Representing Waves on Graphs
A displacement-time graph tracks the motion of a single particle at a fixed position over a period of time. It describes how that specific point oscillates up and down or back and forth.
The time period () is the duration required for one complete oscillation of that particle. This is measured as the horizontal distance between two consecutive peaks on the time axis.
The frequency () can be derived from this graph using the reciprocal relationship . This value indicates how many complete cycles occur every second (measured in Hertz).
While longitudinal waves involve oscillations parallel to the direction of travel, they are often represented using sinusoidal graphs for ease of analysis. In these cases, the y-axis represents the displacement of particles from their mean position.
On a displacement-distance graph for a longitudinal wave, compressions and rarefactions occur at points of zero displacement. Specifically, a compression is located where particles on either side are displaced toward that point.
Alternatively, pressure-distance graphs can be used. In these, the peaks represent maximum pressure (compressions) and the troughs represent minimum pressure (rarefactions). Note that the pressure graph is 90^\\circ out of phase with the displacement graph.
It is vital to distinguish between the two types of x-axes to avoid fundamental calculation errors.
| Feature | Displacement-Distance Graph | Displacement-Time Graph |
|---|---|---|
| X-Axis Represents | Position in space () | Time elapsed () |
| Horizontal Interval | Wavelength () | Time Period () |
| Physical Meaning | A snapshot of the whole wave | History of one single point |
| Derived Value | Wave Speed (if is known) | Frequency () |
Check the Units: Always verify the units on the x-axis. Examiners often use milliseconds () or centimeters (), which must be converted to standard SI units ( or ) before using the wave equation.
Identify the Center: Ensure you measure amplitude from the equilibrium line (the x-axis), not from the peak to the trough. Peak-to-trough distance is exactly .
Multiple Cycles: To increase accuracy when reading a graph, measure the total distance or time for 3-5 complete cycles and then divide by the number of cycles to find a more precise or .
Wave Type Identification: Do not assume a sine wave is transverse. Read the context to see if it represents the displacement of a longitudinal wave (like sound).