Progressive waves

A progressive wave is a disturbance that transfers energy from one place to another, without transferring matter. The particles of the medium oscillate about their equilibrium positions as the wave passes through, but do not travel with the wave.

Wave motion in ropes, springs, and ripple tanks

• In a rope or spring, a wave can be produced by moving one end up and down. The disturbance travels along the rope, but each section of rope only moves up and down (not along the rope).
• In a ripple tank, dropping an object into water creates ripples (waves) that move outward. The water itself moves up and down, but the wave travels across the surface.

Key terms

• Displacement ($x$): The distance of a particle from its equilibrium (rest) position, at any instant, in a specified direction.
• Amplitude ($A$): The maximum displacement from the equilibrium position.
• Period ($T$): The time taken for one complete oscillation (cycle).
• Frequency ($f$): The number of oscillations per second. Measured in hertz (Hz).
• Wavelength ($\lambda$): The distance between two consecutive points in phase (e.g., crest to crest).
• Wave speed ($v$): The speed at which the wave profile (energy) travels through the medium.
• Phase difference: The fraction of a cycle by which one point on a wave leads or lags another, often measured in degrees or radians.

The wave equation

Derivation

• In one period $T$, a wave travels one wavelength $\lambda$.
• Speed $v = \frac{\text{distance}}{\text{time}} = \frac{\lambda}{T}$.
• Since $f = \frac{1}{T}$, then $v = f\lambda$.

Using a cathode-ray oscilloscope (CRO)

• Time-base: Controls the horizontal scale (time per division), allowing measurement of the period and thus frequency.
• Y-gain: Controls the vertical scale (voltage per division), allowing measurement of amplitude.

To determine frequency:

1. Measure the time for one cycle on the screen.
2. Use $f = \frac{1}{T}$.

To determine amplitude:

1. Measure the vertical height (peak-to-peak) and divide by 2 for amplitude.

Energy transfer in progressive waves

• Progressive waves transfer energy through a medium.
• The particles oscillate, but do not move along with the wave.

Intensity of a progressive wave

• Intensity ($I$): The power transmitted per unit area, perpendicular to the direction of wave travel.

• For a progressive wave, intensity is proportional to the square of the amplitude:

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These are the essential points for understanding progressive waves at AS Level.