Waves

A float bobs almost in place while a crest travels away. Later, a reflected wave returns. How can we describe the pattern's speed, the delay between oscillations, and the distance to the place where reflection occurred?

In a mechanical wave, neighbouring parts of the medium pass a disturbance to each other. In a simple model they oscillate about their positions while energy travels onward. Wave speed v describes the motion of a phase, such as a crest, and differs from particle velocity. A float may stop momentarily at its highest point while a crest keeps moving. Real surface waves can cause additional water displacement; the simple model isolates the main oscillatory transfer.

Take a snapshot of a periodic wave. Wavelength λ is the distance along propagation between the nearest points in the same phase. In one period T a crest reaches the previous position of the next crest, travelling λ: v=\lambda/T=\lambda\nu. With λ in metres, T in seconds and ν in hertz, v is in m/s. Period is repetition in time at one point; wavelength is repetition in space at one instant. These are different measurements.

Let the source oscillate twice as often while wave speed stays unchanged. Crests are produced more often but travel at the same speed, so their spacing shrinks: \lambda=v/\nu is halved. This conclusion assumes fixed v. In a dispersive medium, speed itself depends on frequency, so the earlier rule alone cannot determine the wavelength change. The relation v=λν still holds, but the comparison must specify all relevant conditions.

Light carries oscillations of electric and magnetic fields and can travel without a material medium. In vacuum, c\approx3\cdot10^8\,\text{м/с} and c=\lambda\nu. At the same speed, higher frequency means shorter wavelength. In matter, use the speed of light in that material: using c would give the vacuum wavelength. This distinction will be important when studying refraction.

Two points may have the same period but reach a peak at different times. Phase difference Δφ, measured in radians, describes this offset. A change of 2π is a complete cycle: the states agree again. A shift of π gives opposite states, such as a crest and trough. States also agree for zero shift or any whole multiple of 2π. Phase compares both position and progress through the oscillation cycle.

Two coherent waves reach a point along paths of different lengths. The longer path delays arrival of the same phase. Each extra λ adds a complete cycle, so path difference d contributes a phase difference \Delta\varphi=2\pi d/\lambda. d is in metres and d/λ is dimensionless. Doubling d doubles the shift. At d=λ the states are in phase again even though one path really is longer.

This expression compares waves of the same frequency in one medium that started in phase. The sign depends on the order in which paths are subtracted. At d=λ/2 the shift has magnitude π. Add any initial source phase difference with a consistent sign; different media require their different phase accumulation to be considered. Equal geometric path lengths therefore do not always mean equal arrival phases.

A radar sends a pulse and measures the time t until its reflection returns. If transmitter and receiver are close together and the stationary target is at distance S, the signal travels approximately 2S. Thus S=ct/2, with S in metres and t in seconds; the speed in air is approximately c. Doubling the delay doubles the distance found. Division by two comes from the route geometry, not a loss of speed on reflection.

For a sound echo the same geometry gives S=vt/2, but v is now the speed of sound in the medium. Replacing it by light speed would change the answer by many orders of magnitude. If the instrument adds a known delay, subtract it first to leave only propagation time. With separated transmitter and receiver or a moving target, check the route before simply halving the distance travelled.

An echo returns 0.40 s after a short sound signal. The speed of sound is 340 m/s. Find the distance to the stationary wall.

The signal travels to the stationary wall and back to a listener close to the source. The time of 0.40 s covers both parts together.

Total distance travelled: vt = 340 × 0,40 = 136 м.

One-way distance: r = vt/2 = 136/2 = 68 м.

Check: 68/340 = 0,20 с for one direction, so the round trip takes 0.40 s. Reflection in the same medium does not halve the speed.

r = 68 м to the wall. In 0.40 s the sound travelled 136 m along the outward and return route.

A wave transfers a disturbance, so crest motion and particle motion have different velocities.

v=λν links repetition in space and time; an extra path changes phase.

Reflected-signal time covers the entire route. In the simple arrangement, the stationary target is half that route length away.

Next we will study how light rays change direction at a boundary between media.