1-5 Speed of a sinusoidal traveling wave
The figure shows a sinusoidal wave at t = t1. After a time
interval
Dt, the wave moves a distance
Dx.
Another derivation for the speed of a traveling sinusoidal wave v=
lf
Thus the speed of the wave is
Let the displacement
be y
0 at t = t
1 at some position x = x
1. That is
y(x1, t1) = y0.
Thus,
kDx -
wDt = 0.
Or
Then, the displacement at t = t
1+
Dt at x = x
1+
Dx should
also be y
0. That is
y(x1+Dx,
t1+Dt) = y(x1, t1)
= y0.
We can write this as
ym sin(k(x1+Dx) -
w(t1+Dt) +
f) = ym sin(kx1 - wt1
+ f).
So the phases of the sine functions should be
equal,
k(x1+Dx) - w(t1+Dt)
+
f = kx1 - wt1 +
f.