A transverse mechanical wave propagates in the positive
x -direction through a spring (as shown in
[link] (a)) with a constant wave speed, and the medium oscillates between
and
around an equilibrium position. The graph in
[link] shows the height of the spring (
y ) versus the position (
x ), where the
x -axis points in the direction of propagation. The figure shows the height of the spring versus the
x -position at
as a dotted line and the wave at
as a solid line. (a) Determine the wavelength and amplitude of the wave. (b) Find the propagation velocity of the wave. (c) Calculate the period and frequency of the wave.
A transverse wave shown at two instants of time.
Strategy
The amplitude and wavelength can be determined from the graph.
Since the velocity is constant, the velocity of the wave can be found by dividing the distance traveled by the wave by the time it took the wave to travel the distance.
The period can be found from
and the frequency from
Solution
Read the wavelength from the graph, looking at the purple arrow in
[link] . Read the amplitude by looking at the green arrow. The wavelength is
and the amplitude is
Characteristics of the wave marked on a graph of its displacement.
The distance the wave traveled from time
to time
can be seen in the graph. Consider the red arrow, which shows the distance the crest has moved in 3 s. The distance is
The velocity is
The period is
and the frequency is
Significance
Note that the wavelength can be found using any two successive identical points that repeat, having the same height and slope. You should choose two points that are most convenient. The displacement can also be found using any convenient point.
Check Your Understanding The propagation velocity of a transverse or longitudinal mechanical wave may be constant as the wave disturbance moves through the medium. Consider a transverse mechanical wave: Is the velocity of the medium also constant?
In a transverse wave, the wave may move at a constant propagation velocity through the medium, but the medium oscillates perpendicular to the motion of the wave. If the wave moves in the positive
x -direction, the medium oscillates up and down in the
y -direction. The velocity of the medium is therefore not constant, but the medium’s velocity and acceleration are similar to that of the simple harmonic motion of a mass on a spring.
A wave is a disturbance that moves from the point of origin with a wave velocity
v .
A wave has a wavelength
, which is the distance between adjacent identical parts of the wave. Wave velocity and wavelength are related to the wave’s frequency and period by
Mechanical waves are disturbances that move through a medium and are governed by Newton’s laws.
Electromagnetic waves are disturbances in the electric and magnetic fields, and do not require a medium.
Matter waves are a central part of quantum mechanics and are associated with protons, electrons, neutrons, and other fundamental particles found in nature.
A transverse wave has a disturbance perpendicular to the wave’s direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.
