Water is about a thousand times more dense than air, yet the speed of sound in water is greater than in air. How is this possible?

The bulk modulus of a material is a critical factor that influences the speed of sound within it. It indicates how resistant a material is to compression when subjected to an external pressure. Essentially, a higher bulk modulus signifies that the material is less compressible, meaning it's harder to shrink or change its volume.

Imagine squeezing a spring: a stiffer spring (with a higher 'spring constant', analogous to the bulk modulus) is harder to compress than a softer one. Similarly, materials with a high bulk modulus, such as water, do not compress easily when sound waves pass through them. This resistance to compression facilitates the rapid transfer of sound waves, leading to a higher sound speed in such materials compared to those with a lower bulk modulus.

The formula for the speed of sound in a medium is given by the equation:
\[ v = \sqrt{\frac{B}{\rho}} \]
where \( v \) is the sound speed, \( B \) denotes the bulk modulus, and \( \rho \) represents the density of the medium. This equation beautifully sums up the relationship between the speed of sound and the physical properties of the medium.

In essence, this means that the sound speed in a material depends on two factors: how tight or loose the material is packed (its density) and how much it resists changing shape (its elasticity, represented by the bulk modulus). It implies that even if a medium is dense, such as water, the sound speed can still be high if the material is sufficiently rigid (where 'rigid' refers to a high bulk modulus).

One might assume that a denser medium would always slower the sound speed because more mass would impede the motion of the sound waves. However, while density is indeed a determinant of sound velocity, it's not the only one. As the sound speed formula shows, the velocity also heavily depends on the bulk modulus.

Density — a measure of how much mass is packed into a given volume — increases the inertia that sound waves need to overcome to propagate. However, a high-density material can still transmit sound quickly if it has an even higher bulk modulus, showing that density alone does not dictate the agility with which sound can traverse a medium.

Compressibility is essentially the 'give' of a material, or how much its volume decreases under pressure. It's inversely related to the bulk modulus; a material with a low bulk modulus is more compressible. Hence, highly compressible materials will not transmit sound as efficiently as materials that are less compressible.

An intuitive way to understand this is considering a row of balls - if these balls are easily squished together (highly compressible), it will take longer for a push on one end to transfer to the other end, compared to a row of balls that are hard to compress. In the case of sound waves, less compressible materials allow the waves to jump from molecule to molecule more swiftly, leading to a faster sound spread through the medium.