Progressive waves

Progressive Transverse Wave

We speak of a transverse wave when the particles move perpendicular to the direction of the wave motion. An example of a transverse wave is e.g. a wave on a long tensioned string that is swung on one side by hand. The wave propagates along the string while the string particles oscillate perpendicular to the string as shown in the figure below.

Figure 1: The wave propagates along a string and the particles oscillate perpendicularly

Sinusoidal transverse wave

A sinusoidal transverse wave is obtained by exciting the beginning of a string according to a sine function of time. The wave that is created on the string at a certain moment of observation (let it be at the moment ) looks like the figure below:

Figure 2: Transverse wave in time

The wave in Figure 2 is not stationary but travels along the x-axis with speed . If we were to observe only one particle of the string at some distance from the origin, we would see that this particle oscillates sinusoidally in the direction of the y-axis with a certain period , frequency and amplitude (the listed terms are described in the chapter, Simple harmonic motion).

Before moving on to the mathematical notation of the equations, let's first write down the basic concepts of transverse waves. We will explain the listed terms in detail later.

A harmonic or sinusoidal transverse wave is a wave where particles oscillate perpendicular to the direction of wave propagation and their displacement from the equilibrium position can be described as a sine function of distance and time.

The amplitude of a transverse wave is the maximum or peak displacement of the wave from the equilibrium position (from the x-axis in Figure 2). The maximum displacement is also called the crest (hill) of the wave and in the opposite direction the trough (valley) of the wave.

The period is the time when two crests or two troughs of the wave follow each other in the observed distance . (The term period has the same meaning as the time taken to complete a cycle or the oscillation period in an oscillation).

The Wavelength is the length of one wave, for example, the distance between two crests or troughs of a wave.

The frequency of a wave is the reciprocal of the period . It tells how many times per second the wave oscillates.

The speed of a wave is denoted by . A wave moves uniformly along the x-axis, i.e. in a direction perpendicular to the oscillation of the particles.

Let's excite the string in Figure 1 e.g., with a constant sinusoidal oscillation of the arm. The displacement of the string from the equilibrium position can therefore be described by the same equation as we learned in the chapter, Simple harmonic motion:

The oscillations caused at the beginning of the string propagate along the string with a uniform speed . The same equation applies to the distance as in uniform motion:

In this case, the distance of one wavelength is equal to the product of:

  • the wave propagation velocity , and

  • the period .

We write the equation:

We note that the period is the reciprocal of the frequency :



The frequency and wavelength are inversely proportional. They are related by the formula:

where is the speed of the wave.

We have described the progressive transverse wave. The string oscillates transversely, and the wave propagates longitudinally along the string. The instantaneous displacement of the string from its equilibrium position at a distance from the origin of the string will depend on the time and the distance .

Progressive transverse wave

With the equation:

we can determine:

  • the displacement from the equilibrium position for any time at the starting position ,

  • or displacement from the equilibrium position for any place in time if we replace with since and get:

However, we would like an equation that could be used to determine the displacement from the equilibrium position:

  • at any time

  • for any place at the same time.

Thus, we derive the equation by first imagining a wave that has moved (see Figure 3). At time , the wave has moved by:

Figure 3: Progressive wave

With our equation, let's try to track a point that is not far from the equilibrium position. Obviously, we can do this with the equation:

Let's check: if we insert into the equation, the argument of the sine function is equal to 0 and the amplitude is also 0 - as it should be. The equation already allows us to choose any moment of the wave. But since we want to choose an arbitrary coordinate at the same time, we introduce the coordinate via time .

Let's derive the equation from the equation above:

The sine argument in the above equation shows the dependence of the displacement from the equilibrium on two terms:

  • the time , and

  • the distance .

If we put into the equation, we get the dependence on time (see Figure 4 left):

If we put into the equation, we get the dependence on the distance (see Figure 4 right):

Figure 4: Graphs of the dependence of a progressive wave on time (left) and distance (right)

The equation for a progressive transverse wave is given as:

where is any moment of the wave and is also any coordinate of the wave.


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Wave speed on a tensioned string

The wave speed on a stretched string or string is calculated according to the equation:

where is the density of the string which is given as:

and is the tensile stress which is given as:

Let's therefore obtain the common form of the formula of the wave speed in a tensioned spring by inserting equations 5 and 6 into equation 4:

material editor: Noble Menedi Firima