Standing waves

A standing wave (also called stationary wave) is a special case of de interference of waves that happens when two interfering waves of equal amplitude, wavelength and frequency travel in opposite directions through the same medium. Standing waves are for example the waves produced by stringed musical instruments whose strings have two fixed points.

In order to find the wave function of the wave resulting from the superposition of the two waves we use the principle of superposition. It states that:

The wave resulting from the superposition of two or more waves is the sum of the individual waves.

We are going to find the resultant wave caused by the interference of two harmonic waves of equal amplitude, frequency and wave length moving in opposite directions along a string. Their wave functions are given by:

where the plus sign in the argument of y2 means that it travels in opposite direction to y1, A is the amplitude of both waves, ω their angular frequency and k their wavenumber.

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The resultant wave is their sum:

To simplify the expression above we can use the following trigonometric identity:

where:

Now, substituting into the resultant wave function:

The minus sign in the argument of the cosine function has been removed because cosine is an even function.

From the expression for y(x,t) we can see that each point of the string undergoes vertical simple harmonic motion. For any fixed x-value, x0, the equation above gives us the position of only that point of the string against time:

which is precisely the equation describing simple harmonic motion.

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

You can also see (in black in the figure) the point of the string of x-coordinate x0. That point undergoes simple harmonic motion of amplitude 2A sin(kx0).

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On the other hand, you can also see that some points of the string (in red in the figure) undergo simple harmonic motion of maximum amplitude 2A, whereas others (in blue) don’t move at all.

The points that undergo the maximum displacement 2A are called antinodes.

The antinodes are located at positions xA for which sine is maximum (or minimum):

Substituting the wavenumber k and isolating we get:

where n is an integer.

The points of the string that stay at rest are called nodes.

The nodes are located at positions xN for which sine is zero:

And isolating we get:

where n is an integer.

The position of both nodes and antinodes does not change over time. This is the reason why this type of interference pattern is called stationary wave.

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