Calculate wave displacement, angular frequency, time, phase shift or amplitude from x = A sin(ωt + φ) by choosing a solve mode and entering the known values.

Sinusoidal model x = A sin(ωt + φ). Amplitude is nonnegative; time and angular rate may be signed coordinates.

Related Calculators

Wave Amplitude Formula

The calculator uses the sinusoidal wave displacement equation. Choose the value to solve for, then enter the known values. Inverse modes also offer a sine branch and integer cycle index.

x = Asin(ω t + φ)

Rearranged forms used by the calculator:

A = (x) / (sin(ω t + φ))
ω = (sin⁻¹(x / A) - φ) / (t)
t = (sin⁻¹(x / A) - φ) / (ω)
φ = sin⁻¹(x / A) - ω 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 divides displacement by the sine term and requires a nonnegative answer. Opposite signs do not fit the fixed phase; a zero or numerically near-zero sine term cannot reliably determine amplitude.

The displayed inverse formulas above show the default principal branch with cycle index 0. More generally, set θ = asin(x/A) + 2πk or θ = π − asin(x/A) + 2πk, then use ω = (θ − φ)/t, t = (θ − φ)/ω or φ = θ − ωt. Here k is any integer; the calculator supports k from −1,000,000 to 1,000,000 and bounds phase magnitude for numerical reliability. Signed time and angular-rate coordinates are allowed.

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 = 2sin(3(0.5) + 0.2)
x = 2sin(1.7) = 1.9833 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 = (0.5) / (sin(4(0.2) + 0.1))
A = (0.5) / (sin(0.9)) = 0.6383 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 can time, angular frequency, or phase shift have multiple solutions?

Sine has two inverse branches, θ = asin(x/A) + 2πk and θ = π − asin(x/A) + 2πk. Choose a branch and cycle index to select a solution. The default uses the principal branch with k = 0; zero amplitude or a zero inverse denominator may leave the requested value undetermined.

wave amplitude formula