The amplitude of waves generated by a stone thrown into a quiet pool of water decreases with distance according to the inverse square root of the distance, or 1/r^0.5.
To understand this relationship, let's consider the wave phenomena involved. When the stone impacts the water, it creates a disturbance that propagates outward in the form of circular waves. These waves are known as radial or spherical waves because they spread out equally in all directions from the point of impact.
As the waves propagate, they carry energy away from the impact point. The total energy of the waves remains constant, but it spreads out over an increasingly larger area as the waves move outward. This means that the energy per unit area, which is related to the wave amplitude, decreases with increasing distance from the impact point.
The energy carried by the waves spreads out over the surface area of spheres centered at the point of impact. The surface area of a sphere is proportional to the square of its radius (A = 4πr^2). Therefore, as the waves move farther away, the same amount of energy is distributed over a larger surface area.
Since the total energy remains constant, the energy per unit area decreases with increasing distance. Mathematically, if we define the amplitude of the waves as the square root of the energy per unit area, then the amplitude decreases as 1/r^0.5.
In other words, as you move twice as far away from the impact point, the wave amplitude decreases by a factor of 1/√2 (approximately 0.707). This inverse square root relationship holds true for waves propagating in three-dimensional space, such as spherical waves in a quiet pool of water.
So, the amplitude of the waves falls off with distance as 1/r^0.5, indicating that the wave's intensity decreases as you move farther away from the point of impact.