Wave phase value calculator
Find where a travelling wave sits in its cycle at a given place and instant: wave number 2, position 3, angular frequency 4, time 1 and starting phase 0.5 give 2.5 radians. You see the full value and its equivalent within one cycle. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Wave number
- 2
- Position
- 3
- Angular frequency
- 4
- Time
- 1
- Initial phase
- 0.5
- Phase radians
- 2.5
- Wrapped phase radians
- 2.5
The wave phase value calculator evaluates the phase of a sinusoidal traveling wave at one position and instant. Enter wave number, position, angular frequency, time, and initial phase to apply the standard expression kx − omega t + phi zero. The result is reported in radians as both an unwrapped value and an equivalent value within one complete cycle. It is useful for optics, acoustics, electromagnetism, vibration analysis, and any calculation that needs the state of an oscillation before evaluating its sine or cosine.
What instantaneous wave phase represents
A sinusoidal wave repeats, but its argument records exactly where the oscillation sits in that repeating cycle. For a wave traveling in the positive position direction, this calculator uses phase = kx − omega t + phi zero. The wave number k describes how quickly phase changes with position, while angular frequency omega describes how quickly it changes with time.
Position x and time t select the event being examined, and the initial phase supplies the offset at the coordinate origin when time is zero. The returned phase is an angle in radians, not a displacement, intensity, electric field, or probability. You can insert that angle into sine or cosine according to the convention used by your wave equation.
Because phase is dimensionless, the products kx and omega t must each resolve to radians: position must match the reciprocal length unit used for wave number, and time must match the reciprocal time unit used for angular frequency. Consistent units are essential even though the calculator does not attach a specific meter, second, centimeter, or millisecond label to the numbers.
How to read unwrapped and wrapped results
Angles separated by an integer multiple of two pi represent the same point of a sinusoidal cycle, so both results produce the same sine and cosine apart from ordinary floating-point precision. For example, a negative unwrapped phase can have a positive wrapped equivalent near the end of a cycle. Choose the unwrapped value for propagation, continuity, phase accumulation, or differences between samples.
Choose the wrapped value for dials, cycle labels, plots colored by phase, or comparisons that should ignore complete revolutions. The formula follows the common positive-direction traveling-wave convention. If your source writes kx + omega t, it describes the opposite sign convention or propagation direction; represent that convention by supplying a negative angular frequency rather than silently changing the equation after calculation.
Input checks, precision, and automation
Provide all five inputs as finite numbers. Results are rounded to twelve decimal places so repeated calls and browser execution remain stable. That rounding is appropriate for general engineering and educational calculations, but it does not create experimental precision that was absent from the inputs.
The calculation is deterministic, uses no network request, and stores no wave data. In automated work, preserve the sign of wave number and angular frequency, record the unit convention beside the result, and avoid wrapping before calculating a phase difference when the number of elapsed cycles matters.
What you can do with it
Track an optical traveling wave
Evaluate the phase at a detector position and timestamp before using the angle in an electric-field or interference expression.
Compare sensor locations
Calculate unwrapped phases at two positions to determine their spatial phase difference under one wave number and frequency.
Drive a simulation or animation
Convert position and time samples into stable wrapped phase values for a cyclic display, shader, or oscillation model.
FAQ
Which phase formula does the calculator use?
It uses phase = kx − omega t + phi zero, the standard convention for a sinusoidal wave traveling in the positive position direction.
What units should I use?
Use reciprocal units consistently: if k is radians per meter, x must be meters; if omega is radians per second, t must be seconds. Initial phase is in radians.
Can wave number or angular frequency be negative?
Yes. Signed values are allowed and can represent a different propagation or sign convention, provided you interpret them consistently with the stated formula.
What is the difference between the full phase and the phase within one cycle?
The full value keeps every rotation the wave has accumulated, which helps when you compare events many cycles apart. The second value maps the same angle into one cycle, from zero up to just under 2π.
Is the phase the same as the wave's displacement?
No. The phase is an angle in radians that says where the oscillation is in its cycle. To get the actual state of the wave, you take the sine or cosine of that angle afterwards.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/optics/wave-phase-value \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"wave_number":2,"position":3,"angular_frequency":4,"time":1,"initial_phase":0.5}'const res = await fetch("https://api.kit.forhosting.com/optics/wave-phase-value", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"wave_number": 2,
"position": 3,
"angular_frequency": 4,
"time": 1,
"initial_phase": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/wave-phase-value",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"wave_number": 2,
"position": 3,
"angular_frequency": 4,
"time": 1,
"initial_phase": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/wave-phase-value", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"wave_number":2,"position":3,"angular_frequency":4,"time":1,"initial_phase":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"wave_number":2,"position":3,"angular_frequency":4,"time":1,"initial_phase":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/wave-phase-value", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"wave_number": 2,
"position": 3,
"angular_frequency": 4,
"time": 1,
"initial_phase": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.wave_phase_value",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
How to read unwrapped and wrapped results
The primary phase_radians result preserves the full value of kx − omega t + phi zero. This unwrapped phase is valuable when comparing events over many cycles because it retains the number and direction of accumulated rotations. The wrapped_phase_radians result maps the same angle into the interval from zero inclusive to two pi exclusive.
Input checks, precision, and automation
The calculator accepts ordinary decimal or scientific-notation strings as well as JSON numbers, and it recognizes concise aliases such as k, x, omega, t, and phi_0. Values with absolute magnitude greater than one trillion are rejected to keep the public input range explicit, and a computation that overflows to a non-finite phase is rejected instead of returning misleading JSON. The browser tool can be used interactively, while an API request costs $0.002 for a successful item. Invalid inputs return an invalid_input error that identifies the problematic field rather than a fabricated phase.
Technical questions
Why are two phase values returned?
phase_radians preserves accumulated cycles, while wrapped_phase_radians gives the equivalent angle from zero inclusive to two pi exclusive.
How much does an API calculation cost?
A successful API calculation costs $0.002 per item. The calculation is also available in the browser without sending wave data over the network.