# Can a wave propagate in 1D

## Mechanical waves¶

In the following, waves are considered that have a spatially sinusoidal propagation pattern. The wave starts at the origin of the coordinates with the deflection , the result is a wave propagation as shown in the following figure.

At a distance of an integral multiple of the wavelength The wave pattern is repeated in each case. From a spatial point of view, the wave has a period of length ; at the same time every sine function has a period of . The wave can thus be characterized by the following formula:

Here referred to the amplitude of the wave. Is an integral multiple of , the argument of the sine function becomes an integral multiple of . The wave begins at with the value , the above equation is sufficient to describe the wave, otherwise an initial phase angle must be added to the argument of the sine function to be added.

With the exception of standing waves, wave patterns do not stay in place, but move on over time. For example, if the wave moves in a positive direction -Direction, so the wave pattern moves in time about the length further.

For "shifting" the wave around applies:

This relationship is useful for determining the state of deflection of a sinusoidal wave at any given location and to be determined at any time: At the time got the shaft in place namely exactly the same deflection as it was at the time at the point would have. The following applies:

This has been simplified for the difference between the time and the starting point written. The equation can be further transformed by looking for the wave relationship uses:

In the second calculation step, the factor was multiplied into the inner bracket. If you write in this form for the frequency , the spatial and temporal period of the wave becomes clear:

The wave starts over again and again when a multiple of the wavelength is (spatial period), or if a multiple of the period of oscillation is (time period).

For practical calculations it is even more “handy”, including the factor to multiply in the argument of the sine function into the brackets. One obtains:

In this representation, the term just the angular frequency the wave; this indicates the speed at which the shaft oscillates in the pointer representation. The term is called accordingly as so-called "circular wave number" . This results in the following "simple" form of the equation for the state of deflection of a shaft:

The circular wavenumber indicates how many waves in a certain unit of length (for example or ) fit in. So the shorter the wavelength of a wave, the bigger it is -Value. For microwaves, for example on the order of about ever , with light waves in the order of magnitude of over ever .

Remarks: