Transverse harmonic wave

A harmonic wave is a disturbance produced by a simple harmonic oscillator which then propagates along a certain direction. In this page we will focus on the study of a transverse mechanical wave propagating along a rope of infinite length.

In the figure below the rope with its left end attached to a spring is shown. The origin of the coordinate system is at this point.

The end of the rope undergoes a simple harmonic motion of amplitude A, and therefore its vertical position as a function of time y(t) is given by:

Where ω is the angular frequency of this simple harmonic motion (as well as that of the resulting wave) and T is its period.

The energy produced by the simple harmonic motion travels to the right along the rope and at any instant t, all the “particles” of the rope will vibrate vertically in simple harmonic motion. The rope will have the shape shown in the figure below.

The wave thus produced is a transverse wave, because the points of the medium vibrate vertically whereas the energy carried by the wave travels horizontally. This wave will have the same amplitude A than that of the simple harmonic motion undergone by the end of the rope attached to the spring.

The wavelength λ of the wave is the distance between two successive points that are in the same vibrational state. Its SI unit is meter (m).

The points of the rope with the maximum displacement from the equilibrium position (+A) are called crests of the wave. The points with the minimum displacement (-A) are called troughs. The wavelength is therefore the distance between two successive crests or troughs.

In the figure below you can see “snapshots” of the rope in fixed time instants, depicted in different colors.

As shown in the picture, any point P will undergo a phase shifted version of the simple harmonic motion undergone by the left end of the rope. This shift depends on two things: the x-coordinate of the point and the time it takes for the energy to reach it.

On the other hand, the time it takes for the energy to reach P will depend on its propagation velocity or speed, v. The wave velocity (also known as phase velocity) depends on the physical properties of the medium through which the wave propagates.

The wave velocity is given by:

Where T is the period of the wave, defined as the time it takes for a point to complete one cycle. This period is equal to that of the simple harmonic oscillator that produces the wave.

The vertical displacement y(t) of any point P has to be a function of x and t:

Moreover, since the waves travels at speed (v), the energy entering the rope at t = 0 will reach the point P with a certain delay t’ given by:

And its vertical displacement will be:

And by substituting v as a function of the wavelength λ:

Simplifying:

Where k is called the wavenumber, defined as the number of waves or cycles per unit distance:

In order to make the expression more general we add a phase shift; the wave function of a harmonic wave is given by:

If we give a numerical value to x in the function above we will obtain an equation describing the simple harmonic motion of that particular point of the rope; on the other hand, if we give values to t we will obtain a function giving the position of all the points of the rope at that particular time (what we called a “snapshot” of the rope).

When the wave propagates in the negative x direction, its wave function is given by:

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