Trigonometric functions such as sin ( θ ) and cos ( θ ) are commonly used to model the oscillating motion of traveling waves. In the diagram above, a complete wave cycle occurs from crest-to-crest, trough-to-trough, or along the x-axis as indicated by the blue line. Since the diagram above can be modeled with the counterclockwise motion of a unit circle with a radius of ( r ) meters, a complete cycle of any oscillating motion ( angular or linear ) is said to have moved through 2𝜋 radians.
Recall that a radian ( θ rad ) is the distance along the circumference of a circle that is equal in length to the circle’s radius ( r ); When the radius of a circle has traced a distance of 𝜋/2 radians in the counterclockwise direction along the circumference of the circle, the radius is positioned at an angle of 900 with respect to the +x-axis. When the radius of a circle has traced a distance of 𝜋 radians of the circle’s circumference, the radius has moved 1800 with respect to its original position along the +x-axis. A radius that have moved through 3𝜋/2 radians is positioned 2700 from its initial position along the +x-axis. Finally, a radius that has traveled through 2𝜋 radians has moved 3600, and it has completed a full revolution.
Let’s assume that the y-axis above is located in some arbitrary region of space. When the radius in the diagram above moves through 𝜋/2 radians ( 900 ), a traveling sinusoidal waveform’s crest will have reached the +y-axis. When the radius has moved through 𝜋 radians ( 1800 ), a traveling wave’s phase will coincide with the x/y origin ( 0/0 ). At 3𝜋/2 radians ( 2700 ), a traveling waveform’s trough will have reached the -y-axis. Finally, a radius that has traced the entire circumference of a circle ( 2𝜋 radians ) has moved 3600 from its original position along the +x-axis. The wavenumber ( k ) of traveling waves is mathematically described as follows:
k = 2𝜋/𝝀
The number of complete revolutions ( k ) of a given waveform in a region of space is inversely proportional to the wavelength of the waveform; a waveform of smaller 𝝀 will complete more revolutions within a given region of space than a waveform with a larger 𝝀, and vice versa.
Let’s assume that the waveform above is 1 nm ( 10-9 m ) in length. How many ½ nm waveforms will move through 1 nm along the x-axis? We can intuitively surmise that twice as many ½ nm waveforms will move through the same region of space:
k = 2𝜋/( ½ 𝝀 ) = ( 2 )( 2𝜋/𝝀 ) = 2k. Thus, the wavenumber ( k ) has doubled for the ½ nm waveform in question. This shows that the number of complete waveforms that pass an arbitrary point in space is inversely proportional to the wave’s wavelength.
The number of radians ( cycles ) completed by a wave of wavenumber k can be rewritten as a function of distance ( x ) along the x-axis as follows:
kx = ( 2𝜋/𝝀 )( x ) = 2𝜋x/𝝀 = θ radians
When the distance along the x-axis is equal to 𝝀, a complete wave cycle of 2𝜋 radians will have passed an arbitrary point in space. If the distance ( x ) is greater than 𝝀, the wavefront will have completed multiple cycles of radians. The wavenumber can be regarded as the spatial frequency of a wave. For a fixed distance along x, smaller wavefronts will go through more complete revolutions than a wavefront with a longer wavelength.
Taking the amplitude ( A ) of a wave into consideration, kx can be substituted into a trigonometric function of choice as follows:
y = A sin( kx )
We now have an expression that can be used to determine the phase of a wave with a wavenumber of k as a function of distance ( x ). Additionally the phase of wavefronts with different values of k can be compared to one another with convenience. In higher-level physics, wave packets ( carrier waves ) are also evaluated with derived values of k:
y = A sin( kx ) is well suited to model a moving wavefront at some instant in time; however, a wave that moves to the right must be modeled in such a way that a “ shift “ in the wave’s position occurs:
If sin ( x ) = y, and y = 0, then x = 0. If x = 0, notice that the sin ( x – 𝜋/2 ) will give us the same output value as sin ( 3𝜋/2 ) = -1. Thus, by subtracting from the x-input value, we are able to “ shift “ the wavefront to the right. To model a wavefront that moves continuously as a function of time ( t ), we must take a wave’s angular speed ( ⍵ ) into account. Omega ( ⍵ ) is expressed in units of radians/s. As a consequence, ⍵t will shift the wavefront to the right continuously using the following equation:
y( x,t ) = A sin( kx – ⍵t )
Angular speed in radians/s is expressed as follows:
⍵ = 2𝜋/T
T is the period of time ( s ) traversed when a traveling waveform completes a full revolution at some point in space. The structure of ( ⍵ ) = 2𝜋/T is very similar to that of the wavenumber k = 2𝜋/𝝀. The wavenumber ( k ) describes the spatial frequency of a wave; the angular speed ( ⍵ ) describes wave frequency as a function of time ( t ) in seconds. As a consequence, a traveling wave may also be expressed as follows:
y = A sin( ⍵T – 𝜙 )
RENAISSANCE PHYSICS 