Here we examine the displacements and pressure variations associated with a sinusoidal sound wave traveling through air. We will also use sound waves to show that waves can be divided into two general categories:
traveling and standing waves.
To show the displacement s( x, t ) of a traveling sound wave as a function
of position x and time t, we use
either a sine function or a cosine function. Below we use a cosine
function to produce a traveling sound wave in the simulation that shows
how the displacement of the elements of the wave vary with time.
Below the traveling longitudinal wave are graphs that illustrate the relationship
between the displacement of the elements of the wave and the air pressure
at any position along the wave. Note that the displacement and pressure
variations are 90 degrees out of phase. For example, the pressure variation
at any point along the wave is zero when the displacement there is an extremon
(maximum or minimum).
Experiment with the simulation to get a better understanding of this relationship.
Pause the simulation when the displacement of the elements in the yellow box is a maximum, minimum, or zero.
Below is a simulation of how to produce a standing longitudinal
Wave. Standing longitudinal waves are produced when two
waves that are traveling in opposite directions with the same amplitude and
wavenumber combine.
Experiment with the simulation to develop a better understanding of standing waves. Set
the wavenumber and amplitude values to be the same for the
first two oppositely traveling waves (blue and green plots). These two waves are added to form
the standing wave in the bottom red plot.
How is this qualitatively different from a traveling wave? Focus on the motion in time of one part of the waveform. Do parts of a traveling wave ever reach the maximum the same amplitude? Is this the case for a traveling wave?
Adjust the wavenumber and phase to change the position of the displacement nodes (places where the displacement is always zero). What happens if the two waves that are being added don not have: the same wavelength? the same amplitude?
