A Classical Approach of Newtonian Mechanics

13 WAVE MOTION 13.2 Waves on a stretched string

μ

Figure 109: A sinusoidal wave propagating down the x-axis. The solid, dotted, dashed, and dot-
dashed curves show the wave displacement at four successive and equally spaced times.

In other words, the wave peaks all propagate along the x-axis with uniform speed
ω
v =. (13.20)
k

It is easily demonstrated that the wave troughs, y = −y 0 , propagate with the
same speed. Thus, it is fairly clear that the whole wave pattern moves with speed

v—see Fig. 109. Equations (13.14), (13.17), and (13.20) yield

v = f λ : (13.21)

i.e., a wave’s speed is the product of its frequency and its wavelength. This is true

for all types of (sinusoidal) wave.

Equations (13.12) and (13.20) imply that

v =

‚

., T

. (13.22)

In other words, all waves that propagate down a stretched string do so with the

same speed. This common speed is determined by the properties of the string: i.e.,
its tension and mass per unit length. Note, from Eq. (13.7), that the wavelength

v