# 1-5 Speed of a sinusoidal traveling wave

The figure shows a sinusoidal wave at t = t_{1}. 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(x_{1}, t_{1}) = y_{0}.

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(x_{1}+Dx,
t_{1}+Dt) = y(x_{1}, t_{1})
= y_{0}.

We can write this as

y_{m} sin(k(x_{1}+Dx) -
w(t_{1}+Dt) +
f) = y_{m} sin(kx_{1} - wt_{1}
+ f).

So the phases of the sine functions should be
equal,

k(x_{1}+Dx) - w(t_{1}+Dt)
+
f = kx_{1} - wt_{1} +
f.