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Sound Wave Patterns

A violin string is held fixed at two points, and vibrates in between. Let’s simplify it so that it only vibrates up and down. 

A violin string is held fixed at two points, and vibrates in between. Let’s simplify it so that it only vibrates up and down. 

The string is held fixed at two points, where the waves must begin and end. 

So only whole numbers of complete waves can fit. allowing certain numbers of waves (1, 2, 3, 4 and so on) and forbidding other numbers (such as 9/10 of a wave or 1.3 of a wave, and other such fractions).

There’s a fascinating relationship between these standing wave patterns and probability distribution functions of an electron. Subscribe for more on that.

Vibration patterns of standing waves on square plates. These show an interesting resemblance to patterns of electron clouds, probability distributions of an electron. Here is the probability cloud or distribution of the wave functions of the electron of a hydrogen atom, for different energy levels. Brighter points represent greater probability of observing the electron at that point. 

The patterns in music, electrons, and many other domains, have underlying sine waves. 

Sine Waves

The below animation shows a spring oscillation which is described by a sine wave function. 

The sine (red) corresponds to the height of the right triangle (height on y-axis). The cosine (blue) corresponds to the width of the triangle (length on x-axis). This works on a unit circle where the size of the radius is 1.  Imagine it as a clock hand going counter-clockwise. The tip of the clock hand forms the end of a right triangle, changing the angle and the relative sizes of the two sides, one side measured by the x-axis, the other side measured by the y-axis. The remaining side (hypotenuse) is always size 1, since this side is the radius (hand of the clock, with length = 1). 

The angle (θ)is a maximum of 360 degrees. The circumference of the circle is 2 π . The length of travel on the circle is proportional to the size of the angle (θ). 

The following animation is a slight variation on the above circle animation. This highlights the relative size and shape of the sine and cosine wave. The yellow circle shows angle (θ).

Wave Interference Patterns

Waves can create interesting patterns when they interact. Waves add up to larger crests or cancel each other out. Below are water waves travelling through two openings. Such patterns were discovered over a century ago. Similar patterns occur when sending electrons through two slits, showing the wavelike nature of electrons. Then the final patterns of electrons create banded patterns, directly reflecting the wave interference. This is an optional topic available with a subscription. This preview shows a fascinating application of wave science.