Formula Used
For a displacement function, total distance is the sum of absolute position changes
across every monotonic part of the interval.
Total distance = Σ |x(tᵢ) - x(tᵢ₋₁)|
For advanced numerical modes, the calculator integrates absolute velocity.
Total distance = ∫ from a to b |v(t)| dt
For the sine displacement model:
x(t) = d + A sin(ωt + φ),
velocity is v(t) = Aω cos(ωt + φ).
Turning points occur where v(t) = 0.
For the cosine displacement model:
x(t) = d + A cos(ωt + φ),
velocity is v(t) = -Aω sin(ωt + φ).
For damped oscillation:
x(t) = d + A e^(-λt) sin(ωt + φ).
The calculator estimates distance using Simpson integration.
How to Use This Calculator
- Select the oscillating function model.
- Enter amplitude, angular frequency, phase, and offset.
- Enter the start and end time for the interval.
- Use damping only for the damped sine model.
- Increase samples when using numerical integration.
- Click the calculate button.
- Review total distance, net displacement, and graph values.
- Export the result as CSV or PDF.
Example Data Table
| Model |
A |
ω |
φ |
Interval |
Expected Use |
| Sine displacement |
5 |
2 |
0 |
0 to 10 |
Exact total travel over repeated waves |
| Cosine displacement |
3 |
1.5 |
0.5 |
1 to 8 |
Phase-shifted oscillating position |
| Velocity sine |
4 |
3 |
0 |
0 to 6 |
Total distance from absolute velocity |
| Damped sine |
6 |
2.2 |
0.2 |
0 to 12 |
Decaying oscillation with friction |
Understanding Total Distance in Oscillating Functions
What Total Distance Means
Oscillating motion moves back and forth. Net displacement can be small.
Total distance can still be large. This difference matters in maths,
physics, engineering, and signal analysis. A wave may end near its
starting point. Yet it may have traveled through many peaks and valleys.
This calculator measures that full path.
Why Turning Points Matter
A sine or cosine displacement changes direction at each turning point.
These points occur where velocity becomes zero. The calculator splits
the interval at those points. It then adds each absolute position change.
This gives an exact result for standard sine and cosine displacement.
It avoids a common mistake. That mistake is using only final minus
initial position.
Numerical Distance Method
Some oscillating functions are harder. Damped motion does not repeat
with equal height. A velocity wave with drift can cross zero in unusual
places. For those cases, the calculator uses Simpson integration. It
estimates the area under absolute velocity. More samples usually give
a smoother estimate. Very small intervals need fewer samples. Long
intervals need more samples.
Good Input Choices
Use radians for phase. Use angular frequency for omega. If you have
ordinary frequency, multiply it by two pi first. Set offset to shift
the graph up or down. In the velocity model, offset acts like steady
drift. For damped motion, damping should normally be zero or positive.
A larger damping value makes the wave shrink faster.
Reading the Output
Total distance shows the complete traveled path. Net displacement shows
the final position change. Average speed divides total distance by
elapsed time. The graph shows position, velocity, and cumulative
distance. Use the table export for records. Use the report export for
homework, lab notes, and technical checks.
FAQs
1. What is total distance for an oscillating function?
Total distance is the full path traveled by the function value. It adds all forward and backward movement during the chosen interval.
2. Is total distance the same as displacement?
No. Displacement only compares final and starting position. Total distance counts every movement between them, including reversals.
3. Why does the calculator find turning points?
Turning points split the curve into one-direction sections. Summing absolute position changes across those sections gives accurate distance.
4. What unit should I use?
Use any distance unit that matches your amplitude. The calculator keeps the same unit in the distance result.
5. What is angular frequency?
Angular frequency is omega. It measures radians per time unit. It equals ordinary frequency multiplied by two pi.
6. When should I increase samples?
Increase samples for damped motion, long intervals, high frequency, or velocity models. More samples improve numerical accuracy.
7. Can this handle damped oscillation?
Yes. Select the damped sine model. Enter damping lambda. The calculator estimates distance using absolute velocity integration.
8. Why can distance be larger than amplitude?
Amplitude is one maximum reach from the center. Total distance includes repeated travel across many cycles.