Calculate wave displacement, angular frequency, time, phase shift or amplitude from x = A sin(ωt + φ) by entering 4 known values to find the missing one.
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Wave Amplitude Formula
The calculator uses the sinusoidal wave displacement equation. Enter any 4 values to solve for the missing value.
x = A\sin(\omega t+\phi)
Rearranged forms used by the calculator:
A = \left|\frac{x}{\sin(\omega t+\phi)}\right|\omega = \frac{\sin^{-1}(x/A)-\phi}{t}t = \frac{\sin^{-1}(x/A)-\phi}{\omega}\phi = \sin^{-1}(x/A)-\omega t- x = displacement at a specific time
- A = wave amplitude, the maximum displacement from equilibrium
- ω = angular frequency in radians per second
- t = time
- φ = phase shift
The displacement function calculates the wave position from amplitude, angular frequency, time, and phase shift.
The amplitude function solves for the non-negative amplitude using the absolute value of displacement divided by the sine term.
The angular frequency, time, and phase shift functions use inverse sine. These return the principal solution. Other valid wave solutions can exist because sine repeats every 2π radians.
Unit Conversions and Wave Value Checks
| Quantity | Allowed units | Base unit used in calculation |
|---|---|---|
| Displacement | m, cm, mm, km, ft, in | meter |
| Amplitude | m, cm, mm, km, ft, in | meter |
| Angular frequency | rad/s | rad/s |
| Time | s, min, h | second |
| Phase shift | rad, deg | radian |
| Check | Meaning |
|---|---|
| |x| ≤ |A| | Displacement cannot be larger than amplitude in this sinusoidal model. |
| A ≥ 0 | Amplitude is treated as a magnitude, so it is non-negative. |
| sin(ωt + φ) ≠ 0 when solving for A | If the sine term is zero, amplitude cannot be found from x divided by that term. |
Examples
Example 1: Calculate displacement
Given amplitude A = 2 m, angular frequency ω = 3 rad/s, time t = 0.5 s, and phase shift φ = 0.2 rad:
x = 2\sin(3(0.5)+0.2)
x = 2\sin(1.7) = 1.9833\text{ m}The displacement is about 1.9833 m.
Example 2: Calculate amplitude
Given displacement x = 0.5 m, angular frequency ω = 4 rad/s, time t = 0.2 s, and phase shift φ = 0.1 rad:
A = \left|\frac{0.5}{\sin(4(0.2)+0.1)}\right|A = \left|\frac{0.5}{\sin(0.9)}\right| = 0.6383\text{ m}The amplitude is about 0.6383 m.
FAQ
What is wave amplitude?
Wave amplitude is the maximum displacement from the equilibrium position. If a wave moves 3 cm above and 3 cm below the center line, its amplitude is 3 cm.
Why can the calculator reject a displacement value?
For the equation x = A sin(ωt + φ), the sine value must stay between -1 and 1. That means displacement cannot have a greater magnitude than amplitude. If |x| is greater than |A|, the inputs do not fit this wave model.
Why does solving for time, angular frequency, or phase shift give a principal solution?
Sine is periodic. The same displacement can happen at many different angles separated by multiples of 2π radians. The inverse sine function gives one main angle, called the principal value, so the result is one valid solution, not always the only solution.
