Physics: The Wave Equation

Understanding Physics: The Wave Equation

A traveling wave is a repeating disturbance. The disturbance moves through a material or through empty space, but the individual parts of a material usually do not travel along with it. A cork on water mainly moves up and down while the pattern moves across the surface.

A point on a rope moves sideways or up and down while the pulse travels along the rope. This distinction prevents a common mistake.

Wave speed describes the motion of the pattern, not the speed of one piece of rope or one water particle. For sound, air molecules vibrate back and forth in small motions while the sound pattern moves outward.

The link between speed, frequency, and wavelength comes from counting cycles over a set time. During one cycle, a wave pattern advances by one wavelength. If it completes more cycles each second, it covers more wavelengths in that second.

The distance covered per second is therefore frequency times wavelength. This reasoning works only when the wave has a regular repeating pattern, or when measurements refer to a particular section that is nearly regular. It is useful to picture a row of crests passing a fixed marker.

Count how many crests pass each second, then measure the spacing between crests. Those two observations determine how quickly the crests travel.

In a given medium, wave speed is often set by the properties of that medium. A sound wave travels faster in warm air than in cold air because the particles transfer vibrations differently. Sound usually travels much faster in water and solids than in air.

On a stretched string, greater tension usually gives a faster wave. For light in a vacuum, the speed stays the same for every color. This explains an important pattern.

If the speed is fixed, increasing frequency makes wavelength shorter. Radio waves, visible light, and X rays can have very different wavelengths because their frequencies differ, even though they move at the same speed in a vacuum.

Students often mix up wavelength with amplitude because both are read from a wave drawing. Wavelength is a horizontal spacing along the direction the pattern travels. Amplitude is a vertical measure of displacement from the middle position.

A louder sound usually has greater amplitude, while a higher musical pitch has greater frequency. These are separate properties. On diagrams, first identify the equilibrium line.

Measure wavelength between matching points in neighboring cycles, such as two crests. Check units before calculating. Hertz means cycles per second, so multiplying hertz by meters gives meters per second.

If a problem gives period instead of frequency, find frequency by taking one divided by the period. Keep the medium in mind, since changing medium can change wave speed and wavelength while frequency is usually set by the source.