Sinusoidal Wave Function Calculator
Evaluate a sinusoidal wave function y = A·sin(B(x − C)) + D at any x value given its amplitude, frequency, phase shift, and vertical shift. Perfect for physics, signal processing, and math coursework.
About this calculator
A general sinusoidal wave is described by y = A × sin(B × (x − C)) + D, where A is the amplitude (peak displacement from center), B is the angular frequency (controlling how compressed or stretched the wave is), C is the phase shift (horizontal offset), and D is the vertical shift (baseline displacement). The period of the wave is T = 2π / B. Amplitude determines the height of the peaks, while the vertical shift moves the entire wave up or down. The phase shift moves the wave left or right along the x-axis. This calculator evaluates y = amplitude × sin(frequency × (x − phaseShift)) + verticalShift for a user-supplied x value, giving the instantaneous wave height at that point.
How to use
Set amplitude A = 3, frequency B = 2, phase shift C = 1, vertical shift D = 0.5, and evaluate at x = 2. The formula gives: y = 3 × sin(2 × (2 − 1)) + 0.5 = 3 × sin(2 × 1) + 0.5 = 3 × sin(2) + 0.5. sin(2 radians) ≈ 0.9093, so y = 3 × 0.9093 + 0.5 = 2.728 + 0.5 ≈ 3.228. Enter these five values into the calculator to confirm the result. The period of this wave is 2π/2 ≈ 3.14 units.
Frequently asked questions
What does the amplitude parameter control in a sinusoidal wave function?
Amplitude A controls the maximum displacement of the wave from its midline. A wave with A = 3 will oscillate between −3 and +3 (or between D − 3 and D + 3 when a vertical shift D is present). It does not affect the period, frequency, or horizontal position of the wave — only its height. In physics, amplitude relates directly to the energy carried by a wave: doubling the amplitude quadruples the energy in many wave types, such as sound or electromagnetic radiation.
How do I find the period of a sinusoidal wave from its frequency parameter?
The period T is the horizontal distance for one complete cycle of the wave. It is calculated as T = 2π / B, where B is the frequency (angular frequency) parameter entered into the calculator. For example, if B = 4, the period is 2π/4 = π/2 ≈ 1.571 units. A larger B compresses the wave (shorter period, more cycles per unit), while a smaller B stretches it. Note that B here is angular frequency in radians per unit, not cycles per unit (Hz).
What is the difference between phase shift and vertical shift in a sine wave?
Phase shift C moves the wave horizontally along the x-axis: a positive C shifts the wave to the right, while a negative C shifts it to the left. Vertical shift D moves the entire wave up or down, changing the midline from y = 0 to y = D. Neither shift changes the shape, amplitude, or period of the wave. In signal processing, phase shift is critical for understanding timing differences between two waves, while vertical shift corresponds to a DC offset added to an AC signal.