Adding Alternating Waveforms

Adding Alternating Waveforms (Same Frequency)

Add two sinusoidal waveforms of the same angular frequency:

$y_1(t) = A_1 \sin(\omega t + \alpha_1)$
$y_2(t) = A_2 \sin(\omega t + \alpha_2)$
$y_{sum}(t) = y_1(t) + y_2(t) = A_{sum} \sin(\omega t + \alpha_{sum})$

Wave 1 Parameters:

Wave 2 Parameters:

Common Angular Frequency:

Basic Trigonometric Waveforms

Visualizing the fundamental shapes of sine, cosine, and tangent functions plotted against an angle (in radians).

• $y = \sin(x)$: Starts at 0, peaks at 1, crosses 0, troughs at -1, returns to 0 over a period of $2\pi$.
• $y = \cos(x)$: Starts at 1, crosses 0, troughs at -1, crosses 0, returns to 1 over a period of $2\pi$. It leads the sine wave by $\pi/2$.
• $y = \tan(x)$: Represents $\sin(x)/\cos(x)$. It has a period of $\pi$ and vertical asymptotes where $\cos(x) = 0$.

Note: Tangent asymptotes occur at $x = \frac{\pi}{2} + n\pi$. The plot shows values approaching $\pm\infty$.

General Sinusoidal Form: $y(t) = A \sin(\omega t + \alpha)$

This form allows us to describe modified sine waves:

• $A$: Amplitude (controls height).
• $\omega$: Angular Frequency (controls period/frequency - how compressed the wave is).
• $\alpha$: Phase Angle (controls horizontal shift).
  • $+\alpha$: Left shift (Lead) by $\alpha/\omega$.
  • $-\alpha$: Right shift (Lag) by $|\alpha|/\omega$.

Relationship: Period $T = \frac{2\pi}{\omega}$, Frequency $f = \frac{\omega}{2\pi}$.

Adjust Parameters:

Harmonic Synthesis

Complex periodic waves can be constructed by adding a fundamental sine wave ($f_1$) and its harmonics (integer multiples of $f_1$, like $2f_1, 3f_1, ...$), each with its own amplitude and phase.

$y_{complex}(t) = C_1 \sin(\omega_1 t + \phi_1) + C_2 \sin(2\omega_1 t + \phi_2) + C_3 \sin(3\omega_1 t + \phi_3) + ...$

This example adds a fundamental and its 3rd harmonic.

Adjust Parameters for $y(t) = A_1 \sin(\omega t) + A_3 \sin(3\omega t + \alpha_3)$: