2. General Overview: Waves?
• A pulse is generated by a disturbance, or ‘work
done’, on a medium, that gives energy to this
medium.
• A wave is a series of pulses
• A wave carries the energy through the medium
• Analogy: at a sports game, a very enthusiastic fan
shouts to start the wave! He gets the people on the
aisle seats to stand up and sit back down quickly.
o Work done = the shouting/encouraging of the first fan.
o The wave travels as the person beside those in the aisle notice, and they
stand and sit back down. The people beside them, then, stand and sit,
and so on.
3. Motion in a wave
• Motion of the wave travels along the medium
o i.e. From person to person, in our rough analogy
• Motion of particles, in transverse waves, travel
perpendicular to the wave
o i.e. The standing and sitting of individual people.
Snapshots from Phet simulation of a wave on a string:
1) No pulse, all particles at equilibrium
2) Pulse has started and the first particle is ‘pushed’ down by tension force on acting on the string to the left of it.
3) Pulse has moved to the right, and tension is now acting on the second particle as well.
4) A wave is a series of pulses. The arrows show the direction of motion of each particle. The fourth particle from the left of
this snapshot is at its maximum negative displacement, so it is changing directions (instantaneous velocity is zero).
4. Displacement and Speed of Point x
in the Medium
• The displacement of a given point x in a medium
depends on time
• Key Equation – the displacement of a point x when
a pulse travelling towards increasing x (from left to
right) is described by
𝐷 𝑥, 𝑡 = 𝐷(𝑥 − 𝑣𝑡)
Where D(x) is the function of the waveform at t=0
o Speed of the pulse/wave is given by v, which we will explore later on
o Thus, given a point and a time, we can find the displacement of that
point.
5. Applying D(x,t)
A boy is holding a long, thin rope and flicks it once so
that a pulse travels along the rope from left to right.
When the pulse is initiated, it can be described with
the following:
𝐷 𝑥 =
4.5
𝑥2+4.0
, in metres.
The pulse travels with a speed of 2.0 m/s.
a) Write the wave function describing the displacement in terms of x
and time t.
b) The boy marked the point x = 0.25m along the rope before flicking it
and watched its motion. Find the displacement of this point at t =
0.80s.
c) What is the maximum displacement of a point on the string caused
by this pulse? When does this occur for point x = 2.0m?
6. Solutions
a) We substitute x with x-vt. We subtract vt because the pulse is
going towards increasing x. If it were travelling the opposite
direction, we would describe it as x+vt.
We are also given speed v = 2.0m/s.
𝐷 𝑥 =
4.5
𝑥2+4.0
𝐷 𝑥, 𝑡 =
4.5
(𝑥−2.0𝑡)2+4.0
b) We solve for D(0.25m, 0.80s) by plugging the values into the
equation found in (a)
𝐷 0.25,0.80 =
4.5
(0.25−2.0(0.80))2+4.0
= 0.77m
7. Solutions
c) We want to maximize the value of D(x,t), and the only way to
do that is to vary x and/or t. The numerator, 4.5, is constant, so we
want to minimize the denominator in order to maximize the
overall function.
The denominator is minimized when
𝑥 − 2.0𝑡 = 0
Thus, maximum displacement
𝐷 𝑚𝑎𝑥 =
4.5
0+4.0
= 1.12 m
At point x = 2.0m, this occurs at
𝐷 𝑚𝑎𝑥 = 1.12𝑚 =
4.5
(2.0 − 2.0𝑡)2+4.0
𝑡 = 0.933𝑠
8. Wave Speed
• How fast the wave travels through the medium (not
the up/down motion of particles anymore)
• Completely due to properties of the medium
o Elasticity affects tension, which is the force responsible for causing the
up/down motion of particles
o Linear Mass Density, μ, affects inertia, or the tendency to resist the motion
caused by tension force.
𝑣 =
𝑇𝑠
𝜇
in m/s and 𝜇 =
𝑚𝑎𝑠𝑠
𝑙𝑒𝑛𝑔𝑡ℎ
in kg/m
o Intuitively, we notice an increase in tension (Ts) increases force pulling on
particles, thus increasing speed. An increase in the density increases
inertia, which resists movement, thus increase in density decreases speed.
9. Checking Our Understanding
A few children are playing a game where two of
them hold a jump rope flat on the ground. One of
them begins to move his end of the rope side to side,
making a wave that is parallel to the ground. The
other kids have to jump across without their feet
touching the moving rope.
a) The boy thinks that it would be harder for the other
kids to jump across if the waves of the rope moved
faster. If he kept his arm moving with the same
force, what could he change to make the wave
faster?
b) Would moving his arm further from side to side
change the speed?
10. Solutions
a) According to the formula , we are told to assume that
the work done on the rope remains the same. The speed of the
wave depends, then, only on the material of the rope. He can
use a lighter rope, so that in , mass is decreased so the
linear mass density decreases. Plug this back into the first formula
and with a smaller denominator 𝜇, velocity of the wave
increases.
b) Moving his arm further would change the maximum
displacement of particles (increases maximum amplitude of the
wave), but does not affect wave speed because the term is not
found in the formula. Kids would have to jump further, though, so
it would make the game harder!
𝑣 =
𝑇𝑠
𝜇
𝜇 =
𝑚𝑎𝑠𝑠
𝑙𝑒𝑛𝑔𝑡ℎ
