Wave Equation Solver
Introduction to the fixed-end wave equation on a string
A fixed-end string is one of the clearest settings for seeing the wave equation in action, because the starting shape f(x) and starting motion g(x) fully determine how the disturbance begins to move. With this solver, you describe that initial state, choose the wave speed c, and control the spatial and temporal resolution with the grid and step settings so the browser can march the solution forward.
The calculations on this page use a simple explicit finite-difference update, so the spacing of the grid and the selected time steps affect both the appearance of the trace and the amount of detail you can see. A smooth pluck may look almost identical on a coarse grid, while a sharper profile can become jagged unless you give the string enough sample points.
The sections below explain what the solver is doing, how to choose expressions that match a vibrating-string problem, how to read the sampled displacement, and which simplifying assumptions matter most before you rely on the output.
What fixed-end vibrating-string problem does this calculator solve?
This wave equation solver is designed for a one-dimensional string with fixed ends, where the function f(x) supplies the initial displacement and g(x) supplies the initial velocity. The page turns that starting state into a sequence of displacement samples so you can see how the string evolves over the interval from x = 0 to x = 1.
If you are deciding whether to use this calculator, it helps to phrase the question as a string problem instead of a generic math problem. For example, you might ask how a plucked string moves when the initial motion is zero, how much the trace changes if the initial velocity is not flat, or how many grid points are needed before the curve looks smooth enough to trust. Questions like those line up directly with the fields on the form.
This is especially useful when you want to compare one vibration setup against another. A change in f(x) affects the shape you start from, a change in g(x) alters the initial push or pull, and a change in c changes how quickly the disturbance travels through the mesh.
How to use the wave equation solver for a fixed-end string
Using the fixed-end wave equation solver starts with a physical picture of the string at t = 0. Enter the starting shape in f(x), enter the starting vertical motion in g(x), and then choose a positive propagation speed. The grid and time-step fields describe the numerical experiment rather than new physical properties of the string.
- Enter Initial displacement f(x) as the expression that describes the string’s starting shape.
- Enter Initial velocity g(x) as the expression that describes how the string is moving at t = 0.
- Enter Wave speed c as the propagation speed for the medium.
- Enter Grid points to choose how finely the string is sampled in space.
- Enter Time steps to choose how many updates the simulation should run.
- Select Simulate to refresh the displacement profile and result panel.
- Check the output’s shape, sign, and overall size before comparing scenarios.
Keep the expressions and resolution consistent with the problem you want to study. The function fields should return real values for the entire x-domain, the wave speed must be positive, and the grid and step counts should be large enough to show the motion without turning the result into a blur. When comparing multiple runs, record the exact f(x), g(x), c, grid, and step settings so you can reproduce the same string later.
Wave solver inputs: choosing f(x), g(x), c, grid points, and time steps
The inputs on this wave equation solver describe the string before the simulation begins and the numerical mesh used to update it. Most mistakes come from choosing a function that does not match the physical situation, or from using a mesh that is too coarse for the shape you want to inspect.
Initial displacement f(x) is the profile you would see if you froze the string at the opening instant. The default, sin(pi*x), is a single smooth hump that is zero at x = 0 and x = 1, so it fits fixed ends naturally. Initial velocity g(x) describes the vertical velocity at that same instant. Setting it to 0 models a release from rest; a nonzero expression adds an initial push or pull.
- Units: keep the x-scale and wave speed in one consistent unit system before entering them, so the travel speed has a clear meaning.
- Wave speed c: use a positive value. Raising c makes a wave traverse the same spatial mesh more quickly, while the solver correspondingly selects a smaller time increment.
- Grid points: this is the number of spatial intervals used to track the string. More points can resolve sharper bends, but also make the printed sample list longer.
- Time steps: this is the number of update cycles. More steps move the numerical solution farther forward in time.
For this solver, c must be positive, the grid needs at least 10 points, and at least one time step is required before the string can move forward. Those limits are practical guardrails for a well-defined finite-difference march, not universal physical rules. For a quick qualitative picture, begin with a smooth starting shape and then increase the resolution only if the trace looks too blocky.
Formulas for the fixed-end finite-difference wave update
Most wave solvers begin by sampling the initial shape and velocity, then updating the interior points of the string one step at a time. This page follows that pattern for a uniform one-dimensional string with fixed ends: it evaluates f(x) and g(x) across the grid, computes a time increment from the spacing and wave speed, and advances the interior samples while leaving the endpoints pinned.
