The Wave Formula
We can modify the basic sine wave to describe any physical wave (sound, light, or matter). The standard equation is:
\[y = A \sin(Bx)\]
- Amplitude (\(A\)): The height of the wave from the center. It measures "intensity" or "volume."
- Period (\(P\)): The distance for one full cycle. \(P = \frac{2\pi}{B}\).
- Frequency (\(f\)): How many waves per unit distance. \(f = 1/P\).
Worked Examples
Example 1: Scaling Height
Graph \(y = 5\sin(x)\).
- The amplitude is 5. Instead of peaking at 1, the wave peaks at 5 and bottoms at -5.
Example 2: Scaling Length
Find the period of \(y = \sin(2x)\).
- \(B = 2\).
- Period \(P = \frac{2\pi}{2} = \pi\).
- This wave is "squashed"—it repeats twice as fast as the original sine wave.
Example 3: Combined Scaling
Find the period and amplitude of \(y = \frac{1}{2}\cos(4x)\).
- Amplitude = \(1/2\).
- Period = \(2\pi/4 = \pi/2\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Amplitude (\(A\)) determines the probability of finding a particle. Specifically, the probability is proportional to \(|A|^2\). The Period is related to the particle's Momentum. A wave with a short period (high frequency) represents a particle moving with high speed. This relationship, \(p = h/\lambda\), is the core of the De Broglie hypothesis. By looking at a quantum wave's amplitude and period, you are literally looking at the particle's energy and speed.