Why does the speed of sound in water increase with pressure?

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I’ve heard some explanations before which just sound intuitively wrong – mostly that “higher pressure = higher density = particles closer together = easier propagation of pressure wave”. I don’t like this firstly because water is only negligibly compressible and also because a higher density would surely lead to a lower speed of sound ceteris paribus?
Happy that speed of sound is a function of bulk modulus but can’t wrap my head around why the bulk modulus would be different at different pressures. Thanks for any help!

In: Physics