To design an equation that generates a single sine wave with varying amplitudes, you can introduce a time-varying function to modulate the amplitude of the sine wave. This function, often referred to as an envelope or modulation function, controls how the amplitude changes over time.
One common approach is to multiply the time-varying envelope function with a basic sine wave. The resulting equation represents a modulated sine wave with varying amplitudes. Here's a general form of the equation:
y(t) = A(t) * sin(2πf t + φ)
Where:
- y(t) represents the output value (instantaneous amplitude) of the modulated sine wave at time t.
- A(t) is the envelope function that determines the amplitude variation over time.
- f represents the frequency of the sine wave.
- t is the time variable.
- φ is the phase offset (if needed).
The envelope function, A(t), can be defined in various ways to achieve the desired amplitude variation. Here are a few examples:
Linear Amplitude Variation: A(t) = m * t + b
In this case, the amplitude increases linearly with time. 'm' represents the rate of change (slope), and 'b' is the initial amplitude at t = 0.
Exponential Amplitude Variation: A(t) = A₀ * e^(k * t)
Here, the amplitude grows exponentially with time. 'A₀' represents the initial amplitude at t = 0, and 'k' controls the rate of exponential growth.
Modulating Function: A(t) = g(t) * A₀
Instead of a simple mathematical function, you can use a more complex modulating function, g(t), to control the amplitude variation. This function can be defined based on specific requirements or desired effects.
These are just a few examples, and the choice of the envelope function will depend on the specific amplitude variation you intend to achieve. Feel free to adapt or combine these approaches based on your needs and desired effects.