation from a moving point source in a fluid, accounting for the Doppler effect. Utilize the Doppler shift equations to calculate changes in frequency, wavelength, and amplitude as perceived by an observer. The displacement of air molecules can be expressed as a cosine function incorporating these changes.
Step-by-step explanation:
You are inquiring about an analytical solution that describes the pressure field during sound wave propagation from a point source in a moving fluid, considering Doppler shift effects, and assuming one-dimensional fluid flow that is uniform with the fluid extending to infinity in both directions. The appropriate model for this is the Doppler effect in the context of acoustic waves.
The Doppler effect is a change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. For a point source moving in a static fluid, the Doppler effect causes shifts in observed frequency due to the movement of the source and/or the observer.
To model this analytically, you can consider a cosine function for the displacement of air molecules, given as s(x, t) = Smax Cos(kx - wt + ϕ), where s is the displacement, Smax is the maximum displacement, k is the wave number, w is the angular frequency, and ϕ is the initial phase. The Doppler effect for sound waves modifies the observed frequency and wave number based on the relative motion of source and observer.
To determine the frequency or amplitude of sound waves perceived by a stationary observer from a moving source, the classic Doppler shift formulas could be used, accounting for the speed of the source relative to the surrounding fluid.
Enduring Understanding 6.B from the Principles of Physics helps explain this relationship, asserting that the frequency of sound depends upon the relative motion between source and observer, leading to changes in wavelength and amplitude.