Key Points from this Chapter:
We can write an equation for a wave as a function of space and time: for a 1-dimensional wave, y = f(x, t).
A sinusoidal wave, travelling to the right, can be described by a function of the form
y = Asin(2π(x-vt)/λ)
To enable us to use calculus with a function of more then one variable, we introduced the idea of a partial derivative, which is worked out with respect to one variable while others are held constant. For example, ∂y/∂x is the derivative of y with respect to variable x, while other variables are held constant.
The wave equation is a very important, second-order partial differential equation which must be satisfied by any one dimensional wave:
Superposition of two or more waves with the same velocity produces a function which is still a solution of the wave equation and therefore is another wave.
A standing wave can be formed by superposing two waves of equal wavelength and amplitude travelling in opposite directions. If the component waves are simple sinusoids, the superposition wave function can be written in the form
y = 2Asin(2πx/λ)cos(2πvt/λ).
An important feature of this is that the variables are separated: distinct time and displacement-dependent terms are simply multiplied together.