The wave equation is derived by combining Newton’s second law with Hooke’s law, focusing on a small segment of a wave.

To begin the derivation of the wave equation, we examine a small element of a wave traveling along a string. This element experiences two forces due to the tension in the string on either side. When the string is not perfectly straight, these tension forces do not cancel each other out, resulting in a net force acting on the element. According to Newton’s second law, this net force will induce acceleration in the element.

To determine the net force acting on the element, we resolve the tension forces into their vertical and horizontal components. While the horizontal components cancel each other, the vertical components sum up, yielding a net vertical force. This force is proportional to the curvature of the string, which can be mathematically represented as the second derivative of the string’s displacement with respect to position.

Next, we analyze the acceleration of the element. According to Hooke’s law, the acceleration of the element is proportional to the net force acting on it. This relationship can be expressed mathematically as the second derivative of the string’s displacement with respect to time.

By equating these two expressions, we arrive at the wave equation: the second derivative of displacement with respect to position equals the second derivative of displacement with respect to time. This equation describes how the shape of the wave evolves over both time and space.

In mathematical notation, if $y(x, t)$ represents the displacement of the string at position $x$ and time $t$, the wave equation can be expressed as:

where $v$ denotes the speed of the wave. This equation illustrates that the curvature of the wave at any given point is proportional to the rate of change of the wave’s velocity at that same point. This fundamental result in wave physics is applicable to all types of waves, not just those traveling along a string.