12
WAVE MOTION
12.1 The basic form of a wave
Wave motion in a medium is a collective phenomenon that involves local
interactions among the particles of the medium. Waves are characterized by:
1) a disturbance in space and time.
2) a transfer of energy from one place to another,
and
3) a non-transfer of material of the medium.
(In a water wave, for example, the molecules move perpendicularly to the velocity
vector of the wave).
Consider a kink in a rope that propagates with a velocity V along the +x-axis, as
shown
y
Displacement
V , the velocity of the waveform
x
x at time t
Assume that the shape of the kink does not change in moving a small distance ∆x in a short
interval of time ∆t. The speed of the kink is defined to be V = ∆x/∆t. The displacement
in the y-direction is a function of x and t,
, 168 WAVE MOTION
y = f(x, t).
We wish to answer the question: what basic principles determine the form of the argument
of the function, f ? For water waves, acoustical waves, waves along flexible strings, etc. the
wave velocities are much less than c. Since y is a function of x and t, we see that all points
on the waveform move in such a way that the Galilean transformation holds for all inertial
observers of the waveform. Consider two inertial observers, observer #1 at rest on the x-
axis, watching the wave move along the x-axis with constant speed, V, and a second
observer #2, moving with the wave. If the observers synchronize their clocks so that
t1 = t2 = t0 = 0 at x1 = x 2 = 0, then
x2 = x 1 – Vt.
We therefore see that the functional form of the wave is determined by the form of the
Galilean transformation, so that
y(x, t) = f(x – Vt), (12.1)
where V is the wave velocity in the particular medium. No other functional form is
possible! For example,
y(x, t) = Asink(x – Vt) is permitted, whereas
y(x, t) = A(x2 + V2t) is not.
If the wave moves to the left (in the –x direction) then
y(x, t) = f(x + Vt). (12.2)
We shall consider waves that superimpose linearly. If, for example, two waves
move along a rope in opposite directions, we observe that they “pass through each other”.
WAVE MOTION
12.1 The basic form of a wave
Wave motion in a medium is a collective phenomenon that involves local
interactions among the particles of the medium. Waves are characterized by:
1) a disturbance in space and time.
2) a transfer of energy from one place to another,
and
3) a non-transfer of material of the medium.
(In a water wave, for example, the molecules move perpendicularly to the velocity
vector of the wave).
Consider a kink in a rope that propagates with a velocity V along the +x-axis, as
shown
y
Displacement
V , the velocity of the waveform
x
x at time t
Assume that the shape of the kink does not change in moving a small distance ∆x in a short
interval of time ∆t. The speed of the kink is defined to be V = ∆x/∆t. The displacement
in the y-direction is a function of x and t,
, 168 WAVE MOTION
y = f(x, t).
We wish to answer the question: what basic principles determine the form of the argument
of the function, f ? For water waves, acoustical waves, waves along flexible strings, etc. the
wave velocities are much less than c. Since y is a function of x and t, we see that all points
on the waveform move in such a way that the Galilean transformation holds for all inertial
observers of the waveform. Consider two inertial observers, observer #1 at rest on the x-
axis, watching the wave move along the x-axis with constant speed, V, and a second
observer #2, moving with the wave. If the observers synchronize their clocks so that
t1 = t2 = t0 = 0 at x1 = x 2 = 0, then
x2 = x 1 – Vt.
We therefore see that the functional form of the wave is determined by the form of the
Galilean transformation, so that
y(x, t) = f(x – Vt), (12.1)
where V is the wave velocity in the particular medium. No other functional form is
possible! For example,
y(x, t) = Asink(x – Vt) is permitted, whereas
y(x, t) = A(x2 + V2t) is not.
If the wave moves to the left (in the –x direction) then
y(x, t) = f(x + Vt). (12.2)
We shall consider waves that superimpose linearly. If, for example, two waves
move along a rope in opposite directions, we observe that they “pass through each other”.
