De Broglie Waves: Understanding the Relationship Between Displacement and Time

[Von Neumann]

Inquiry:
When introducing de Broglie waves, the author of my modern physics book first makes the analogy of a wave propagating on a string. Naturally the displacement at any time is given by y=Acos2$\pi$ft, where A is amplitude of the vibrations and f is the frequency. Then he argues that a more complete description should tell us what y is at any point on the string at any time, inferring that what is needed is a function in both x and t. He uses the example of shaking the string at x=0 when t=0, so the wave is propagating in the +x direction. The distance traveled in a time t is x=wt, where w is the speed of the wave; making the time interval between the formation of the wave at x=0 and its arrival at the point x to be x/w.

The next part I found confusing. He says "Accordingly the displacement y of the string at x at any time t is exactly the same as the value of y at x=0 at the earlier time t-x/w. By simply replacing t in the equation y=Acos2$\pi$ft with t-x/w, then, we have the desired formula [ie. y=Acos2$\pi$f(t-$\frac{x}{w})$=Acos2$\pi$(ft-$\frac{fx}{w}$)] giving y in terms of both x and t." I understand the relevance of getting the equation into the desirable form of containing both x and t, but I don't understand its equivalency; other than y reducing to the original form y=Acos2$\pi$ft when x=0.

 

[dx]

The wave is propagating in the +x direction with speed w, so if you know y₀(t) at x = 0, then you can calculate y(x,t) as y₀(t - x/w), since the part of the wave which is at x at time t would have been at x = 0 at time t - x/w.

 

[Von Neumann]

The wave is propagating in the +x direction with speed w, so if you know y₀(t) at x = 0, then you can calculate y(x,t) as y₀(t - x/w), since the part of the wave which is at x at time t would have been at x = 0 at time t - x/w.

Oh, this makes perfect sense. At first it sounded like something was being obtained from nothing.