{
  "id": "wave-speed",
  "version": "84115fd71b90",
  "status": "published",
  "name": "Wave Speed Calculator",
  "question": "What is the wave speed?",
  "summary": "Wave speed calculator: v = f × λ from frequency and wavelength, λ ÷ T from the period, distance ÷ time, or √(F ÷ μ) on a stretched string, with the period, frequency and string fundamental.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/wave-speed-calculator",
  "markdown": "https://www.acalculator.org/physics/wave-speed-calculator.md",
  "kind": "function",
  "method": "v = f × λ = λ ÷ T; v = distance ÷ time; on a string v = √(F_T ÷ μ), with f₁ = v ÷ 2L and λ₁ = 2L for a string of length L; T = 1 ÷ f, ω = 2πf, k = 2π ÷ λ.",
  "assumptions": [
    "A wave of one frequency travelling at a steady speed; the string is uniform and perfectly flexible.",
    "For a string, the fundamental assumes both ends are fixed (nodes at each end)."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "From",
        "description": "What you know about the wave.",
        "type": "string",
        "enum": [
          "fl",
          "period",
          "dt",
          "string"
        ]
      },
      "f": {
        "title": "Frequency (f)",
        "description": "How many waves pass a point each second.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "frequency",
        "minimum": 1e-9,
        "maximum": 1e+21
      },
      "wl": {
        "title": "Wavelength (λ)",
        "description": "The distance from one crest to the next.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-15,
        "maximum": 1000000000000
      },
      "p": {
        "title": "Period (T)",
        "description": "The time for one full wave to pass a point.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-15,
        "maximum": 1000000000000
      },
      "d": {
        "title": "Distance",
        "description": "How far a crest travels.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-15,
        "maximum": 1000000000000
      },
      "t": {
        "title": "Time",
        "description": "How long the crest takes to travel that distance.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-15,
        "maximum": 1000000000000
      },
      "F": {
        "title": "Tension (F_T)",
        "description": "The pulling force in the string.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "mu": {
        "title": "Linear density (μ)",
        "description": "The mass of the string per unit length, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-9,
        "maximum": 1000000
      },
      "muu": {
        "title": "Linear density unit",
        "description": "The unit of the linear density you type.",
        "type": "string",
        "enum": [
          "kg-m",
          "g-m",
          "g-cm",
          "lb-ft",
          "lb-in"
        ]
      },
      "len": {
        "title": "String length (L)",
        "description": "Optional: the length between the two fixed ends, for the fundamental frequency.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.000001,
        "maximum": 1000000
      }
    }
  },
  "outputs": {
    "speed": {
      "label": "Wave speed",
      "description": "How fast the wave travels, in metres per second.",
      "format": "quantity",
      "unit": "mps"
    },
    "kmh": {
      "label": "Wave speed (km/h)",
      "description": "The same speed in kilometres per hour.",
      "format": "number"
    },
    "fps": {
      "label": "Wave speed (ft/s)",
      "description": "The same speed in feet per second.",
      "format": "number"
    },
    "mph": {
      "label": "Wave speed (mph)",
      "description": "The same speed in miles per hour.",
      "format": "number"
    },
    "frequency": {
      "label": "Frequency (Hz)",
      "description": "f: as typed, 1 ÷ T from the period, or the fundamental v ÷ 2L of a string of length L.",
      "format": "number"
    },
    "period": {
      "label": "Period (s)",
      "description": "T = 1 ÷ f, the time for one full wave.",
      "format": "number"
    },
    "wavelength": {
      "label": "Wavelength (m)",
      "description": "λ: as typed, or 2L for the fundamental of a string of length L.",
      "format": "number"
    },
    "omega": {
      "label": "Angular frequency (rad/s)",
      "description": "ω = 2π × f.",
      "format": "number"
    },
    "k": {
      "label": "Wave number (rad/m)",
      "description": "k = 2π ÷ λ.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "fl",
      "f": "2 Hz",
      "wl": "2.5 m",
      "p": "4 s",
      "d": "30 m",
      "t": "6 s",
      "F": "56.4 N",
      "mu": 0.000309,
      "muu": "kg-m"
    },
    "outputs": {
      "speed": 5,
      "kmh": 18,
      "fps": 16.404199475065617,
      "mph": 11.184681460272012,
      "frequency": 2,
      "period": 0.5,
      "wavelength": 2.5,
      "omega": 12.566370614359172,
      "k": 2.5132741228718345
    },
    "text": "The wave speed is 5 m/s."
