1
00:00:00,000 --> 00:00:03,223
A wave is not an object sliding along a string. It is

2
00:00:03,323 --> 00:00:07,613
a pattern of motion, carried by many tiny pieces passing a disturbance

3
00:00:07,713 --> 00:00:12,380
from neighbor to neighbor. The material mostly stays put; the message moves.

4
00:00:12,483 --> 00:00:15,540
Watch one bead. It mostly moves up and down, while

5
00:00:15,640 --> 00:00:19,265
the shape itself slides sideways through the string. Like a

6
00:00:19,365 --> 00:00:23,495
stadium wave, motion travels without every piece traveling with it.

7
00:00:23,600 --> 00:00:27,703
The secret is local. Each small piece only feels its nearby pieces.

8
00:00:27,803 --> 00:00:31,844
Where the string bends, the middle is pulled back toward the shape

9
00:00:31,944 --> 00:00:35,044
around it. No piece needs to know the whole string.

10
00:00:35,150 --> 00:00:39,835
That rule becomes the wave equation: acceleration in time equals wave speed

11
00:00:39,935 --> 00:00:44,365
squared times curvature in space. The left side measures how the height

12
00:00:44,465 --> 00:00:47,938
speeds up; the right side measures how sharply it bends.

13
00:00:48,050 --> 00:00:51,825
The constant c is the wave speed. Change it, and the rule does

14
00:00:51,925 --> 00:00:56,326
not change; only how quickly the disturbance hands itself down the line.

15
00:00:56,435 --> 00:00:58,899
Now turn time into a second direction. The

16
00:00:58,999 --> 00:01:02,438
moving string becomes a surface, where every cross-section

17
00:01:02,538 --> 00:01:04,879
is the string at one particular instant.

18
00:01:04,982 --> 00:01:08,722
Slide a bright slice through that surface. The slice is not a

19
00:01:08,822 --> 00:01:12,562
different object; it is the same string, seen a little later.

20
00:01:12,676 --> 00:01:15,980
The same idea scales up. On a drumhead, every point talks

21
00:01:16,080 --> 00:01:20,520
to nearby points, and the equation becomes a living two-dimensional surface.

22
00:01:20,633 --> 00:01:23,555
So the wave equation is a recipe for propagation:

23
00:01:23,655 --> 00:01:28,549
curvature creates acceleration, and acceleration carries the disturbance forward.