y(x,t) = A·sin(kx ± ωt + φ) — compute the displacement, break down the equation, and watch the wave move.
Compute displacement y(x,t)
Simpangan y = —
Besaran
Value
Fase θ = kx ± ωt + φ
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Wavelength λ = 2π/k
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Frekuensi f = ω/2π
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Periode T = 2π/ω
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Wave speed v = ω/k
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Arah rambat
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Verified examples
Example 1. A = 0.5; k = 3 rad/m; ω = 4 rad/s; φ = 0; minus sign; x = 1; t = 0.5.
Phase θ = 3·1 − 4·0.5 = 1 radian, so y = 0.5·sin(1) ≈ 0.5 × 0.8415 ≈ 0.4207. Derived: λ = 2π/3 ≈ 2.0944 m; f = 4/(2π) ≈ 0.6366 Hz; T = 2π/4 ≈ 1.5708 s; v = 4/3 ≈ 1.3333 m/s toward +x.
Example 2. A = 2; k = π/2 rad/m; ω = π rad/s; φ = π/6; tanda −; x = 1; t = 1.
θ = π/2 − π + π/6 = −π/3, sehingga y = 2·sin(−π/3) = −√3 ≈ −1.7321.
Example 3 (equation breakdown). y = 2 sin(5x + 2t + π/3) → A = 2; k = 5; ω = 2; φ = π/3 ≈ 1.0472; the + sign means it travels toward −x. λ = 2π/5 ≈ 1.2566 m; f = 2/(2π) ≈ 0.3183 Hz; T = π ≈ 3.1416 s; v = 2/5 = 0.4 m/s toward −x.
Wave equation breakdown
Paste an equation of the form y = A sin(kx ± ωt ± φ), misalnya y = 0.5 sin(3x - 4t). Konstanta π didukung (mis. π/3). Bagian φ boleh dihilangkan.
Parameter
Value
Amplitudo A
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Bilangan gelombang k
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Frekuensi sudut ω
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Konstanta fase φ
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Tanda / arah rambat
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λ = 2π/k
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f = ω/2π
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T = 2π/ω
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v = ω/k
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Visualisasi gelombang berjalan
The curve shows y(x) at time t. Press play to watch the wave propagate (parameters follow the calculator tab).
Pertanyaan umum
What is the harmonic wave equation?
The equation y(x,t) = A·sin(kx ± ωt + φ) describes the displacement y of a point at position x and time t for a one-dimensional traveling wave. A is the amplitude (maximum displacement), k is the wavenumber, ω is the angular frequency, φ is the phase constant, and the ± sign determines the direction of propagation.
What is the difference between wavenumber k and angular frequency ω?
k measures how densely packed the wave is in space: k = 2π/λ in rad/m — the larger k, the shorter the wavelength. ω measures how fast it oscillates in time: ω = 2πf = 2π/T in rad/s — the larger ω, the faster the vibration. The two are linked by the wave speed v = ω/k.
How do I determine the wave’s direction of travel from the ± sign?
(kx − ωt) travels toward +x (to the right), while (kx + ωt) travels toward −x (to the left). A memory aid: follow one point of constant phase, e.g. a wave crest. For (kx − ωt) to stay constant, if t increases then x must increase too — so the wave moves right.
How do I compute the wave speed v from k and ω?
Use v = ω/k. Since ω = 2πf and k = 2π/λ, this is equivalent to v = f·λ. Example: k = 3 rad/m and ω = 4 rad/s give v = 4/3 ≈ 1.3333 m/s.
How is k related to wavelength λ, and ω to frequency f and period T?
k = 2π/λ so λ = 2π/k; ω = 2πf = 2π/T so f = ω/(2π) and T = 2π/ω. Example: k = 3 gives λ = 2π/3 ≈ 2.0944 m; ω = 4 gives f = 4/(2π) ≈ 0.6366 Hz and T = 2π/4 ≈ 1.5708 s.
What is the phase constant φ?
φ (in radians) shifts the wave’s initial condition: y(0,0) = A·sin(φ). φ = 0 means the wave starts at the equilibrium point moving upward; φ = π/2 means it starts at a crest, equivalent to a cosine function.
For y = 0.5 sin(3x − 4t), what are A, k, ω, λ, f, and v?
A = 0.5; k = 3 rad/m; ω = 4 rad/s; φ = 0; the minus sign means the wave travels toward +x. Derived: λ = 2π/3 ≈ 2.0944 m; f = 4/(2π) ≈ 0.6366 Hz; T ≈ 1.5708 s; v = ω/k = 4/3 ≈ 1.3333 m/s.
What is y at x = 1 and t = 0.5 for y = 0.5 sin(3x − 4t)?
Phase θ = 3·1 − 4·0.5 = 1 radian, so y = 0.5·sin(1) ≈ 0.5 × 0.8415 ≈ 0.4207.
What is the difference between the harmonic wave equation and the general wave equation?
The general wave equation ∂²y/∂x² = (1/v²)·∂²y/∂t² is satisfied by every function of the form y = f(x ± vt), including non-periodic pulses. The harmonic wave equation y = A·sin(kx ± ωt + φ) is its particular sinusoidal solution: a single periodic wave with exactly one frequency.
What units are used for k and ω?
k uses radians per meter (rad/m) and ω uses radians per second (rad/s). If a problem gives angles in degrees, convert to radians first. Amplitude A uses the same displacement unit as y, e.g. meters.
Batasan cakupan. This tool solves the one-dimensional traveling wave equation. To convert frequency, wavelength, and period units (Hz, nm, ms, etc.), use wavelength–frequency calculator or konverter frekuensi.
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