Of Particular Significance

# Standing Waves of Two Types

In musical instruments of many types, standing waves play a central role. (This is discussed in Chapter 11 of Waves in an Impossible Sea.) These familiar standing waves have less familiar cousins that play a central role in our existence. The fundamental difference between these two types of standing waves lies in how their frequencies change with the size of the box that contains them.

Click here for a graph showing how their frequencies depends on box size

For the unfamiliar standing waves,

• $f_{box}^2 = f_0^2 + C^2/L^2$

where $C$ is a constant that depends on the details of the box and on whatever is vibrating. Familiar standing waves are merely the unfamiliar ones with $f_0=0$, so that

• $f_{box} = C/L$

Some years ago, I wrote a series of pages entitled Particles and Fields (with Math) in which I used first-year university physics and math to explain the basic math and concepts of particle physics. The fourth webpage in this series specifically addresses what I called “Class 0” and “Class 1” wave equations. The solutions to these two classes of equations include the two types of standing waves discussed here.

In fact, you’ll see that Figure 1 on that page looks essentially identical to the graph above, although it relates frequency to the wavelength of traveling waves instead of to the size of the box containing a standing wave. Despite this difference, the math is essentially the same, and so Class 1 fields can have standing waves even without a box, while Class 0 fields cannot.

If you read the remainder of that series, you’ll see how the difference between familiar and unfamiliar standing waves becomes, in particle physics, the difference between particles with zero rest mass and particles with non-zero rest mass.

## Comparing the Two Standing Waves

Below are shown the simplest standing waves of the two types, in boxes of different sizes. Each such wave has alternately a single crest (drawn in blue) or a single trough (drawn in red). [If the animations do not load automatically for you, click each box to reveal its animation; you’ll see the frequencies of the two waves differ dramatically for the largest boxes.]

#### Familiar

As the box grows, the frequency of the wave decreases toward zero

For a large box, both the shape and frequency of the wave depend on the size and shape of the box

#### Unfamiliar

As the box grows, the frequency of the wave reaches a minimum: the resonance frequency

For a large box, only the shape (not the frequency) of the wave depends on the size and shape of the box. We can even remove the box; the wave will still vibrate at its resonance frequency.

### 2 Responses

1. Bob Brzozowski says:

The spherical shape of the standing wave in your graphic makes me think of an s atomic orbital. Does an s orbital have a similar time-dependence? And if so, is it the resonant frequency as in your illustration?

1. It is but with a resonance frequency shifted differently.

For example, in the ground state of hydrogen, the frequency of the electron is (mc^2 – B)/h, where B=13.6 eV is the binding energy of hydrogen. This shift is one part in 50000. In other words, it is not merely the frequency of an electron in a box, because there is an inward attractive electric force which shifts the frequency downward.

An electron in an atom-sized box, with no electric pull inward, would have a slightly raised frequency instead.

There are much bigger effects in the innermost electrons of atoms such as radium or uranium.

Search