Plane Wave - Mathematical Formalisms

Mathematical Formalisms

Two functions that meet the above criteria of having a constant frequency and constant amplitude are the sine and cosine functions. One of the simplest ways to use such a sinusoid involves defining it along the direction of the x-axis. The equation below, which is illustrated toward the right, uses the cosine function to represent a plane wave travelling in the positive x direction.

In the above equation:

• is the magnitude or disturbance of the wave at a given point in space and time. An example would be to let represent the variation of air pressure relative to the norm in the case of a sound wave.
• is the amplitude of the wave which is the peak magnitude of the oscillation.
• is the wave’s wave number or more specifically the angular wave number and equals 2π/λ, where λ is the wavelength of the wave. has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a given distance at a particular point in time.
• is a point along the x-axis. and are not part of the equation because the wave's magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are.
• is the wave’s angular frequency which equals 2π/T, where T is the period of the wave. has the units of radians per unit time and is a measure of how rapidly the disturbance changes over a given length of time at a particular point in space.
• is a given point in time
• is the phase shift of the wave and has the units of radians. Note that a positive phase shift, at a given moment of time, shifts the wave in the negative x-axis direction. A phase shift of 2π radians shifts it exactly one wavelength.

Other formalisms which directly use the wave’s wavelength, period, frequency and velocity are below.

To appreciate the equivalence of the above set of equations note that and