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?

## Answer:

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.