The interior update used by the calculator can be written in standard finite-difference form as:
In that notation, u represents the displacement at grid position j and time level n. The factor involving c, Δt, and Δx controls how strongly neighboring samples influence the next update, and the fixed-end assumption means the boundary points stay anchored while the interior evolves. Because the page derives Δt from the grid spacing, changing the number of grid points also changes the time increment used in the march.
That relationship matters when interpreting a comparison. If you refine the grid, the string is sampled more densely and the displayed list becomes longer; if you coarsen the grid, the output is shorter but may hide curvature that matters. The solver is therefore best thought of as a controlled numerical approximation to the classical wave equation, not as a complete physical instrument.
Worked example: a sine-shaped fixed-end string released from rest
A useful worked example for this wave equation solver is the default sine-shaped displacement with zero initial velocity, because it matches the fixed-end boundary condition and gives you a clean baseline for comparison. Keep f(x) = sin(pi*x), set g(x) to 0, use c = 1, and leave the grid and time-step settings at their defaults. The string starts in a smooth single-hump shape before the explicit update moves it through time.
- Initial displacement f(x): sin(pi*x)
- Initial velocity g(x): 0
- Wave speed c: 1
- Grid points: 50
This fixed-end sine-string setup gives you a simple visual check of the solver’s behavior. The ends should remain pinned, the sampled profile should stay smooth enough to follow, and the output should look symmetric when the initial conditions are symmetric. If the result looks irregular, the likely causes are an expression that does not evaluate cleanly across the domain or a grid that is too coarse for the shape you entered.
Once the baseline looks right, change only one input at a time. A different f(x) changes the starting wave shape, a nonzero g(x) adds motion at t = 0, and a different c changes how quickly the disturbance advances through the mesh. This one-at-a-time approach makes it easier to identify why two snapshots differ.
Comparison table: grid resolution in the fixed-end wave solver
Grid resolution is usually the first setting to compare when a vibrating-string trace looks too rough or too dense. The table below describes how the same wave problem changes when you alter the number of spatial samples while leaving the other expressions alone.
| Scenario | Grid setting | What changes in the trace | What to watch |
|---|---|---|---|
| Coarser grid | Fewer grid points | The displacement list is shorter, and sharp bends can look blocky or stepped. | Check whether the main peak and the zero crossings are still easy to identify. |
| Baseline grid | Current setting | This is the reference run you should compare against before changing other inputs. | Use it to see whether a later change improves clarity or just changes the display density. |
| Finer grid | More grid points | The string is sampled more densely, so the curve usually looks smoother and more detailed. | Watch whether the shape itself changes or whether only the resolution improves. |
Because the solver ties the time step to the spatial spacing, a finer grid also changes the physical time represented by each update. A more detailed trace can improve clarity, but compare runs with the same f(x), g(x), c, and step count when you want a fair side-by-side check.
How to interpret the fixed-end vibrating-string displacement result
The result panel for this wave equation solver prints the displacement samples after the chosen number of steps, not a full animation. Read it as a snapshot of the string’s state at the end of the explicit march: large positive values indicate upward displacement, negative values indicate downward displacement, and values near zero show where the string has returned close to equilibrium.
As you inspect the output, focus on three questions. Does the overall shape make sense given f(x) and g(x)? Do the endpoints stay fixed as a fixed-end string should? If you change one major input, does the result move in the direction you expected? Those checks reveal more than looking for a single sample value in isolation.
If you want to save a run, use the Copy Result button to capture the displayed summary. That preserves the displacement snapshot exactly as the solver printed it, which is useful when comparing two parameter sets in notes, coursework, or a lab record.
Limitations and assumptions of the fixed-end wave model
Like any fast browser-based wave approximation, this calculator makes simplifying assumptions so the string can be updated quickly. It models a uniform one-dimensional medium with fixed ends and a linear explicit update. That makes it useful for teaching, quick checks, and comparing expressions, but it does not represent every real vibrating system.
- Input interpretation: read f(x) and g(x) literally; changing either function changes the physical setup.
- Unit conversions: keep the spatial scale and wave speed in a consistent system before you enter them.
- Linearity: this quick solver assumes a linear wave equation on a uniform medium; strong damping, stiffness, or changing material properties are not included.
- Rounding: displayed displacement values are rounded, so tiny differences between runs are normal.
- Missing effects: damping, energy loss, nonlinear motion, and higher-dimensional wave effects may not be represented.
If you use the output for lab notes, classroom work, or design checks, treat it as a numerical estimate and confirm the setup with a more detailed model or experiment before acting on it. The value of a wave calculator is that it makes the assumptions visible: you can see which input changed the motion, which grid settings affected the display, and where a more refined analysis may be needed.