Analysis of waves in shallow water (depth much less than wavelength) yields the following wave equation: $$\frac{\partial^{2} y}{\partial x^{2}}=\frac{1}{g h} \frac{\partial^{2} y}{\partial t^{2}}$$,where \(h\) is the water depth and \(g\) the gravitational acceleration. Give an expression for the wave speed.

Shallow water waves are a type of wave occurring in a body of water where the depth is significantly less than the wavelength of the wave. These waves are affected by the seafloor because the depth of the water influences the wave's movement and speed. Unlike deep water waves, where the water depth is greater than half the wavelength, shallow water waves' speed depends on the water depth rather than the wave frequency or wavelength.

When analyzing shallow water waves, it's important to understand that they behave differently from deep water waves. They tend to have a longer wavelength and can travel vast distances. Their speed is calculated based on both the water's depth and the force of gravity acting upon the water.

Gravitational acceleration, denoted as \(g\), is the acceleration that is imparted to objects due to the force of gravity exerted by a massive body, like Earth. Near the Earth's surface, it is approximately \(9.81 \; \text{m/s}^2\) and is considered a constant in many physics problems involving motion. However, \(g\) can vary slightly depending on the location on Earth, due to factors like altitude and geological formations.

The concept of gravitational acceleration is crucial in understanding the behavior of waves, as it influences the rate at which waves propagate through different media, such as water in the case of shallow water waves. Whether dealing with celestial bodies or ocean waves, gravitational acceleration is a foundational force that underpin the equations and principles that govern motion and dynamics.

The wave speed derivation from a wave equation represents how fast the wave travels through a medium. In the context of the shallow water wave equation, the derivation follows from setting the wave equation in its standard form in comparison to the specific scenario given.

To derive wave speed, we compare the given wave equation \(\frac{\partial^{2} y}{\partial x^{2}}=\frac{1}{g h} \frac{\partial^{2} y}{\partial t^{2}}\) to the standard wave equation and identify what component of the given equation represents the reciprocal of the wave speed squared. In our scenario, the term \(\frac{1}{g h}\) equates to \(\frac{1}{v^2}\), with \(v\) being the wave speed. Solving for \(v\) gives us the expression \(v = \sqrt{g h}\), where \(v\) is the wave speed, \(g\) is the gravitational acceleration, and \(h\) is the depth of the water. This derivation is essential for understanding how physical characteristics like gravity and depth influence the speed at which shallow water waves travel.