Make the plate ring
Slide slowly: only some frequencies make a clear pattern.
What is happening?
When the plate rings, most of it shakes up and down very fast, too fast to see. The shaking parts bounce the sand away. But along some lines the plate stays still. The sand keeps bouncing until it lands on one of those still lines, and there it stays. The lines it draws are called nodal lines.
Each pitch makes its own pattern. A higher note makes more lines, closer together. Between the notes that fit the plate, the plate barely moves and the sand just sits.
Nobody draws these patterns. They come from the plate's size, shape and stiffness and the pitch it is played at. It is a wonderful thing that so much order is hidden in a simple ringing plate.
A thin plate held at the center and driven at frequency f vibrates as a standing wave. At a resonant frequency one mode dominates, with displacement Z(x, y) cos(2πft). Grains on the plate are thrown off wherever |Z| is large and do a random walk whose step size grows with |Z|, so they collect where Z = 0: the nodal lines.
Square plate (0 ≤ x, y ≤ 1): Z = cos(nπx) cos(mπy) ± cos(mπx) cos(nπy). The ± pair share a frequency in this model; real plates split them slightly.
Round plate (radius 1): Z = Jm(αr) cos(mθ), where Jm is a Bessel function and α makes Jm′(α) = 0 at the rim. The pattern has m straight nodal lines through the center and a ring wherever Jm(αr) = 0.
Bending waves are dispersive: their frequency grows with the square of the wavenumber, so here f ∝ n² + m² (square) or f ∝ α² (round). Doubling the number of lines takes about four times the frequency, not twice.
A little history
The German musician and scientist Ernst Chladni scattered sand on metal plates and stroked their edges with a violin bow. In 1787 he published the patterns in a book on the theory of sound, and they became famous as “Chladni figures.”
Chladni later toured Europe with his plates. After he showed them in Paris, Napoleon was so impressed that he paid for a prize of 3,000 francs, which the scientific academy in Paris (then the First Class of the Institut de France) announced in 1809 for a mathematical theory of vibrating plates. The problem was hard. Only one person sent in an answer: Sophie Germain, who had taught herself mathematics. Her first try was turned down and her second earned an honorable mention, but she kept working, and in January 1816 her third paper won the prize. She was the first woman to win a prize from the Paris academy, and her work helped begin the science of elasticity used today for bridges and buildings.
Move the sound source
Sound stays off until you turn it on. The pass-by is a short, quiet sketch of what a listener beside the track would hear.
What is happening?
A sound source sends out waves in rings, like ripples from a pebble in a pond. If the source moves, it chases the rings it sent forward and runs away from the ones behind it. In front, the rings get squeezed close together, so the sound is higher. Behind, they spread apart, so it is lower.
That is why a fire truck's siren or a train horn sounds high as it comes toward you and drops lower the moment it passes. This is called the Doppler effect.
If the source goes as fast as sound itself, it catches up with its own rings and they pile up into a wall. Go faster still and the rings trail behind in a cone. When the cone sweeps over you, you hear all that piled-up sound at once: a sonic boom.
Source moving at speed v through still air, sound speed c (343 m/s at 20 °C), Mach number M = v/c. A listener straight ahead hears
f = f0 / (1 − v/c)
and straight behind
f = f0 / (1 + v/c)
Example: f0 = 440 Hz at v = 0.9c gives 440/0.1 = 4400 Hz approaching and 440/1.9 ≈ 232 Hz receding.
At M = 1 the forward formula blows up: every front reaches the listener at the same moment as the source. For M > 1 the fronts lie inside a Mach cone with half-angle
sin θ = 1/M
so at M = 1.5, θ = sin−1(2/3) ≈ 41.8°, and at M = 2, θ = 30°. The cone's edge is the shock wave heard on the ground as the boom.
Real examples
- A passing siren or train horn. A train at 30 m/s (about 67 mph) is near Mach 0.09. Its horn sounds about 10% higher coming and 8% lower going: about a minor third between the two.
- A jet's sonic boom. A jet flying faster than sound drags its Mach cone along the ground behind it. Everyone the cone passes over hears the boom, not just at the moment the jet “breaks” the barrier. That is why airliners do not fly faster than sound over land.
- Chuck Yeager, October 14, 1947. Flying the rocket-powered Bell X-1 over the Mojave Desert in California, Captain Yeager reached about Mach 1.06 at roughly 43,000 feet, the first confirmed flight faster than sound in level flight.
- A cracking whip. The crack is a tiny sonic boom: the very tip of the whip moves faster than sound.
Try it (high school)
- A train horn sounds at 300 Hz. The train moves at 30 m/s; take c = 343 m/s. What do you hear as it approaches, and as it moves away?
Show the answer
Approaching: 300 / (1 − 30/343) ≈ 329 Hz. Receding: 300 / (1 + 30/343) ≈ 276 Hz.
- A jet flies at Mach 2. What is the half-angle of its Mach cone?
Show the answer
sin θ = 1/2, so θ = 30°.
- An ambulance siren heard ahead of it is 1.05 times its true pitch. How fast is it going, as a fraction of c?
Show the answer
1/(1 − v/c) = 1.05, so 1 − v/c = 1/1.05 ≈ 0.952 and v ≈ 0.048c, about 16 m/s (37 mph).