One coordinate moving back and forth
- Displacement u(t)
- Distance since t = 0
- Amplitude / peak-to-peak
Its time history, not a path in space
Blue: u(t) Coral: current instantSame displacement is not always the same state
A complete cycle restores the whole repeating pattern. Equal position alone is insufficient: at the two crossings the rates have opposite signs.
u(t + T) = u(t), f = 1/T
For these nonconstant records, T is the smallest positive repeat time. Intervals 2T and 3T also reproduce the pattern, but are not new fundamental periods.
A fair frequency comparison: the same one second
The horizontal coordinate is time in seconds, not position. A fraction of a cycle can fit inside one second even when no whole cycle finishes. Playback speed changes neither record's physical period nor its frequency.
Reading the experiment
Zero is a reference, not accumulated travel
The marker has one coordinate relative to equilibrium, assigned u = 0. Its sign identifies a side of the chosen axis. Reversing that axis changes u and its rate of change, while the physical marker and its traveled distance stay the same.
The motion starts at equilibrium. With A = 2 cm, going to the positive extreme and back takes a distance of 4 cm, even though the displacement is zero again. That is only half a cycle. A full cycle returns to equilibrium in the original direction and covers 8 cm.
Distance counts every outward and return segment. It never decreases, whereas signed displacement can increase, decrease, or vanish.
Amplitude and timing answer different questions
A is the largest displacement magnitude relative to zero. The span between the two extremes is 2A. A larger amplitude changes the vertical range; it does not automatically change this record's period.
T is measured in seconds per cycle. Its reciprocal f is measured in cycles per second, or hertz. T = 0.25 s means f = 4 Hz; f = 20 Hz means T = 0.05 s.
Counting every equilibrium crossing as a cycle doubles the answer for these records. Crossings alternate direction; complete cycles restore the original direction as well as the position.
The graph records motion; it is not the motion's road
The marker moves only along its straight axis. The blue curve is a record of its coordinate at successive times. The point on that graph advances horizontally because the clock advances, not because anything is traveling horizontally through a medium.
No wavelength, wavenumber, propagation speed, angular frequency or phase parameter is needed here. The rounded and uneven records use smooth polynomial interpolation; the triangle is an idealized trace with sharp turnarounds. None assumes Hooke's law.
At A = 0 the record is constant. It then has no smallest positive repeat time: the displayed T setting is only nominal, and no observed frequency or cycle count can be inferred from that flat line.