Wave Equation Solver Calculator

Calculator

Responsive inputs: 3 columns large, 2 medium, 1 mobile.

All fields are unit-aware.

Mode

Choose a solution family and input style.

Speed method

Wave speed supports multiple physics inputs.

Amplitude A

Displacement amplitude in your chosen units.

Phase φ (rad)

Use radians for phase offsets.

Wave speed v (m/s)

Example: 340 for air, 100 for a string.

Tension T (N)

Used with μ to get v = √(T/μ).

Linear density μ (kg/m)

Mass per unit length of the medium.

Wavelength λ (m)

Used with f to get v = fλ.

Frequency f (Hz)

Required for traveling waves; optional for standing waves.

Length L (m)

Domain length for standing or numerical modes.

Mode number n

n=1 fundamental, n=2 second harmonic, etc.

Position x (m)

Value used for u(x,t) in the summary.

Time t (s)

Time used for profile and point evaluation.

Profile x-start (m)

Start of the chart domain.

Profile x-end (m)

Leave blank for an automatic range.

Profile points

Higher values improve resolution.

Spatial step Δx (m)

Smaller Δx increases accuracy and cost.

Time step Δt (s)

For stability, aim for vΔt/Δx ≤ 1.

Total time (s)

Total simulation duration.

Snapshot time (s)

Table shows this time; chart also shows final.

Initial velocity scale (units/s)

Applied to the same sine mode shape.

Results appear above this form after submission.

Example data table

Sample inputs and typical outputs for a string mode. Use these to sanity-check your setup.

Mode	Speed method	Inputs	Typical outputs
Standing	Tension / linear density	T=120 N, μ=0.012 kg/m, L=1.2 m, n=2, A=0.01	v≈100 m/s, λ=1.2 m, f≈83.3 Hz
Traveling	Direct speed	v=340 m/s, f=440 Hz, A=1, φ=0	λ≈0.773 m, ω≈2764.6 rad/s
Numerical	Direct speed	v=50 m/s, L=1 m, Δx=0.01 m, Δt=0.00018 s	r≈0.9, stable propagation, bounded u(x)

Formula used

The one-dimensional wave equation is:

∂²u/∂t² = v² ∂²u/∂x²

Traveling wave (analytic)

u(x,t) = A sin(kx − ωt + φ)

With ω = 2πf, k = 2π/λ, and v = ω/k = fλ.

Standing wave (fixed ends)

u(x,t) = A sin(nπx/L) cos(ωt + φ)

Mode k = nπ/L, ω = nπv/L, and λ = 2L/n.

Speed from string properties

v = √(T/μ)

T is tension, and μ is linear mass density.

Numerical solver

uᵢⁿ⁺¹ = 2uᵢⁿ − uᵢⁿ⁻¹ + r²(uᵢ₊₁ⁿ − 2uᵢⁿ + uᵢ₋₁ⁿ)

Courant number r = vΔt/Δx should be ≤ 1 for stability.

How to use this calculator

Pick a mode: traveling, standing, or numerical.
Select a speed method and enter the required fields.
Set amplitude and phase to match your signal.
Choose x and t for the point displacement summary.
For numerical runs, keep vΔt/Δx at or below one.
Press Solve to view values, table, and chart.
Use the download buttons for CSV or PDF outputs.

Wave Equation Solver: Practical Notes

Inputs and unit discipline

This calculator is most reliable when every value uses SI units: meters (m), seconds (s), newtons (N), and kilograms per meter (kg/m). Typical lab amplitudes for a string are 0.001 to 0.02 m. Keep phase in radians to compare runs without hidden unit conversions.

Common material wave speeds

Direct-speed entry should match a realistic medium. Sound in dry air at 20 C is about 343 m/s. In water, the speed is roughly 1480 m/s near room temperature. For many steels, longitudinal wave speed is often around 5000 m/s. These references help you catch order-of-magnitude mistakes fast.

Speed from tension and linear density

For a stretched string, the solver uses v = sqrt(T/mu), where T is tension and mu is linear mass density. Example: T = 120 N and mu = 0.012 kg/m gives v about 100 m/s. Doubling T increases v by sqrt(2), while doubling mu decreases v by the same factor.

Traveling-wave relationships

In Traveling mode, wavelength follows lambda = v/f and the phase is kx - omega t + phi. The solver reports k = 2*pi/lambda and omega = 2*pi*f. With v = 343 m/s and f = 440 Hz, lambda is about 0.780 m and omega is about 2764.6 rad/s, matching standard acoustics.

Standing-wave harmonics with fixed ends

In Standing mode, allowed wavelengths are lambda = 2L/n, so the wave fits an integer number of half-wavelengths in the length L. If L = 1.2 m, then n = 2 gives lambda = 1.2 m. Frequencies scale approximately linearly with n when speed is constant, producing harmonic series behavior.

Profile resolution and sampling

The output table samples u(x,t) on a grid. For smooth curves, use at least 100 points across the plotted distance. For higher modes or short wavelengths, 200 to 500 points is safer. A practical guideline is to keep the spatial sampling smaller than about lambda/20 to avoid aliasing peaks and nodes.

Numerical stability and the CFL condition

The finite-difference solver uses an explicit update that depends on the Courant number r = v*dt/dx. Stable runs typically require r less than or equal to 1. If v = 50 m/s and dx = 0.01 m, then dt should be about 0.0002 s or smaller. When r exceeds 1, results may grow without bound.

Interpreting outputs for reports

Use the derived panel for headline quantities (v, f, lambda, k, omega) and record the snapshot time in numerical mode. The table is ready for plotting displacement versus position in spreadsheets or scripts. Export PDF for quick lab submission, and attach CSV when you need reproducible analysis later.

FAQs

1) Which mode should I choose?

Traveling gives a sinusoidal analytic wave, Standing gives fixed-end harmonics, and Numerical evolves the field on a grid. Choose Numerical when you need stability checks, snapshot control, or a time-stepped demonstration.

2) Why does wavelength change when I edit frequency?

In traveling waves, the solver uses lambda = v/f. With the same medium, v stays constant, so increasing f reduces lambda proportionally. That is the standard dispersion-free relationship for an ideal string or uniform medium.

3) What does the phase phi do?

Phase shifts the wave without changing amplitude or speed. It helps match a measured signal that starts at a nonzero displacement, or align two runs so their peaks and zero crossings occur at the same reference time.

4) Can I override the standing-wave frequency?

Yes. If you enter a positive frequency in Standing mode, the calculator uses it to set omega = 2*pi*f while keeping k = n*pi/L. This is useful for comparing theory against a measured drive frequency.

5) What Courant number is recommended?

Keep r = v*dt/dx at or below 1, and many users prefer 0.6 to 0.9 for margin. If you see noisy oscillations or growing amplitudes, reduce dt or increase dx to lower r.

6) Why are the endpoints zero in Standing and Numerical modes?

Those modes assume fixed boundary conditions, u(0,t) = u(L,t) = 0, like a string clamped at both ends. If your setup has free ends or damping, the boundary conditions and solution form will differ.

7) How should I cite results from this calculator?

Save your input values, the derived outputs, and the snapshot time for numerical runs. Include the exported CSV or PDF as an appendix so plots and calculations can be reproduced later without re-entering parameters.