Speed of a dynamic hydraulic system

We all have noticed that changing the temperature of the water in the shower is not instantaneous, rather the result is felt when the water that was in the tap works its way up to the showerhead. However, changing the pressure does feel instantaneous. I wonder at what speed the change in pressure propagates in the direction of flow in a flowing fluid. My guess would be at the speed of sound in the medium, based on the fact that it is faster than the speed of the flow of the water (like the temperature is), yet obviously at or below C. I am obviously missing the correct keywords to google for as I cannot find any references to this phenomenon. What is the answer to the query, or better yet, how could I have found this information (short of empirically)?

Thanks.

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The pressure wave does indeed travel with the speed of sound of the media. This is much higher for water then for air with a speed of $\approx$ 1500 m/s.