Calculate the displacement of one sinusoidal wave component, y = A sin(ωt + φ), or solve for its amplitude, angular frequency, time or phase shift.

This widget evaluates one sinusoidal component, y = A sin(ωt + φ). For superposition, evaluate components at the same point and time, then add their signed displacements. Inverse solutions depend on the selected sine branch and cycle.

Related Calculators

Superposition Formula

In a linear wave system, the net output at a given instant is the algebraic sum of the individual wave displacements evaluated at the same observation point. For sinusoidal components, the combined response is:

SP = Σᵢ₌₁ⁿ Aᵢ sin(ωᵢ t + φᵢ)

If you are evaluating only one sinusoidal component, the expression reduces to:

SP = A sin(ω t + φ)
Term Meaning Typical Units
SP Net superposition output or instantaneous displacement m, cm, mm
Ai Amplitude of the i-th wave component m, cm, mm
ωi Angular frequency of the i-th component rad/s
t Time at which the wave is evaluated s
φi Phase shift of the i-th component rad or deg
n Number of wave components being combined unitless

How to Calculate Superposition

  1. Select Component displacement and enter the amplitude of one component.
  2. Enter that component’s angular frequency in radians per second.
  3. Use the same time value for every component so the waves are compared at the same instant.
  4. Enter each phase shift using a consistent angle unit.
  5. Calculate each component separately and add the signed displacements to obtain the total; the widget does not automatically sum multiple components.

This calculator is most useful when the system is linear, meaning the total response is equal to the sum of the individual responses.

Helpful Relationships

If your information is given in frequency instead of angular frequency, convert it first:

ω = 2π f

The period corresponding to an angular frequency is:

T = (2π) / (ω)

If phase shift is entered in degrees, convert it to radians before evaluating the sine function:

φrad = φdeg × (π) / (180)

What the Output Means

  • Positive SP: the net displacement is above the reference equilibrium at that instant.
  • Negative SP: the net displacement is below the reference equilibrium.
  • SP = 0: the waves may be canceling at that moment; it does not necessarily mean no waves are present.
  • Larger magnitude: the instantaneous displacement has a larger absolute value; this alone does not establish constructive interference.

Constructive and Destructive Interference

Superposition explains interference. When waves line up in phase, they reinforce each other. For two waves with the same angular frequency and zero phase, the amplitudes add directly:

SP = (A₁ + A₂)sin(ω t)

When two equal-amplitude waves are separated by a phase difference of π radians, they cancel completely:

Asin(ω t) + Asin(ω t + π) = 0

Most real cases fall between these extremes, producing partial reinforcement or partial cancellation depending on relative phase.

Example

For a wave with amplitude 4 m, angular frequency π rad/s, time 0.5 s, and zero phase shift, the instantaneous output is:

SP = 4sin(π × 0.5 + 0) = 4sin((π) / (2)) = 4

If a second wave contributes -1 m at the same instant, the total superposition becomes:

SPₜₒₜₐₗ = 4 + (- 1) = 3

Common Mistakes

  • Using frequency in hertz when the calculation requires angular frequency in rad/s.
  • Mixing degrees and radians for phase shift.
  • Adding amplitudes directly without evaluating the sine term.
  • Comparing components at different times instead of the same time.
  • Applying simple superposition to nonlinear systems, where direct addition may not hold.

Where Superposition Is Used

This type of calculation appears in acoustics, vibration analysis, waveform synthesis, optics, electrical signals, and any linear system where multiple oscillations combine into one measurable response. In many simplified wave models, position is embedded in the phase term, which is why time, angular frequency, amplitude, and phase shift are enough to determine the instantaneous output.