Plane Wave Equation

## Question:

A plane wave is represented by y=4sin(50t-5x) where y is the displacement in mm,t is the time in second,and x is the distance in m measured from a fixed origin leftward.

(a) Does the wave travel from leftward or rightward?
(b) Describe the time variation of displacement of the particle at x = pi/2
(c) Calculate the speed of the wave.

In (a), I don't know how to determine leftward or rightward from the equation. I think it has something to do with the -5x.

In (b), I have tried to substitute in x=pi/2, I get: y = 4sin(50t - 2.5 pi) and then I don't know how to go on.

In (c), I get the equation 50 = 2 X pi X f therefore, frequency = 7.96 Hz the wavelength = 2 X pi = 6.28 m, then, the speed of wave = frequency X wavelength = 7.96 X 6.28 ms*-1 = 50
I don't know why I got the value 50 again. Is it the wavelength substituted is wrong?

The general mathematical expression for a plane traveling wave is y=A*sin(kx-wt-f) for a wave of frequency f=2*p*w traveling in the positive x direction with wavelength 2*p/k and phase constant f. The sign on the w term determines the direction of travel. When you have a wave equation it is good practice to put it into this form to determine its characteristics.

Since the sine function is symmetrical about zero, we may multiply the argument of the sine by -1 without changing the value of y. So y=4*sin(-50t+5x). Rearranging the terms in ( ) we get y=4*sin(5x-50t), making the standard form parameters A=4, k=5, w=50 and f=0 in your case. This gives us a wave traveling in the positive x direction, based on the sign of w=50. Since in your coordinate system x is measured leftward then the wave is traveling to the left.

In part (b) your approach is correct. Fix x at p/2 and you have described mathematically the displacement of a particle at that point. It moves in accordance with y=4*sin(5*p/2-50*t).

In part c, the wavelength is 2*p/k = 1.26m.