The Wave Function

The Wave Function

• The wave function, also known as a sinusoidal function, is a mathematical function that describes a smooth periodic oscillation.

• It can be modelled using the sine, cosine, and tangent functions in trigonometry.

• The properties of a wave function are determined by four parameters: amplitude, period, phase and vertical shift.

Amplitude

• The amplitude represents the wave’s peak from its mean position. In a mathematical description, it’s the vertical distance between a peak or a trough and the equilibrium position.

• In a function given as y = A sin(θ), A represents the amplitude. A larger amplitude would mean higher peaks and deeper troughs.

• The amplitude of a basic sine or cosine function is 1.

Period

• The period is the length of one complete cycle. In trigonometry, the period of basic sine and cosine functions is 2π.

• The period can be adjusted by multiplying the angle by a factor. The function y = sin(Bθ) has a period of 2π/B.

• The frequency of the wave is the reciprocal of the period. A higher frequency means more cycles in a given timeframe.

Phase and Vertical Shift

• The phase shift refers to the horizontal displacement of the wave. In a function y = sin(θ + C), C is the phase shift. A positive value for C will shift the graph to the left, while a negative value shifts it to the right.

• The vertical shift, on the other hand, shifts the wave up or down on the graph. It’s represented by a constant D in the function y = sin(θ) + D.

Knowing about these properties of the wave function is important for understanding the behaviour of waves, and can be manipulated to model various types of periodic phenomena from vibrating strings to alternate current.