How do you derive the wave equation?

The wave equation is derived from combining Newton's second law with Hooke's law for a small segment of a wave.

To derive the wave equation, we start by considering a small element of a wave on a string. This element is subject to two forces: the tension in the string on either side of the element. If the string is not perfectly straight, these forces will not cancel out, and there will be a net force on the element. According to Newton's second law, this net force will cause the element to accelerate.

The net force on the element can be found by resolving the tension forces into vertical and horizontal components. The horizontal components will cancel out, but the vertical components will add up to give a net vertical force. This force is proportional to the curvature of the string, which can be described mathematically as the second derivative of the displacement of the string with respect to position.

Next, we consider the acceleration of the element. According to Hooke's law, the acceleration of the element is proportional to the net force on it. This can be described mathematically as the second derivative of the displacement of the string with respect to time.

Setting these two expressions equal to each other, we get the wave equation: the second derivative of displacement with respect to position is equal to the second derivative of displacement with respect to time. This equation describes how the shape of the wave changes over time and space.

In mathematical terms, if y(x, t) is the displacement of the string at position x and time t, the wave equation is ∂²y/∂x² = (1/v²) ∂²y/∂t², where v is the speed of the wave. This equation tells us that the curvature of the wave at a given point is proportional to the rate of change of the wave's velocity at that point. This is a fundamental result in wave physics, and it applies to all types of waves, not just waves on a string.