Waves carry energy through a medium without transporting the medium itself. This motion of energy is called a wave. A pulse is a short-duration disturbance from equilibrium that can move as a wave through a medium. The displacement of a pulse from equilibrium over time and position can be described mathematically. The equation D(x,t) defines the displacement D of a pulse at position x and time t. This allows calculating the displacement over time as a pulse moves through a medium at a given velocity.

1. Key Points:
*Waves carry energy
As a wave travels along a string, air or any medium
form, energy is created and travels through the medium.
We can consider this as a wave is a motion of energy
through a medium
*The wave travels through the medium
A wave does not carry its medium with it, instead,
the medium willmove out of its equilibrium position.
For example, during an earthquake, damage to
surroundings is caused due to the ground moving out of
equilibrium.

2. Consider:
 What is a pulse?
A pulse can be defined as an equilibrium
disturbance in a wave for a short duration. For example,
a single knock on a door creates a sound pulse that lasts
a short amount of time.
If a string is held under constant tension, we can
analyze the motion of the string with respect to a point.

3. Calculatingvariables in a Pulse In 1D
*The equilibrium of a pulse is denoted by
D(x, t)
D = displacement from equilibrium (m)
x = position (m)
t= time (s)
*We can use this equation to describe a waveform. At a
certaintime during a pulse, different waveforms are
present.
*Consider the function D(x, t) =
6
𝑥2+4
*If a pulse has a position of x = 2.0m at 0.0s, what is the
equation?
Answer:
x=2.0m, t=0.0s:
D(2.0m, 0.0s) =
6
(2.0𝑚)2+4

4. DISPLACEMENT FUNCTION
*How about at 1.5s if it is moving at 3.0m/s to the left?
At any speed, we can substitute x for (x-vt).
Why?
If at t=0, the wave of a pulse is moving at an increasing x
with speed v, then we can make this substitution. This is
called the displacement function
Answer:
D(x, 1.5s) =
6
( 𝑥−𝑣𝑡)2+4
Plug in t=1.5s and v = -3.0m/s
D(x, 1.5s) =
6
( 𝑥+3.0(1.5))2+4
*Remember, since x-vt represents a positive velocity,
x+vt will represent a negative velocity.
peak
Example of a waveform
function when t=1.0s

5. Try:
A travelling pulse is given by:
D(x, t) =
2.0
( 𝑥−3.0𝑡)4+5.0
a) What is the velocity of the pulse?
b) What is the displacement of a point on the medium
at x = 1.5m at t=2.0s
Answer:
a) remember, x-vt
velocity = 3.0m/s to the right
b)Plug in values:
D(1.5, 2.0) =
20.0
(1.5−3.0(2.0))4+5.0
D = 4.8e-2m