  },
  "examples": [
    {
      "given": {
        "by": "dt",
        "d": 30,
        "t": 6
      },
      "expect": {
        "speed": 5,
        "kmh": 18
      },
      "source": "OpenStax, University Physics Volume 1, §16.1 Traveling Waves (Equation 16.1, v = λ ÷ T = λf; Examples 16.1 and 16.2), https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves (retrieved 2026-10-18), Example 16.1: v = 30.00 m ÷ 6.00 s = 5.00 m/s"
    },
    {
      "given": {
        "by": "fl",
        "f": 2,
        "wl": 2.5
      },
      "expect": {
        "speed": 5,
        "p": 0.5,
        "f": 2,
        "wl": 2.5,
        "omega": 12.566370614359172
      },
      "source": "OpenStax, University Physics Volume 1, §16.1 Traveling Waves (Equation 16.1, v = λ ÷ T = λf; Examples 16.1 and 16.2), https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves (retrieved 2026-10-18), Example 16.1: T = 1 ÷ 2.00 Hz = 0.50 s and λ = vT = 2.50 m, so v = fλ = 5.00 m/s"
    },
    {
      "given": {
        "by": "period",
        "wl": 0.08,
        "p": 4
      },
      "expect": {
        "speed": 0.02,
        "f": 0.25,
        "k": 78.53981633974483
      },
      "source": "OpenStax, University Physics Volume 1, §16.1 Traveling Waves (Equation 16.1, v = λ ÷ T = λf; Examples 16.1 and 16.2), https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves (retrieved 2026-10-18), Example 16.2: λ = 8.00 cm, T = 4.00 s, v = 2.00 cm/s, f = 0.25 Hz"
    },
    {
      "given": {
        "by": "string",
        "F": 56.4,
        "mu": 0.000309,
        "muu": "kg-m"
      },
      "expect": {
        "speed": 427.2285943668333
      },
      "source": "OpenStax, University Physics Volume 1, §16.3 Wave Speed on a Stretched String (Equation 16.8, v = √(F_T ÷ μ); Example 16.5), https://openstax.org/books/university-physics-volume-1/pages/16-3-wave-speed-on-a-stretched-string (retrieved 2026-10-18), Example 16.5: v = √(56.40 N ÷ 3.09 × 10⁻⁴ kg/m) = 427.23 m/s",
      "tolerance": 1e-12
    },
    {
      "given": {
        "by": "string",
        "F": 19.6,
        "mu": 0.006,
        "muu": "kg-m",
        "len": 2
      },
      "expect": {
        "speed": 57.154760664940824,
        "f": 14.288690166235206,
        "wl": 4
      },
      "source": "OpenStax, University Physics Volume 1, §16.6 Standing Waves and Resonance (Equations 16.15 and 16.16, λ₁ = 2L, f₁ = v ÷ 2L; Example 16.7), https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance (retrieved 2026-10-18), Example 16.7: v = √(19.6 ÷ 0.006) = 57.15 m/s, λ₁ = 4.00 m, f₁ = 14.29 Hz",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §16.1 Traveling Waves: Equation 16.1, v = λ ÷ T = λf, f = 1 ÷ T; Examples 16.1 and 16.2, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves",
    "OpenStax, University Physics Volume 1, §16.3 Wave Speed on a Stretched String: Equation 16.7, μ = m ÷ l; Equation 16.8, v = √(F_T ÷ μ); Example 16.5, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-1/pages/16-3-wave-speed-on-a-stretched-string",
    "OpenStax, University Physics Volume 1, §16.6 Standing Waves and Resonance: Equations 16.15 and 16.16, λₙ = 2L ÷ n, fₙ = nv ÷ 2L; Example 16.7, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 mi = 1,609.344 m, 1 lb = 0.45359237 kg, 1 lbf = 4.4482216152605 N, 1 kgf = 9.80665 N, retrieved 2026-10-18. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "wavelength",
    "frequency",
    "speed"
  ],
  "changelog": []
}