ONE-DIMENSIONAL WAVES: MATHEMATICAL EXPRESSION AND EXAMPLES - PHYSICAL - 2025

One- dimensional waves are those that propagate in only one direction, regardless of whether the vibration occurs in the same direction of propagation or not. A good example of these is the wave that travels through a taut string like that of a guitar.

In a transverse plane wave, the particles vibrate in a vertical direction (they rise and fall, see the red arrow in figure 1), but it is one-dimensional because the disturbance travels in only one direction, following the yellow arrow.

Figure 1: The image represents a one-dimensional wave. Note that the ridges and valleys form lines parallel to each other and perpendicular to the direction of propagation. Source: self made.

One-dimensional waves appear quite frequently in everyday life. In the next section some examples of them and also of waves that are not one-dimensional are described, to clearly establish the differences.

Examples of one-dimensional waves and non-one-dimensional waves

One-dimensional waves

Here are some examples of one-dimensional waves that can be easily observed:

- A sound pulse that travels through a straight bar, since it is a disturbance that spreads along the entire length of the bar.

- A wave that travels through a channel of water, even when the displacement of the water surface is not parallel to the channel.

- Waves that propagate on a surface or through three-dimensional space can also be one-dimensional, as long as their wave fronts are planes parallel to each other and travel in only one direction.

Non-one-dimensional waves

An example of a non-one-dimensional wave is found in waves that form on a still water surface when a stone is dropped. It is a two-dimensional wave with a cylindrical wavefront.

Figure 2. The image represents an example of what a one-dimensional wave IS NOT. Note that the crests and valleys form circles and the direction of propagation is radial outward, it is then a circular two-dimensional wave. Source: Pixabay.

Another example of a non-one-dimensional wave is the sound wave that a firecracker generates by exploding at a certain height. This is a three-dimensional wave with spherical wave fronts.

Mathematical expression of a one-dimensional wave

The most general way of expressing a one-dimensional wave that propagates without attenuation in the positive direction of the xy axis with velocity v is, mathematically:

In this expression y represents the disturbance at position x at time t. The shape of the wave is given by the function f. For example, the wave function shown in figure 1 is: y (x, t) = cos (x - vt) and the wave image corresponds to the instant t = 0.

A wave like this, described by a cosine or sine function, is called a harmonic wave. Although it is not the only waveform that exists, it is of utmost importance, because any other wave can be represented as a superposition or sum of harmonic waves. It is the well-known Fourier theorem, so widely used to describe signals of all kinds.

When the wave travels in the negative direction of the x-axis, simply change v to -v in argument, leaving:

Figure 3 shows the animation of a wave traveling to the left: it is a form called the Lorentzian function and its mathematical expression is:

In this example the speed of propagation is v = 1, -one unit of space for each unit of time-.

Figure 3. Example of a Lorentzian wave traveling to the left with speed v = 1. Source: Prepared by F. Zapata with Geogebra.

One-dimensional wave equation

The wave equation is a partial derivative equation, the solution of which is of course a wave. It establishes the mathematical relationship between the spatial part and the temporal part of it, and has the form:

Worked example

The following is the general expression y (x, t) for a harmonic wave:

a) Describe the physical meaning of the parameters A, k, ω and θo.

b) What meaning do the ± signs have in the cosine argument?

c) Verify that the given expression is indeed the solution of the wave equation of the previous section and find the velocity v of propagation.

Solution to)

The characteristics of the wave are found in the following parameters:

Second derivative with respect to t: ∂ ² and / ∂t ² = -ω ². A ⋅ cos (k ⋅ x ± ω ⋅ t + θo)

These results are substituted into the wave equation:

Both A and the cosine are simplified, since they appear on both sides of the equality and the argument of the cosine is the same, therefore the expression reduces to:

Which allows to obtain an equation for v in terms of ω and k:

References

1. E-educational. Equation of one-dimensional harmonic waves. Recovered from: e-ducativa.catedu.es
2. The corner of Physics. Wave classes. Recovered from: fisicaparatontos.blogspot.com.
3. Figueroa, D. 2006. Waves and Quantum Physics. Series: Physics for Science and Engineering. Edited by Douglas Figueroa. Simon Bolivar University. Caracas Venezuela.
4. Physics Lab. Wave motion. Recovered from: fisicalab.com.
5. Peirce, A. Lecture 21: The one dimensional Wave Equation: D'Alembert's Solution. Recovered from: ubc.ca.
6. Wave equation. Recovered from: en.wikipedia.com