String wave average power calculator
A wave with 0.05 kg/m, 20 m/s, 10 rad/s and a 0.02 m amplitude carries 0.02 W along the string. Enter those four values for your own wave and see the average power, with each squared term shown. Free in your browser.
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- Your data stays on your device
Run — free
Worked example
- Linear density
- 0.05
- Wave speed
- 20
- Angular frequency
- 10
- Amplitude
- 0.02
- Omega squared
- 100
- Amplitude squared
- 0.0004
- Average power
- 0.02
- Average power unit
- W
A sinusoidal wave traveling along a taut string carries energy at a steady average rate set by how heavy the string is, how fast the wave propagates, how rapidly the particles oscillate, and how large the transverse displacement is. Physics students, instrument designers, and lab automation scripts all need that average power without re-deriving the product of linear density, wave speed, angular frequency squared, and amplitude squared every time. This string wave average power calculator accepts μ, v, ω, and A in SI units, evaluates P_avg equals one-half times μ times v times ω squared times A squared, and returns the watt result with intermediate squares and transparent step strings.
What average power on a string wave means
When a continuous sinusoidal transverse wave travels along an ideal taut string, each segment of the medium oscillates perpendicular to the propagation direction and does work on the neighboring segment. Instantaneous power fluctuates twice per cycle as the product of transverse force and transverse velocity, but the time average over one period is constant for a steady wave.
Standard wave mechanics derives that average power as one-half times the linear mass density μ times the wave speed v times the square of the angular frequency ω times the square of the amplitude A: P_avg = ½ μ v ω² A², with SI units kilograms per metre, metres per second, radians per second, and metres, giving watts. The factor one-half comes from averaging sin² or cos² over a full period.
Doubling the amplitude multiplies average power by four because A appears squared; doubling ω also multiplies power by four; doubling μ or v multiplies power by two. The formula assumes a non-dispersive ideal string with constant tension and no damping, so the wave shape and amplitude do not change with distance—exactly the model used in introductory and intermediate physics courses when they ask for the average power transmitted by a wave on a string.
What you can do with it
Classroom and homework checks
Let students submit μ, v, ω, and A for a sinusoidal string wave and verify P_avg = ½ μ v ω² A² against textbook worked examples.
Exam and LMS auto-grading
Mark average-power items with a deterministic oracle that rejects non-positive linear density and negative speed, frequency, or amplitude.
Instrument and lab notebooks
Estimate average energy transport along driven strings in design scripts without importing a full wave-mechanics library.
FAQ
What is the formula for average power of a wave on a string?
P_avg = ½ μ v ω² A², where μ is linear mass density (kg/m), v is wave speed (m/s), ω is angular frequency (rad/s), and A is amplitude (m). The result is in watts.
What happens if amplitude or wave speed is zero?
Zero amplitude, zero angular frequency, or zero wave speed means no average energy is transmitted; that is valid physics, not an error.
Why must linear density be positive?
A string with non-positive mass per length is not physical in this model.
How much power does a wave with 0.05 kg/m, 20 m/s, 10 rad/s and 0.02 m carry?
0.02 W. The angular frequency squared is 100 and the amplitude squared is 0.0004; half the product of linear density, speed and those two squares gives the average power.
What happens to the power if I double the amplitude?
It becomes four times larger, because amplitude enters squared. Doubling the angular frequency also multiplies the power by four, while doubling the linear density or the wave speed only doubles it.
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/string-wave-average-power \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"linear_density":0.05,"wave_speed":20,"angular_frequency":10,"amplitude":0.02}'const res = await fetch("https://api.kit.forhosting.com/optics/string-wave-average-power", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"linear_density": 0.05,
"wave_speed": 20,
"angular_frequency": 10,
"amplitude": 0.02
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/string-wave-average-power",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"linear_density": 0.05,
"wave_speed": 20,
"angular_frequency": 10,
"amplitude": 0.02
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/string-wave-average-power", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"linear_density":0.05,"wave_speed":20,"angular_frequency":10,"amplitude":0.02}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"linear_density":0.05,"wave_speed":20,"angular_frequency":10,"amplitude":0.02}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/string-wave-average-power", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"linear_density": 0.05,
"wave_speed": 20,
"angular_frequency": 10,
"amplitude": 0.02
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.string_wave_average_power",
"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
Zero wave speed, zero angular frequency, or zero amplitude correctly yields zero average power; non-physical negative values and non-positive linear density fail with invalid_input instead of a silent NaN.
What average power on a string wave means
Related quantities such as wave speed from tension and linear density (v = √(T/μ)) or frequency from angular frequency (f = ω/(2π)) are separate inputs you supply or compute elsewhere; this endpoint focuses solely on average energy transport once μ, v, ω, and A are known.
Input contract, response shape, and validation
Example: μ equals 0.05 kg/m, v equals 20 m/s, ω equals 10 rad/s, and A equals 0.02 m yields ω² equals 100, A² equals 0.0004, and P_avg equals 0.02 W. POST a JSON body with four required finite numbers: linear_density (μ in kg/m, strictly positive), wave_speed (v in m/s, non-negative), angular_frequency (ω in rad/s, non-negative), and amplitude (A in m, non-negative). Each field may arrive as a JSON number or as a numeric string; aliases such as mu, v, omega, and A are accepted. Missing fields, non-finite values, empty strings, and absolute values above 1e12 fail with invalid_input naming the field. Negative wave_speed, angular_frequency, or amplitude are likewise rejected; use non-negative magnitudes as textbooks do. On success the body echoes the cleaned inputs, omega_squared and amplitude_squared for inspection, average_power in watts, average_power_unit set to W, the formula string P_avg = (1/2) · μ · v · ω² · A², a short model tag, a human-readable message, and an ordered steps array you can show next to the answer. Zero v, ω, or A returns average_power of zero with a clear message rather than an error, matching the physics that a stationary or vanishingly small oscillation carries no average energy. Failed validation is not billed; successful calls use the published $0.002 base price.
Where a string-wave power API earns its keep
Classrooms and tutoring platforms use a string wave average power calculator so learners can check homework that starts from linear density, wave speed, angular frequency, and amplitude without hand-multiplying squared terms incorrectly. Lab scripts that drive a string with a known oscillator amplitude and measure speed can log transmitted average power for safety limits on continuous excitation. Keep requests small, pin the rounded watt result in tests, and you get reproducible average power for homework, LMS plugins, and edge workers without standing up wave-simulation infrastructure of your own. Exam generators need a deterministic oracle that rejects negative amplitude or non-positive μ as invalid_input instead of inventing a watt value. Guitar and piano design notebooks often estimate average energy flow along a string segment when comparing tension, gauge, and drive frequency; calling a tiny parse API avoids pulling a full symbolic toolbox for four numbers.
Technical questions
What inputs does the string wave average power calculator need?
Four finite numbers: linear_density (μ > 0), wave_speed (v ≥ 0), angular_frequency (ω ≥ 0), and amplitude (A ≥ 0). Aliases such as mu, v, omega, and A are accepted.
What does the string wave average power calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.