One Dimensional Wave Equation

Many phenomena around us behave like waves.

The sound we hear travels through air as a wave.
A guitar string produces music because it vibrates as a wave.
Water in a pond moves in the form of waves.
Light, which allows us to see, also shows wave behaviour.

To understand and control such phenomena, we need a mathematical relation that connects the quantities governing wave motion. In the seventeenth century, this was extremely difficult. There was no established wave theory and no standard use of partial differential equations.

In 1687, Isaac Newton introduced his second law of motion,

$$F = ma$$

This law suggested a new approach. Instead of analysing an entire string at once, one could apply Newton’s law to a very small segment of the string.

d’Alembert’s idea

Around 1747, Jean le Rond d’Alembert made a crucial observation. Instead of trying to directly guess the equation of a vibrating string, he decided to first construct a partial differential equation using physical laws. Once this equation was obtained, he used mathematics to solve it and determine the function describing the motion of the string.

NOTE: Our Method is to Apply Newton’s second law to a small piece of string. Tension is a Force and since tension arises when a string is pulled, we can Apply Newtons Law. Hence the force that arises from tension, and the resulting acceleration is the vertical acceleration of that piece.

Math For Engineers and Physicists

Available Formula Sheets