Shouldn't the curvature of space-time have effects on the Doppler effect for sound waves?

Sound is a mechanical wave-form, with its speed determined by the properties of the media it travels in -- air, water etc. Light is a relativistic wave-form, with its speed determined by the light speed limitation and the curvature of space-time. The Doppler effect applies to both. This is the equation for the Doppler effect for sound. fo = ((v + vo)/(v + vs))fs where fo is the perceived sound frequency for the observer, vo is the speed of the observer, vs is the speed of the source of the sound, and fs is the actual frequency of the sound. The formula for the Doppler effect for light is ν′ = ν[(1 - (v/c)]/√[1 - (v/c)2] Sound travels in air at 340 m/s. Sound travels in water at 1500 m/s. Shouldn't this very substantial difference in the speed result in a small but measurable -- with atomic clocks anyway -- difference in the size of the Doppler effect for sound in air and water, that would result in a measurable departure from the basic Doppler effect for sound? That is, shouldn't the Doppler effect for light apply, as well as the Doppler effect for sound, and shouldn't the difference be rather greater for sound in water, than for sound in air? Has this experiment ever been done? If not, shouldn't it be done?