To properly describe a vibration or wave motion, we use four fixed characteristics. In this chapter, you will learn how to read these characteristics from a graph and how to calculate them from a formula.

1. Equilibrium position and Amplitude

The equilibrium position and the amplitude together determine the vertical range of the graph (how high and how low the wave goes).

🔹 Equilibrium position ( $y = d$ ): The horizontal line exactly in the middle of the wave.

equilibrium = $\frac{maximum + minimum}{2}$
🔹 Amplitude ( $a$ ): The maximum distance from the graph to the equilibrium position. The amplitude is always a positive number.

amplitude = $\frac{maximum - minimum}{2}$

2. Period and Frequency

These two concepts say something about the horizontal speed: how fast does the wave repeat itself?

Period ( $T$ ): The time or distance required for one full repetition.
Frequency ( $f$ ): The number of repetitions per second (or per unit). The unit is Hertz (Hz).

The calculation rule: Period and frequency are each other's reciprocal:

f = \frac{1}{T}

and

T = \frac{1}{f}

3. The factor $b$ in the formula

In the general form $y = d + a \cdot \sin (b (x - c))$ , the number $b$ is the factor that determines the period.

Calculate period:

period =

\frac{2 π}{b}

Calculate factor

b

:

b =

\frac{2 π}{period}

Example: Given $f (x) = 10 + 4 \sin (0.5 π \cdot x)$

Equilibrium position ( $d$ ) = 10
Amplitude ( $a$ ) = 4
Period ( $T$ ) = $\frac{2 π}{0.5 π}$ = 4
Frequency ( $f$ ) = $\frac{1}{4}$ = 0.25 Hz

Short Overview

Concept	Meaning	In the formula
Equilibrium position	The middle of the wave	$d$
Amplitude	The maximum displacement (half height)	$a$
Period	Length of one full repetition	$\frac{2 π}{b}$
Frequency	Number of repetitions per unit	$\frac{1}{T}$