13 WAVE MOTION 13.2 Waves on a stretched string

Equation (13.7) describes a pattern of motion which is periodic in both space
and time. This periodicity follows from the well-known periodicity property of the

cosine function: namely, cos(θ + 2 π) = cos θ. Thus, the wave pattern is periodic
in space,

with periodicity length

y(x + λ, t) = y(x, t), (13.13)

2 π

λ =. (13.14)

k

Here, λ is known as the wavelength, whereas k is known as the wavenumber. The
wavelength is the distance between successive wave peaks. The wave pattern is

periodic in time,

with period

y(x, t + T) = y(x, t), (13.15)

2 π
T =. (13.16)
ω
The wave period is the oscillation period of the wave disturbance at a given point

in space. The wave frequency (i.e., the number of cycles per second the wave
pattern executes at a given point in space) is written

1

f = =

T

ω

. (13.17)
2 π

The quantity ω is termed the angular frequency of the wave. Finally, at any given

point in space, the displacement y oscillates between +y 0 and −y 0 (since the

maximal values of cos θ are ± 1 ). Hence, y 0 corresponds to the wave amplitude.

Equation (13.7) also describes a sinusoidal pattern which propagates along the

x-axis without changing shape. We can see this by examining the motion of the

wave peaks, y = +y 0 , which correspond to

k x − ω t = n 2 π, (13.18)

where n is an integer. Differentiating the above expression with respect to time,
we obtain
dx ω

dt k

. (13.19)